×

zbMATH — the first resource for mathematics

Approximation methods for solving fractional equations. (English) Zbl 1448.65059
Summary: In this review paper, we are mainly concerned with the numerical methods for solving fractional equations, which are divided into the fractional differential equations (FDEs), time-fractional, space-fractional and space-time-fractional partial differential equations (FPDEs), fractional integro-differential equations (FIDEs) and delay fractional differential and/or fractional partial differential equations (DFDE/DFPDEs). The concept of the variable-order fractional operators will also be reviewed. At the same time, the techniques for improving the accuracy and computation and storage are also introduced.

MSC:
65L03 Numerical methods for functional-differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35R11 Fractional partial differential equations
34K37 Functional-differential equations with fractional derivatives
45J05 Integro-ordinary differential equations
Software:
ma2dfc
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Miller, K. S.; Ross, B., An introduction to the fractional calculus and fractional differential equations (1993), John Wiley and Sons: John Wiley and Sons New York · Zbl 0789.26002
[2] Li, C.; Zeng, F., Finite difference methods for fractional differential equations, Int J Bifurcat Chaos, 22, 04, 1230014 (2012) · Zbl 1258.34018
[3] Li, C.; Zhao, Z.; Chen, Y., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput Math Appl, 62, 3, 855-875 (2011) · Zbl 1228.65190
[4] Magin, R. L.; Abdullah, O.; Baleanu, D.; Zhou, X. J., Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J Mag Reson, 190, 2, 255-270 (2008)
[5] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in highfrequency financial data: an empirical study, Phys A Stat Mech Appl, 314, 1-4, 749-755 (2002) · Zbl 1001.91033
[6] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys Rep, 339, 1, 1-77 (2000) · Zbl 0984.82032
[7] Magin, R. L., Fractional Calculus in Bioengineering, 269-355 (2006), Redding:Begell House
[8] Podlubny, I., Fractional Differential Equations (1999), Acdemic Press: Acdemic Press San Diego · Zbl 0918.34010
[9] Ding, Z.; Xiao, A.; Li, M., Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients, J Comput Appl Math, 233, 8, 1905-1914 (2010) · Zbl 1185.65146
[10] Lynch, V. E.; Carreras, B. A.; del Castillo-Negrete, D.; Ferreira-Mejias, K. M.; Hicks, H. R., Numerical methods for the solution of partial differential equations of fractional order, J Comput Phys, 192, 2, 406-421 (2003) · Zbl 1047.76075
[11] Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Jara, B. M.V., Matrix approach to discrete fractional calculus II: partial, J Comput Phys, 228, 8, 3137-3153 (2009) · Zbl 1160.65308
[12] Rihan, F. A., Computational methods for delay parabolic and time-fractional partial differential equations, Numer Methods Part Differ Equ, 26, 6, 1556-1571 (2010) · Zbl 1204.65114
[13] Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl Math Model, 34, 1, 200-218 (2010) · Zbl 1185.65200
[14] Jumarie, G., Fractional partial differential equations and modified riemannliouville derivative new methods for solution, J Appl Math Comput, 24, 1-2, 31-48 (2007) · Zbl 1145.26302
[15] Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys Lett A, 355, 4-5, 271-279 (2006) · Zbl 1378.76084
[16] Jiang, Y.; Ma, J., High-order finite element methods for time-fractional partial differential equations, J Comput Appl Math, 235, 11, 3285-3290 (2011) · Zbl 1216.65130
[17] Momani, S.; Odibat, Z., A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula, J Comput Appl Math, 220, 1-2, 85-95 (2008) · Zbl 1148.65099
[18] Odibat, Z.; Momani, S., A generalized differential transform method for linear partial differential equations of fractional order, Appl Math Lett, 21, 2, 194-199 (2008) · Zbl 1132.35302
[19] Boling, G.; Xueke, P.; Fenghui, H., Fractional partial differential equations and their numerical solutions (2015), World Scientific · Zbl 1335.35001
[20] Xu, Q.; Hesthaven, J. S., Stable multi-domain spectral penalty methods for fractional partial differential equations, J Comput Phys, 257, 241-258 (2014) · Zbl 1349.35414
[21] Fernandez, A.; Baleanu, D.; Fokas, A. S., Solving PDEs of fractional order using the unified transform method, Appl Math Comput, 339, 738-749 (2018)
[22] Baleanu, D.; Fernandez, A., A generalisation of the malgrangeehrenpreis theorem to find fundamental solutions to fractional PDEs, Electron J Qualit Theory Differ Equ, 2017, 15, 1-12 (2017)
[23] Bin, Z., (G/g)-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun Theoret Phys, 58, 5, 623 (2012) · Zbl 1264.35273
[24] Fernandez, A., An elliptic regularity theorem for fractional partial differential operators, Comput Appl Math, 1-12 (2018)
[25] Yang, X. J.; Srivastava, H. M.; Cattani, C., Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Roman Rep Phys, 67, 3, 752-761 (2015)
[26] Fu, Z. J.; Chen, W.; Ling, L., Method of approximate particular solutions for constant-and variable-order fractional diffusion models, Eng Anal Boundary Elements, 57, 37-46 (2015) · Zbl 1403.65087
[27] Fu, Z. J.; Chen, W.; Yang, H. T., Boundary particle method for laplace transformed time fractional diffusion equations, J Comput Phys, 235, 52-66 (2013) · Zbl 1291.76256
[28] Agrawal, O. P., A general solution for the fourth-order fractional diffusion-wave equation, Fract Calculus Appl Anal, 3, 1, 1-12 (2000) · Zbl 1111.45300
[29] Agrawal, O. P., A general solution for the fourth-order fractional diffusion-wave equation, Fract Calculus Appl Anal, 3, 1, 1-12 (2000) · Zbl 1111.45300
[30] Chen, Y.; Wu, Y.; Cui, Y.; Wang, Z.; Jin, D., Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, J Comput Sci, 1, 3, 146-149 (2010)
[31] Du, R.; Cao, W. R.; Sun, Z. Z., A compact difference scheme for the fractional diffusion-wave equation, Appl Math Model, 34, 10, 2998-3007 (2010) · Zbl 1201.65154
[33] Gu, Y.; Zhuang, P.; Liu, Q., An advanced meshless method for time fractional diffusion equation, Int J Comput Methods, 8, 04, 653-665 (2011) · Zbl 1245.65133
[34] Langlands, T. A.M.; Henry, B. I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J Comput Phys, 205, 2, 719-736 (2005) · Zbl 1072.65123
[35] Liu, Q.; Gu, Y.; Zhuang, P.; Liu, F.; Nie, Y. F., An implicit RBF meshless approach for time fractional diffusion equations, Comput Mech, 48, 1, 1-12 (2011) · Zbl 1377.76025
[36] Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl Math Lett, 9, 6, 23-28 (1996) · Zbl 0879.35036
[37] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Phys A Stat Mech Appl, 278, 1-2, 107-125 (2000)
[38] Murillo, J. Q.; Yuste, S. B., An explicit difference method for solving fractional diffusion and diffusion-wave equations in the caputo form, J Comput Nonlinear Dyn, 6, 2, 021014 (2011)
[39] Pedro, H. T.C.; Kobayashi, M. H.; Pereira, J. M.C.; Coimbra, C. F.M., Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere, J Vib Control, 14, 9-10, 1659-1672 (2008) · Zbl 1229.76099
[40] Sun, H. G.; Chen, W.; Wei, H.; Chen, Y. Q., A comparative study of constantorder and variable-order fractional models in characterizing memory property of systems, Eur Phys J Spec Top, 193, 1, 185 (2011)
[41] Shyu, J. J.; Pei, S. C.; Chan, C. H., An iterative method for the design of variable fractional-order FIR differintegrators, Signal Process, 89, 3, 320-327 (2009) · Zbl 1151.94410
[42] Sun, H.; Chen, W.; Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Phys A Stat Mech Appl, 388, 21, 4586-4592 (2009)
[43] Sun, H.; Chen, Y.; Chen, W., Random-order fractional differential equation models, Signal Process, 91, 3, 525-530 (2011) · Zbl 1203.94056
[44] Sheng, H.; Sun, H.; Chen, Y.; Qiu, T., Synthesis of multi-fractional gaussian noises based on variable-order fractional operators, Signal Process, 91, 7, 1645-1650 (2011) · Zbl 1213.94049
[45] Ostalczyk, P.; Rybicki, T., Variable-fractional-order dead-beat control of an electromagnetic servo, J Vib Control, 14, 9-10, 1457-1471 (2008)
[46] Orosco, J.; Coimbra, C. F.M., On the control and stability of variable-order mechanical systems, Nonlinear Dyn, 86, 1, 695-710 (2016)
[47] Ramirez, L. E.; Coimbra, C. F., A variable order constitutive relation for viscoelasticity, Annalen der Physik, 16, 7-8, 543-552 (2007) · Zbl 1159.74008
[48] Ingman, D.; Suzdalnitsky, J., Control of damping oscillations by fractional differential operator with time-dependent order, Comput Methods Appl Mech Eng, 193, 52, 5585-5595 (2004) · Zbl 1079.70020
[50] Pindza, E.; Owolabi, K. M., Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun Nonlinear Sci Numer Simulat, 40, 112-128 (2016)
[51] Uchaikin, V. V., Fractional derivatives for physicists and engineers (2013), Springer: Springer Berlin · Zbl 1312.26002
[52] Tarasov, V. E., Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (2011), Springer Science & Business Media
[54] Liu, F.; Zhuang, P.; Turner, I.; Burrage, K.; Anh, V., A new fractional finite volume method for solving the fractional diffusion equation, Appl Math Model, 38, 15-16, 3871-3878 (2014) · Zbl 1429.65213
[55] Chen, Y.; Han, X.; Liu, L., Numerical solution for a class of linear system of fractional differential equations by the haar wavelet method and the convergence analysis, CMES: Comput Model Eng Sci, 97, 5, 391-405 (2014) · Zbl 1356.65191
[56] Khodabakhshi, N.; Vaezpour, S. M.; Baleanu, D., Numerical solutions of the initial value problem for fractional differential equations by modification of the adomian decomposition method, Fract Calculus Appl Anal, 17, 2, 382-400 (2014) · Zbl 1308.34015
[57] Hosseini, V. R.; Chen, W.; Avazzadeh, Z., Numerical solution of fractional telegraph equation by using radial basis functions, Eng Anal Boundary Elements, 38, 31-39 (2014) · Zbl 1287.65085
[58] Kim, M. H.; Ri, G. C.; Hyong-Chol, O., Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives, Fract Calculus Appl Anal, 17, 1, 79-95 (2014) · Zbl 1312.34018
[59] Zaky, M. A., A legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations, Comput Appl Math, 37, 3, 3525-3538 (2018) · Zbl 1404.65204
[60] Kumar, D.; Agarwal, R. P.; Singh, J., A modified numerical scheme and convergence analysis for fractional model of lienard’s equation, J Comput Appl Math, 339, 405-413 (2018) · Zbl 1404.34007
[61] Singh, J.; Kumar, D.; Baleanu, D.; Rathore, S., An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl Math Comput, 335, 12-24 (2018)
[62] Kumar, D.; Tchier, F.; Singh, J.; Baleanu, D., An efficient computational technique for fractal vehicular traffic flow, Entropy, 20, 4, 259 (2018)
[63] Esmaeili, S.; Garrappa, R., A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation, Int J Comput Math, 92, 5, 980-994 (2015) · Zbl 1314.65130
[64] Hafez, R. M.; Ezz-Eldien, S. S.; Bhrawy, A. H.; Ahmed, E. A.; Baleanu, D., A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations, Nonlinear Dyn, 82, 3, 1431-1440 (2015) · Zbl 1348.65144
[65] Abd-Elhameed, W. M.; Youssri, Y. H., Generalized Lucas polynomial sequence approach for fractional differential equations, Nonlinear Dyn, 89, 2, 1341-1355 (2017) · Zbl 1384.41003
[66] Saker, M. A.; Ezz-Eldien, S. S.; Bhrawy, A. H., A pseudo-spectral method for solving the time-fractional generalized hirotasatsuma coupled kortewegde vries system, Roman J Phys, 62, 105 (2017)
[67] Kashkari, B. S.; Syam, M. I., Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional order, Appl Math Comput, 290, 281-291 (2016) · Zbl 1410.34021
[68] Ezz-Eldien, S. S., On solving fractional logistic population models with applications, Comput Appl Math, 37, 5, 6392-6409 (2018) · Zbl 1413.34164
[69] Khosravian-Arab, H.; Dehghan, M.; Eslahchi, M. R., Fractional spectral and pseudo-spectral methods in unbounded domains: theory and applications, J Comput Phys, 338, 527-566 (2017)
[70] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for two sided space-fractional partial differential equations, Appl Numer Math, 56, 1, 80-90 (2006) · Zbl 1086.65087
[71] Pal, K.; Liu, F.; Yan, Y.; Roberts, G., Finite difference method for two-sided space-fractional partial differential equations, Proceedings of the International Conference on Finite Difference Methods, 307-314 (2014), Springer: Springer Cham
[72] Pal, K.; Liu, F.; Yan, Y.; Roberts, G., Finite difference method for two-sided space-fractional partial differential equations, Proceedings of the International Conference on Finite Difference Methods, 307-314 (2014), Springer: Springer Cham
[73] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for Twosided space-fractional partial differential equations, Appl Numer Math, 56, 1, 80-90 (2006) · Zbl 1086.65087
[74] Zhang, Y., A finite difference method for fractional partial differential equation, Appl Math Comput, 215, 2, 524-529 (2009) · Zbl 1177.65198
[75] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer Methods Part Differ Equ Int J, 26, 2, 448-479 (2010) · Zbl 1185.65187
[76] Odibat, Z.; Momani, S., A generalized differential transform method for linear partial differential equations of fractional order, Appl Math Lett, 21, 2, 194-199 (2008) · Zbl 1132.35302
[77] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer Methods Part Differ Equ Int J, 26, 2, 448-479 (2010) · Zbl 1185.65187
[78] Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys Lett A, 365, 5-6, 345-350 (2007) · Zbl 1203.65212
[79] Liu, K.; Wang, J.; ORegan, D., Ulam-Hyers-Mittag-Leffler stability for \(ψ\)-hilfer fractional-order delay differential equations, Adv Differ Equ, 2019, 1, 50 (2019) · Zbl 07020825
[80] Bhrawy, A. H.; Zaky, M. A., A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J Comput Phys, 281, 876-895 (2015) · Zbl 1352.65386
[81] Elsaid, A., The variational iteration method for solving Riesz fractional partial differential equations, Comput Math Appl, 60, 7, 1940-1947 (2010) · Zbl 1205.65287
[82] Khalil, H.; Khan, R. A., A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation, Comput Math Appl, 67, 10, 1938-1953 (2014) · Zbl 1366.74084
[83] Khan, R. A.; Khalil, H., A new method based on Legendre polynomials for solution of system of fractional order partial differential equations, Int J Comput Math, 91, 12, 2554-2567 (2014) · Zbl 1328.65253
[84] Wu, J. L., A wavelet operational method for solving fractional partial differential equations numerically, Appl Math Comput, 214, 1, 31-40 (2009) · Zbl 1169.65127
[85] Yang, X. J.; Gao, F.; Tenreiro Machado, J. A.; Baleanu, D., Exact travelling wave solutions for local fractional partial differential equations in mathematical physics, (Tas, K.; Baleanu, D.; Machado, J., Mathematical Methods in Engineering. Nonlinear Systems and Complexity, 24 (2019), Springer: Springer Cham)
[86] Bhrawy, A. H.; Zaky, M. A.; Baleanu, D., New numerical approximations for space-time fractional burgers equations via a Legendre spectral-collocation method, Rom Rep Phys, 67, 2, 340-349 (2015)
[87] Wang, L.; Ma, Y.; Meng, Z., Haar wavelet method for solving fractional partial differential equations numerically, Appl Math Comput, 227, 66-76 (2014) · Zbl 1364.65213
[88] Jaradat, H.; Awawdeh, F.; Rawashdeh, E. A., Analytic solution of fractional integro-differential equations, Ann Uni Craiova Math Comput Sci Ser, 38, 1, 1-10 (2011) · Zbl 1240.45015
[89] Momani, S. M., Local and global existence theorems on fractional integro-differential equations, J Fract Calc, 18, 81-86 (2000) · Zbl 0967.45004
[90] Lepik, U., Solving fractional integral equations by the haar wavelet method, Appl Math Comput, 214, 2, 468-478 (2009) · Zbl 1170.65106
[91] Bhrawy, A. H.; Abdelkawy, M. A.; Baleanu, D.; Amin, A. Z., A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel, Hacet J Math Stat, 47, 553-566 (2018) · Zbl 1417.65222
[92] Eslahchi, M. R.; Dehghan, M.; Parvizi, M., Application of the collocation method for solving nonlinear fractional integro-differential equations, J Comput Appl Math, 257, 105-128 (2014) · Zbl 1296.65106
[93] Momani, S.; Noor, M. A., Numerical methods for fourth-order fractional integro-differential equations, Appl Math Comput, 182, 1, 754-760 (2006) · Zbl 1107.65120
[94] Momani, S.; Qaralleh, R., An efficient method for solving systems of fractional integro-differential equations, Comput Math Appl, 52, 3-4, 459-470 (2006) · Zbl 1137.65072
[95] Rawashdeh, E. A., Numerical solution of fractional integro-differential equations by collocation method, Appl Math Comput, 176, 1, 1-6 (2006) · Zbl 1100.65126
[97] Mittal, R. C.; Nigam, R., Solution of fractional integro-differential equations by Adomian decomposition method, Int J Appl Math Mech, 4, 2, 87-94 (2008)
[98] Yang, Y., Jacobi spectral Galerkin methods for fractional integro-differential equations, Calcolo, 52, 4, 519-542 (2015) · Zbl 1331.65179
[99] Tao, X.; Xie, Z.; Zhou, X., Spectral Petrov-Galerkin methods for the second kind volterra type integro-differential equations, numerical mathematics: theory, Methods Appl, 4, 2, 216-236 (2011) · Zbl 1249.65291
[100] Ahmad, B.; Nieto, J. J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound Value Probl, 2011, 1, 36 (2011) · Zbl 1275.45004
[101] Abbasbandy, S.; Hashemi, M. S.; Hashim, I., On convergence of homotopy analysis method and its application to fractional integro-differential equations, Quaest Math, 36, 1, 93-105 (2013) · Zbl 1274.65229
[102] Zurigat, M.; Momani, S.; Alawneh, A., Homotopy analysis method for systems of fractional integro-differential equations, neural, Parallel Sci Comput, 17, 2, 169 (2009) · Zbl 1180.65181
[103] Ahmad, B.; Ntouyas, S. K., On hadamard fractional integro-differential boundary value problems, J Appl Math Comput, 47, 1-2, 119-131 (2015) · Zbl 1328.34006
[104] Ahmad, B.; Ntouyas, S. K.; Tariboon, J., Existence results for mixed hadamard and Riemann-Liouville fractional integro-differential equations, Adv Differ Equ, 2015, 1, 293 (2015) · Zbl 1422.34015
[105] Anguraj, A.; Karthikeyan, P.; Rivero, M.; Trujillo, J. J., On new existence results for fractional integro-differential equations with impulsive and integral conditions, Comput Math Appl, 66, 12, 2587-2594 (2014) · Zbl 1368.45005
[106] Arikoglu, A.; Ozkol, I., Solution of fractional integro-differential equations by using fractional differential transform method, chaos, Solitons Fractals, 40, 2, 521-529 (2009) · Zbl 1197.45001
[107] Aslefallah, M.; Shivanian, E., Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions, Eur Phys J Plus, 130, 3, 47 (2015)
[108] Balasubramaniam, P.; Vembarasan, V.; Senthilkumar, T., Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in hilbert space, Numer Funct Anal Optim, 35, 2, 177-197 (2014) · Zbl 1288.34074
[109] Balasubramaniam, P.; Tamilalagan, P., The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J Optim Theory Appl, 174, 1, 139-155 (2017) · Zbl 1380.93280
[110] Baleanu, D.; Mousalou, A.; Rezapour, S., A new method for investigating approximate solutions of some fractional integro-differential equations involving the caputo-fabrizio derivative, Adv Differ Equ, 2017, 1, 51 (2017) · Zbl 1422.34219
[111] Liao, C.; Ye, H., Existence of positive solutions of nonlinear fractional delay differential equations, Positivity, 13, 3, 601-609 (2009) · Zbl 1177.34081
[112] Morgado, M. L.; Ford, N. J.; Lima, P. M., Analysis and numerical methods for fractional differential equations with delay, J Comput Appl Math, 252, 159-168 (2013) · Zbl 1291.65214
[113] Babakhani, A.; Baleanu, D.; Khanbabaie, R., Hopf bifurcation for a class of fractional differential equations with delay, Nonlinear Dyn, 69, 3, 721-729 (2012) · Zbl 1258.34155
[114] Gao, Z., A graphic stability criterion for non-commensurate fractional-order time-delay systems, Nonlinear Dyn, 78, 3, 2101-2111 (2014) · Zbl 1345.34123
[115] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput Math Appl, 59, 3, 1326-1336 (2010) · Zbl 1189.65151
[116] Morgado, M. L.; Ford, N. J.; Lima, P. M., Analysis and numerical methods for fractional differential equations with delay, J Comput Appl Math, 252, 159-168 (2013) · Zbl 1291.65214
[117] Saeed, U.; ur Rehman, M.; Iqbal, M. A., Modified chebyshev wavelet methods for fractional delay-type equations, Appl Math Comput, 264, 431-442 (2015) · Zbl 1410.65286
[118] Dehghan, M.; Salehi, R., Solution of a nonlinear time-delay model in biology via semi-analytical approaches, Comput Phys Commun, 181, 7, 1255-1265 (2010) · Zbl 1219.65062
[119] Debbouche, A.; Torres, D. F., Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions, Appl Math Comput, 243, 161-175 (2014) · Zbl 1335.34094
[121] Wang, Z., A numerical method for delayed fractional-order differential equations, J Appl Math (2013)
[122] Wang, Z.; Huang, X.; Zhou, J., A numerical method for delayed fractionalorder differential equations: based on GL definition, Appl Math Inf Sci, 7, 2, 525-529 (2013)
[123] Pandey, R. K.; Kumar, N.; Mohaptra, R. N., An approximate method for solving fractional delay differential equations, Int J Appl Comput Math, 3, 2, 1395-1405 (2017) · Zbl 1397.65100
[124] Mohammadi, F., Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis, Comput Appl Math (2018) · Zbl 1432.65207
[125] Li, M.; Wang, J., Finite time stability of fractional delay differential equations, Appl Math Lett, 64, 170-176 (2017) · Zbl 1354.34130
[126] Moghaddam, B. P.; Mostaghim, Z. S., Modified finite difference method for solving fractional delay differential equations, Boletim da Sociedade Paranaense de Matematica, 35, 2, 49-58 (2017) · Zbl 1424.65094
[127] Muthukumar, P.; Ganesh Priya, B., Numerical solution of fractional delay differential equation by shifted Jacobi polynomials, Int J Comput Math, 94, 3, 471-492 (2017) · Zbl 1388.34070
[128] Ghasemi, M.; Fardi, M.; Ghaziani, R. K., Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel hilbert space, Appl Math Comput, 268, 815-831 (2015) · Zbl 1410.34185
[129] Hosseinpour, S.; Nazemi, A.; Tohidi, E., A new approach for solving a class of delay fractional partial differential equations, Mediterr J Math, 15, 6, 218 (2018) · Zbl 1407.65214
[130] Ouyang, Z., Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput Math Appl, 61, 4, 860-870 (2011) · Zbl 1217.35206
[131] Rihan, F. A., Computational methods for delay parabolic and time-fractional partial differential equations, Numer Methods Part Differ Equ, 26, 6, 1556-1571 (2010) · Zbl 1204.65114
[132] Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J Appl Math, 52, 3, 855-869 (1992) · Zbl 0760.92018
[133] Dehghan, M.; Shakeri, F., The use of the decomposition procedure of adomian for solving a delay differential equation arising in electrodynamics, Phys Scr, 78, 6, 065004 (2008) · Zbl 1159.78319
[134] Arino, O.; Snchez, E., An integral equation of cell population dynamics formulated as an abstract delay equation-some consequences, Math Models Methods Appl Sci, 8, 04, 713-735 (1998) · Zbl 0944.92004
[135] Campbell, S. A.; Edwards, R.; van den Driessche, P., Delayed coupling between two neural network loops, SIAM J Appl Math, 65, 1, 316-335 (2004) · Zbl 1072.92003
[136] Moghaddam, B. P.; Machado, J. A.T.; Behforooz, H., An integro quadratic spline approach for a class of variable-order fractional initial value problems, chaos, Solit Fract, 102, 354-360 (2017) · Zbl 1422.65131
[137] Samko, S., Fractional integration and differentiation of variable order: an overview, Nonlinear Dyn, 71, 4, 653-662 (2013) · Zbl 1268.34025
[138] Zhang, H.; Liu, F.; Phanikumar, M. S.; Meerschaert, M. M., A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput Math Appl, 66, 5, 693-701 (2013) · Zbl 1350.65092
[139] Moghaddam, B. P.; Machado, J. A.T., Extended algorithms for approximating variable order fractional derivatives with applications, J Sci Comput, 71, 3, 1351-1374 (2017) · Zbl 1370.26017
[140] Moghaddam, B. P.; Machado, J. A.T., A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Comput Math Appl, 73, 6, 1262-1269 (2017) · Zbl 1412.65084
[141] Moghaddam, B. P.; Machado, J. A.T., SM-Algorithms for approximating the variable-order fractional derivative of high order, Fundam Inf, 151, 1-4, 293-311 (2017) · Zbl 1377.65031
[142] Moghaddam, B. P.; Machado, J. A.T., A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels, Fract Calculus Appl Anal, 20, 4, 1023-1042 (2017) · Zbl 1376.65159
[143] Bhrawy, A. H.; Zaky, M. A., Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn, 80, 1-2, 101-116 (2015) · Zbl 1345.65060
[144] Bhrawy, A. H.; Zaky, M. A., Numerical algorithm for the variable-order caputo fractional functional differential equation, Nonlinear Dyn, 85, 3, 1815-1823 (2016) · Zbl 1349.65505
[145] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J Numer Anal, 47, 3, 1760-1781 (2009) · Zbl 1204.26013
[146] Lin, R.; Liu, F.; Anh, V.; Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl Math Comput, 212, 2, 435-445 (2009) · Zbl 1171.65101
[147] Sun, H.; Chen, W.; Li, C.; Chen, Y., Finite difference schemes for variable order time fractional diffusion equation, Int J Bifurcat Chaos, 22, 04, 1250085 (2012) · Zbl 1258.65079
[148] Shen, S.; Liu, F.; Chen, J.; Turner, I.; Anh, V., Numerical techniques for the variable order time fractional diffusion equation, Appl Math Comput, 218, 22, 10861-10870 (2012) · Zbl 1280.65089
[149] Shen, S.; Liu, F.; Anh, V.; Turner, I.; Chen, J., A characteristic difference method for the variable-order fractional advection-diffusion equation, J Appl Math Comput, 42, 1-2, 371-386 (2013) · Zbl 1296.65114
[150] Diaz, G.; Coimbra, C. F.M., Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation, Nonlinear Dyn, 56, 1-2, 145-157 (2009) · Zbl 1170.70012
[151] Kobelev, Y. L.; Kobelev, L. Y.; Klimontovich, Y. L., Statistical physics of dynamic systems with variable memory, Doklady Phys, 48, 285-289 (2003)
[152] Yaghoobi, S.; Moghaddam, B. P.; Ivaz, K., An efficient cubic spline approximation for variable-order fractional differential equations with time delay, Nonlinear Dyn, 87, 2, 815-826 (2017) · Zbl 1372.34125
[153] Machado, J. A.T.; Moghaddam, B. P., A robust algorithm for nonlinear variable-order fractional control systems with delay, Int J Nonlinear Sci Numer Simul, 19, 3-4, 1-8 (2018)
[154] Yaghoobi, S.; Parsa Moghaddam, B.; Ivaz, K., A numerical approach for variable-order fractional unified chaotic systems with time-delay, Comput Methods Differ Equ, 6, 4, 396-410 (2018) · Zbl 1438.65163
[155] Moghaddam, B. P.; Yaghoobi, S.; Machado, J. T., An extended predictor corrector algorithm for variable-order fractional delay differential equations, J Comput Nonlinear Dyn, 11, 6, 061001 (2016)
[156] Keshi, F. K.; Moghaddam, B. P.; Aghili, A., A numerical approach for solving a class of variable-order fractional functional integral equations, Comput Appl Math, 1-14 (2018)
[157] Zayernouri, M.; Karniadakis, G. E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J Comput Phys, 293, 312-338 (2015) · Zbl 1349.65531
[158] Chauhan, A.; Dabas, J., Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition, Commun Nonlinear Sci Numer Simul, 19, 4, 821-829 (2014)
[159] Cui, J.; Yan, L., Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J Phys A Math Theor, 44, 33, 335201 (2011)
[160] Dabas, J.; Chauhan, A., Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math Comput Model, 57, 3-4, 754-763 (2013) · Zbl 1305.34132
[161] Debbouche, A.; Baleanu, D., Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput Math Appl, 62, 3, 1442-1450 (2011) · Zbl 1228.45013
[162] Dos Santos, J. P.C.; Cuevas, C., Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl Math Lett, 23, 9, 960-965 (2010) · Zbl 1198.45014
[163] Dos Santos, J. P.C.; Arjunan, M. M.; Cuevas, C., Existence results for fractional neutral integro-differential equations with state-dependent delay, Comput Math Appl, 62, 3, 1275-1283 (2011) · Zbl 1228.45014
[164] Nawaz, Y., Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Comput Math Appl, 61, 8, 2330-2341 (2011) · Zbl 1219.65081
[165] Heydari, M. H.; Hooshmandasl, M. R.; Mohammadi, F.; Cattani, C., Wavelets method for solving systems of nonlinear singular fractional volterra integro-differential equations, Commun Nonlinear Sci Numer Simul, 19, 1, 37-48 (2014) · Zbl 1344.65126
[166] Huang, L.; Li, X. F.; Zhao, Y.; Duan, X. Y., Approximate solution of fractional integro-differential equations by taylor expansion method, Comput Math Appl, 62, 3, 1127-1134 (2011) · Zbl 1228.65133
[167] Jiang, W.; Tian, T., Numerical solution of nonlinear volterra integro-differential equations of fractional order by the reproducing kernel method, Appl Math Model, 39, 16, 4871-4876 (2015)
[168] Khader, M. M.; Sweilam, N. H., On the approximate solutions for system of fractional integro-differential equations using chebyshev pseudo-spectral method, Appl Math Model, 37, 24, 9819-9828 (2013) · Zbl 1427.65419
[169] Lin, A.; Ren, Y.; Xia, N., On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators, Math Comput Model, 51, 5-6, 413-424 (2010) · Zbl 1190.60045
[170] Liu, Z.; Sun, J.; Szanto, I., Monotone iterative technique for Riemann-Liouville fractional integro-differential equations with advanced arguments, Results Math, 63, 3-4, 1277-1287 (2013) · Zbl 1276.26017
[171] Ma, X.; Huang, C., Spectral collocation method for linear fractional integrodifferential equations, Appl Math Model, 38, 4, 1434-1448 (2014) · Zbl 1427.65421
[172] Meng, Z.; Wang, L.; Li, H.; Zhang, W., Legendre wavelets method for solving fractional integro-differential equations, Int J Comput Math, 92, 6, 1275-1291 (2015) · Zbl 1315.65111
[173] Mokhtary, P.; Ghoreishi, F., The l 2-convergence of the Legendre spectral tau matrix formulation for nonlinear fractional integro differential equations, Numer Algor, 58, 4, 475-496 (2011) · Zbl 1270.65078
[174] Mokhtary, P., Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations, J Comput Appl Math, 279, 145-158 (2015) · Zbl 1306.65294
[175] Nazari, D.; Shahmorad, S., Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J Comput Appl Math, 234, 3, 883-891 (2010) · Zbl 1188.65174
[176] Zhu, L.; Fan, Q., Numerical solution of nonlinear fractional-order volterra integro-differential equations by SCW, Commun Nonlinear Sci Numer Simul, 18, 5, 1203-1213 (2013) · Zbl 1261.35152
[177] Saadatmandi, A.; Dehghan, M., A legendre collocation method for fractional integro-differential equations, J Vib Control, 17, 13, 2050-2058 (2011) · Zbl 1271.65157
[178] Saeedi, H.; Moghadam, M. M.; Mollahasani, N.; Chuev, G. N., A CAS wavelet method for solving nonlinear fredholm integro-differential equations of fractional order, Commun Nonlinear Sci Numer Simul, 16, 3, 1154-1163 (2011) · Zbl 1221.65354
[179] Suganya, S.; Arjunan, M. M.; Trujillo, J. J., Existence results for an impulsive fractional integro-differential equation with state-dependent delay, Appl Math Comput, 266, 54-69 (2015) · Zbl 1410.34242
[180] Tarasov, V. E., Fractional integro-differential equations for electromagnetic waves in dielectric media, Theoret Math Phys, 158, 3, 355-359 (2009) · Zbl 1177.78020
[181] Vanani, S. K.; Aminataei, A., Operational tau approximation for a general class of fractional integro-differential equations, Comput Appl Math, 30, 3, 655-674 (2011) · Zbl 1247.65174
[182] Vijayakumar, V.; Selvakumar, A.; Murugesu, R., Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Appl Math Comput, 232, 303-312 (2014) · Zbl 1410.93025
[183] Yan, Z., Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in hilbert spaces, IMA J Math Control Inf, 30, 4, 443-462 (2013) · Zbl 1279.93024
[184] Tai, Z.; Wang, X., Controllability of fractional-order impulsive neutral functional infinite delay integro-differential systems in banach spaces, Appl Math Lett, 22, 11, 1760-1765 (2009) · Zbl 1181.34078
[185] Debbouche, A.; Baleanu, D., Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput Math Appl, 62, 3, 1442-1450 (2011) · Zbl 1228.45013
[187] Yan, Z., On a nonlocal problem for fractional integrodifferential inclusions in banach spaces, Annales Polonici Math, 1, 101, 87-103 (2011) · Zbl 1223.34007
[188] Yin, Y.; Yanping, C.; Huang, Y., Convergence analysis of the Jacobi spectralcollocation method for fractional integro-differential equations, Acta Mathematica Scientia, 34, 3, 673-690 (2014) · Zbl 1313.65343
[189] Chen, Y.; Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for volterra integral equations with a weakly singular kernel, Math Comput, 79, 269, 147-167 (2010) · Zbl 1207.65157
[191] Zhang, X.; Tang, B.; He, Y., Homotopy analysis method for higher-order fractional integro-differential equations, Comput Math Appl, 62, 8, 3194-3203 (2011) · Zbl 1232.65120
[192] Zhao, J.; Xiao, J.; Ford, N. J., Collocation methods for fractional integrodifferential equations with weakly singular kernels, Numer Algor, 65, 4, 723-743 (2014) · Zbl 1298.65197
[193] Zhu, L.; Fan, Q., Solving fractional nonlinear fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun Nonlinear Sci Numer Simul, 17, 6, 2333-2341 (2012) · Zbl 1335.45002
[194] Zhu, L.; Fan, Q., Numerical solution of nonlinear fractional-order volterra integro-differential equations by SCW, Commun Nonlinear Sc Numer Simul, 18, 5, 1203-1213 (2013) · Zbl 1261.35152
[195] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications (1993), Gordon & Breach Science Publishers in Switzerland: Gordon & Breach Science Publishers in Switzerland Philadelphia, Pa., USA · Zbl 0818.26003
[196] Li, C. P.; Deng, W. H., Remarks on fractional derivatives, Appl Math Comput, 187, 777-784 (2007) · Zbl 1125.26009
[197] Li, C. P.; Dao, X. H.; Guo, P., Fractional derivatives in complex planes, Nonlin Anal TMA, 71, 1857-1869 (2009) · Zbl 1173.26305
[198] Li, C. P.; Zhao, Z. G., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput Math Appl (2011)
[199] Li, C. P.; Dao, X. H.; Guo, P., Fractional derivatives in complex planes, Nonlin Anal TMA, 71, 1857-1869 (2009) · Zbl 1173.26305
[200] Li, C. P.; Qian, D. L.; Chen, Y. Q., On Riemann-Liouville and caputo derivatives, Discr Dyn Nat Soc (2011)
[201] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations (2006), Elsevier: Elsevier Netherlands · Zbl 1092.45003
[202] Yuzbasi, S., Bessel polynomial solutions of linear differential, integral and integro-differential equations (2009), Graduate School of Natural and Applied Sciences, Mugla University, Master thesis
[203] Hadamard, J., Essai sur l’etude des fonctions, donnees par leur developpement de Taylor (1892), Gauthier-Villars · JFM 24.0359.01
[204] Butzer, P. L.; Kilbas, A. A.; Trujillo, J. J., Compositions of hadamard-type fractional integration operators and the semigroup property, J Math Anal Appl, 269, 2, 387-400 (2002) · Zbl 1027.26004
[205] Butzer, P. L.; Kilbas, A. A.; Trujillo, J. J., Fractional calculus in the Mellin setting and hadamard-type fractional integrals, J Math Anal Appl, 269, 1, 1-27 (2002) · Zbl 0995.26007
[206] Butzer, P. L.; Kilbas, A. A.; Trujillo, J. J., Mellin transform analysis and integration by parts for hadamard-type fractional integrals, J Math Anal Appl, 270, 1, 1-15 (2002) · Zbl 1022.26011
[207] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J Numer Anal, 47, 1760-1781 (2009) · Zbl 1204.26013
[208] Yang, Q. Q.; Liu, F.; Turner, I., Computationally efficient numerical methods for time- and space-fractional fokkerplanck equations, Phys Scr, 014026 (2009)
[209] Yang, Q. Q.; Turner, I.; Liu, F., Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation, ANZIAM J, C800-C814 (2009) · Zbl 1359.35099
[210] Yang, Q. Q.; Liu, F.; Turner, I., Stability and convergence of an effective numerical method for the time-space fractional fokkerplanck equation with a nonlinear source term, Int J Diff Eqs (2010)
[211] Yang, Q. Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with riesz space fractional derivatives, Appl Math Model, 34, 200-218 (2010) · Zbl 1185.65200
[212] Khalil, R.; Al Horani, M.; Yousef, A.; Sababheh, M., A new definition of fractional derivative, J Comput Appl Math, 264, 65-70 (2014) · Zbl 1297.26013
[213] Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr Fract Differ Appl, 1, 2, 1-13 (2015)
[214] Losada, J.; Nieto, J. J., Properties of a new fractional derivative without singular kernel, Progr Fract Differ Appl, 1, 2, 87-92 (2015)
[215] Atangana, A.; Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm Sci, 18 (2016), . OnLine-First (00)
[216] Atangana, A., On the new fractional derivative and application to nonlinear fishe’s reaction-diffusion equation, Appl Math Comput, 273, 948-956 (2016) · Zbl 1410.35272
[217] Algahtani, O. J.J., Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: allen Cahn model, Chaos Solit Fract, 89, 552-559 (2016) · Zbl 1360.35094
[218] Atanackovi, T. M.; Pilipovi, S.; Zorica, D., Properties of the Caputo-Fabrizio fractional derivative and its distributional settings, Fract Calculus Appl Anal, 21, 1, 29-44 (2018) · Zbl 1392.26010
[219] Ortigueira, M. D.; Machado, J. T., A critical analysis of the Caputo-Fabrizio operator, Commun Nonlinear Sci Numer Simul, 59, 608-611 (2018)
[220] Lorenzo, C. F.; Hartley, T. T., Initialization, conceptualization, and application in the generalized (fractional) calculus, Critical Rev Biomed Eng, 35, 6 (2007)
[221] Ingman, D.; Suzdalnitsky, J.; Zeifman, M., Constitutive dynamic-order model for nonlinear contact phenomena, J Appl Mech, 67, 2, 383-390 (2000) · Zbl 1110.74493
[222] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dyn, 29, 1-4, 57-98 (2002) · Zbl 1018.93007
[223] Coimbra, C. F., Mechanics with variable-order differential operators, Annalen der Physik, 12, 11-12, 692-703 (2003) · Zbl 1103.26301
[224] Sheng, H.; Sun, H. G.; Coopmans, C.; Chen, Y. Q.; Bohannan, G. W., A physical experimental study of variable-order fractional integrator and differentiator, Eur Phys J Spec Top, 193, 1, 93-104 (2011)
[225] Sierociuk, D.; Malesza, W.; Macias, M., Numerical schemes for initialized constant and variable fractional-order derivatives: matrix approach and its analog verification, J Vib Control, 22, 8, 2032-2044 (2016) · Zbl 1365.34135
[226] Sierociuk, D.; Malesza, W.; Macias, M., Switching scheme, equivalence, and analog validation of the alternative fractional variable-order derivative definition, Proceedings of the IEEE 52nd annual conference on decision and control (CDC), 3876-3881 (2013), IEEE
[227] Sun, H. G.; Chen, W.; Wei, H.; Chen, Y. Q., A comparative study of constantorder and variable-order fractional models in characterizing memory property of systems, Eur Phys J Spec Top, 193, 1, 185 (2011)
[228] Valrio, D.; Da-Costa, J. S., Variable-order fractional derivatives and their numerical approximations, Signal Process, 91, 3, 470-483 (2011) · Zbl 1203.94060
[229] Zhao, X.; Sun, Z. Z.; Karniadakis, G. E., Second-order approximations for variable order fractional derivatives: algorithms and applications, J Comput Phys, 293, 184-200 (2015) · Zbl 1349.65092
[230] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for fractional advection dispersion equation, J Comput Appl Math, 172, 65-77 (2004) · Zbl 1126.76346
[231] Metzler, R.; Klafter, J., The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys Rep, 339, 1-77 (2000) · Zbl 0984.82032
[232] Oldham, K.; Spanier, J., The fractional calculus (1974), Acdemic Press: Acdemic Press NY · Zbl 0428.26004
[233] Langlands, T. A.M.; Henry, B. I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J Comput Phys, 205, 719-736 (2005) · Zbl 1072.65123
[234] Aissani, K.; Benchohra, M., Fractional integro-differential equations with state-dependent delay, Adv Dyn Syst Appl, 9, 1, 17-30 (2014)
[235] Yang, X. J.; Machado, J. T.; Srivastava, H. M., A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Appl Math Comput, 274, 143-151 (2016) · Zbl 1410.65415
[236] Atanackovic, T. M.; Stankovic, B., On a numerical scheme for solving differential equations of fractional order, Mech Res Commun, 35, 7, 429-438 (2008) · Zbl 1258.65103
[237] Pooseh, S.; Almeida, R.; Torres, D. F.M., Numerical approximations of fractional derivatives with applications, Asian J Control, 15, 3, 698-712 (2013) · Zbl 1327.93165
[238] Zahra, W. K.; Hikal, M. M., Non standard finite difference method for solving variable order fractional optimal control problems, J Vib Control, 23, 6, 948-958 (2017) · Zbl 1387.93095
[239] Odibat, Z. M., Computational algorithms for computing the fractional derivatives of functions, Math Comput Simul, 79, 7, 20132020 (2009)
[240] Sousa, E., How to approximate the fractional derivative of order \(1 < α\) ≤ 2, Proceedings of the 4-th IFAC workshop fractional differentiation and its applications (Badajoz, Spain) (2010)
[241] Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Y., Algorithms for the fractional calculus: a selection of numerical methods, Comput Meth Appl Mech Eng, 194, 43-773 (2005) · Zbl 1119.65352
[242] Odibat, Z. M., Approximations of fractional integrals and Caputo fractional derivatives, Appl Math Comput, 178, 527-533 (2006) · Zbl 1101.65028
[243] Murio, D. A., On the stable numerical evaluation of Caputo fractional derivatives, Comput Math Appl, 51, 1539-1550 (2006) · Zbl 1134.65335
[244] Schmidt, A.; Gaul, L., On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems, Sign Process, 86, 2592-2601 (2006) · Zbl 1172.65371
[245] Podlubny, I., Matrix approach to discrete fractional calculus, Fract Cal Appl Anal, 4, 359-386 (2000) · Zbl 1030.26011
[246] Sierociuk, D.; Malesza, W.; Macias, M., Numerical schemes for initialized constant and variable fractional-order derivatives: matrix approach and its analog verification, J Vib Control, 22, 8, 2032-2044 (2016) · Zbl 1365.34135
[247] Xu, Y.; He, Z., Existence and uniqueness results for cauchy problem of variable-order fractional differential equations, J Appl Math Comput, 43, 1-2, 295-306 (2013) · Zbl 1296.34041
[248] Samko, S. G.; Ross, B., Integration and differentiation to a variable fractional order, Integral Trans Spec Funct, 1, 4, 277-300 (1993) · Zbl 0820.26003
[249] Ramirez, L. E.; Coimbra, C. F., On the selection and meaning of variable order operators for dynamic modeling, Int J Differ Equ (2010) · Zbl 1207.34011
[250] Lifshits, M.; Linde, W., Fractional integration operators of variable order: continuity and compactness properties, Math Nach, 287, 8-9, 980-1000 (2014) · Zbl 1298.26025
[251] Li, C.; Zeng, F., Finite difference methods for fractional differential equations, Int J Bifurcat Chaos, 22, 04, 1230014 (2012) · Zbl 1258.34018
[252] Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl Math Comput, 273, 948-956 (2016) · Zbl 1410.35272
[253] Al-Khaled, K., Numerical solution of time-fractional partial differential equations using sumudu decomposition method, Rom J Phys, 60, 1-2, 99-110 (2015)
[254] Si-Ammour, A.; Djennoune, S.; Bettayeb, M., A sliding mode control for linear fractional systems with input and state delays, Commun Nonlinear Sci Numer Simul, 14, 5, 2310-2318 (2009) · Zbl 1221.93048
[255] Magin, R. L., Fractional calculus models of complex dynamics in biological tissues, Comput Math Appl, 59, 5, 1586-1593 (2010) · Zbl 1189.92007
[256] Baskonus, H. M.; Hammouch, Z.; Mekkaoui, T.; Bulut, H., Chaos in the fractional order logistic delay system: circuit realization and synchronization, Proceedings of the AIP Conference Proceedings (2016), AIP Publishing
[257] Bhalekar, S., Dynamical analysis of fractional order Ucar prototype delayed system, signal, Image Video Process, 6, 3, 513-519 (2012)
[258] Bhalekar, S., A necessary condition for the existence of chaos in fractional order delay differential equations, Int J Math Sci, 7, 3, 28-32 (2013)
[259] Feliu-Batlle, V.; Rivas-Perez, R.; Castillo-Garcia, F. J., Fractional order controller robust to time delay variations for water distribution in an irrigation main canal pool, Comput Electron Agricul, 69, 2, 185-197 (2009)
[260] Bhalekar, S.; Daftardar-Gejji, V., A predictor corrector scheme for solving nonlinear delay differential equations of fractional order, J Fract Calculus Appl, 1, 5, 1-9 (2011)
[261] Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J Math Anal Appl, 338, 2, 1340-1350 (2008) · Zbl 1209.34096
[262] Wang, Z.; Huang, X.; Zhou, J., A numerical method for delayed fractional order differential equations: based on GL definition, Appl Math Inf Sci,, 7, 2, 525-529 (2013)
[263] Delasen, M., Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays, Proceedings of the Abstract and Applied Analysis, Hindawi (2011)
[264] Delasen, M., About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl, 2011, 1, 867932 (2011) · Zbl 1219.34102
[265] Bhalekar, S.; Daftardar-Gejji, V.; Baleanu, D.; Magin, R., Generalized fractional order Bloch equation with extended delay, Int J Bifurcat Chaos, 22, 04, 1250071 (2012) · Zbl 1258.34156
[266] Wang, Z., A numerical method for delayed fractional-order differential equations, J Appl Math Approx Methods Solv Fract Equ 33 (2013)
[267] Morgado, M. L.; Ford, N. J.; Lima, P. M., Analysis and numerical methods for fractional differential equations with delay, J Comput Appl Math,, 252, 159-168 (2013) · Zbl 1291.65214
[268] Solis-Perez, J. E.; Gomez-Aguilar, J. F.; Atangana, A., Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws, Chaos Solitons Fract, 114, 175-185 (2018)
[269] Zhao, T.; Mao, Z.; Karniadakis, G. E., Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations, Comput Methods Appl Mech Eng, 348, 377-395 (2019)
[270] Avalos-Ruiz, L. F.; Zuniga-Aguilar, C. J.; Gomez-Aguilar, J. F.; Escobar-Jimenez, R. F.; Romero-Ugalde, H. M., FPGA Implementation and control of chaotic systems involving the variable-order fractional operator with mittag-leffler law, Chaos Solit Fract, 115, 177-189 (2018)
[271] Baleanu, D.; Shiri, B., Collocation methods for fractional differential equations involving non-singular kernel, Chaos Solit Fract, 116, 136-145 (2018)
[272] Bhrawy, A. H.; Zaky, M. A., Highly accurate numerical schemes for multi-dimensional space variable-order fractional schrodinger equations, Comput Math Appl, 73, 6, 1100-1117 (2017) · Zbl 1412.65162
[273] Soon, C. M.; Coimbra, C. F.; Kobayashi, M. H., The variable viscoelasticity oscillator, Annalen der Physik, 14, 6, 378-389 (2005) · Zbl 1125.74316
[274] Hu, L.; Ren, Y.; Sakthivel, R., Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, 79, 507 (2009), SpringerVerlag · Zbl 1184.45006
[275] Ren, Y.; Qin, Y.; Sakthivel, R., Existence results for fractional order semilinear integro-differential evolution equations with infinite delay, Integral Equ Operator Theory, 67, 1, 33-49 (2010) · Zbl 1198.45009
[276] Agarwal, R. P.; De-Andrade, B.; Siracusa, G., On fractional integro differential equations with state-dependent delay, Comput Math Appl, 62, 3, 1143-1149 (2011) · Zbl 1228.35262
[277] Yu, Q.; Vegh, V.; Liu, F.; Turner, I., A variable order fractional differential-based texture enhancement algorithm with application in medical imaging, PloS One, 10, 7, e0132952 (2015)
[278] Razminia, A.; Dizaji, A. F.; Majd, V. J., Solution existence for non-autonomous variable-order fractional differential equations, Math Comput Model, 55, 3-4, 1106-1117 (2012) · Zbl 1255.34008
[279] Zhang, S., Existence result of solutions to differential equations of variable-order with nonlinear boundary value conditions, Commun Nonlinear Sci Numer Simul, 18, 12, 3289-3297 (2013) · Zbl 1344.34022
[280] Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, J Comput Phys, 293, 104-114 (2015) · Zbl 1349.65263
[281] Zaky, M. A.; Ezz-Eldien, S. S.; Doha, E. H.; Machado, J. T.; Bhrawy, A. H., An efficient operational matrix technique for multi-dimensional variable-order time fractional diffusion equations, J Comput Nonlinear Dyn, 11, 6, 061002 (2016)
[282] Bhrawy, A. H.; Zaky, M. A., An improved collocation method for multi-dimensional space-time variable-order fractional Schrdinger equations, Appl Numer Math, 111, 197-218 (2017) · Zbl 1353.65106
[283] Cao, J.; Qiu, Y., A high order numerical scheme for variable order fractional ordinary differential equation, Appl Math Lett, 61, 88-94 (2016) · Zbl 1347.65119
[284] Cao, J.; Qiu, Y.; Song, G., A compact finite difference scheme for variable order subdiffusion equation, Commun Nonlinear Sci Numer Simul, 48, 140-149 (2017)
[285] Yepez-Martinez, H.; Gomez-Aguilar, J. F., A new modified definition of caputo-fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM), J Comput Appl Math, 346, 247-260 (2019) · Zbl 1402.26005
[286] Chen, C. M., Numerical methods for solving a two-dimensional variable-order modified diffusion equation, Appl Math Comput, 225, 62-78 (2013) · Zbl 1334.65129
[287] Chen, R.; Liu, F.; Anh, V., Numerical methods and analysis for a multi-term time-space variable-order fractional advection-diffusion equations and applications, J Comput Appl Math, 352, 437-452 (2019) · Zbl 1448.76151
[288] Hu, X.; Liu, F.; Turner, I.; Anh, V., An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation, Numer Algor, 72, 2, 393-407 (2016) · Zbl 1343.65110
[289] Mainardi, F., Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models (2010), World Scientific · Zbl 1210.26004
[290] Zhang, H.; Jiang, X.; Yang, X., A time-space spectral method for the time-space fractional Fokker-planck equation and its inverse problem, Appl Math Comput, 320, 302-318 (2018)
[291] Yu, B.; Jiang, X.; Xu, H., A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation, Numer Algor, 68, 4, 923-950 (2015) · Zbl 1314.65114
[292] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-planck equation, J Comput Appl Math, 166, 1, 209-219 (2004) · Zbl 1036.82019
[293] Escamilla, A. C.; Aguilar, J. F.G.; Torres, L.; Jimenez, R. F.E.; Rodriguez, M. R., Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order, Phys A Stat Mech Appl, 487, 1-21 (2017)
[294] Zuniga-Aguilar, C. J.; Romero-Ugalde, H. M.; Goomez-Aguilar, J. F.; Escobar-Jimenez, R. F.; Valtierra-Rodriguez, M., Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos Solit Fract, 103, 382-403 (2017) · Zbl 1375.34110
[295] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Anh, V.; Li, J., A fast second-order accurate method for a two-sided space-fractional diffusion equation with variable coefficients, Comput Math Appl, 73, 6, 1155-1171 (2017) · Zbl 1412.65072
[296] Chen, M.; Deng, W., Fourth order accurate scheme for the space fractional diffusion equations, SIAM J Numer Anal, 52, 3, 1418-1438 (2014) · Zbl 1318.65048
[297] Ren, J.; Sun, Z. Z.; Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with neumann boundary conditions, J Comput Phys, 232, 1, 456-467 (2013) · Zbl 1291.35428
[298] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Li, J., High-order numerical methods for the Riesz space fractional advection-dispersion equations (2016)
[299] Ding, H.; Li, C., High-order algorithms for Riesz derivative and their applications (III), Fract Calculus Appl Anal, 19, 1, 19-55 (2016) · Zbl 1332.65122
[300] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Gu, Y. T., Finite element method for space-time fractional diffusion equation, Numer Algor, 72, 3, 749-767 (2016) · Zbl 1343.65122
[301] Liu, Y.; Du, Y.; Li, H.; Li, J.; He, S., A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput Math Appl, 70, 10, 2474-2492 (2015)
[302] Zhao, Y.; Bu, W.; Huang, J.; Liu, D. Y.; Tang, Y., Finite element method for two-dimensional space-fractional advection-dispersion equations, Appl Math Comput, 257, 553-565 (2015) · Zbl 1339.65185
[303] Hejazi, H.; Moroney, T.; Liu, F., Stability and convergence of a finite volume method for the space fractional advection-dispersion equation, J Comput Appl Math, 255, 684-697 (2014) · Zbl 1291.65280
[304] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I., Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation, Appl Math Comput, 257, 52-65 (2015) · Zbl 1339.65144
[305] Bhrawy, A. H.; Baleanu, D., A spectral legendregausslobatto wcollocation method for a space-fractional advection diffusion equations with variable coefficients, Rep Math Phys, 72, 2, 219-233 (2013) · Zbl 1292.65109
[306] Zheng, M.; Liu, F.; Turner, I.; Anh, V., A novel high order space-time spectral method for the time fractional Fokker-planck equation, SIAM J Sci Comput, 37, 2, A701-A724 (2015) · Zbl 1320.82052
[307] Zeng, F.; Liu, F.; Li, C.; Burrage, K.; Turner, I.; Anh, V., A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J Numer Anal, 52, 6, 2599-2622 (2014) · Zbl 1382.65349
[308] Moroney, T.; Yang, Q., Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast poisson preconditioners, J Comput Phys, 246, 304-317 (2013) · Zbl 1349.65398
[309] Chen, S.; Liu, F.; Jiang, X.; Turner, I.; Anh, V., A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients, Appl Math Comput, 257, 591-601 (2015) · Zbl 1339.65104
[310] Ghanbari, B.; Gomez-Aguilar, J. F., Modeling the dynamics of nutrient-phytoplankton-zooplankton system with variable-order fractional derivatives, Chaos Solit Fract, 116, 114-120 (2018)
[311] Ma, H. C.; Yao, D. D.; Peng, X. F., Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method, Thermal Sci, 19, 4, 1239-1244 (2015)
[312] Mohyuddin, S. T.; Nawaz, T.; Azhar, E.; Akbar, M. A., Fractional sub-equation method to space-time fractional Calogero-Degasperis and potential Kadomtsev-Petviashvili equations, J Taibah Uni Sci, 11, 2, 258-263 (2017)
[313] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn, 29, 1-4, 3-22 (2002) · Zbl 1009.65049
[314] Gejji, V. D.; Sukale, Y.; Bhalekar, S., A new predictor-corrector method for fractional differential equations, Appl Math Comput, 244, 158-182 (2014) · Zbl 1337.65071
[315] Rosenfeld, J. A.; Dixon, W. E., Approximating the Caputo fractional derivative through the Mittag-Leffler reproducing kernel hilbert space and the Kernelized adams-Bashforth-Moulton method, SIAM J Numer Anal, 55, 3, 1201-1217 (2017) · Zbl 1375.26018
[316] Owolabi, K. M.; Atangana, A., Analysis and application of new fractional adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solit Fract, 105, 111-119 (2017) · Zbl 1380.65120
[317] Asl, M. S.; Javidi, M., An improved PC scheme for nonlinear fractional differential equations: error and stability analysis, J Comput Appl Math, 324, 101-117 (2017) · Zbl 1369.65087
[318] Li, C.; Yi, Q.; Chen, A., Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J Comput Phys, 316, 614-631 (2016) · Zbl 1349.65246
[319] Zhao, L.; Deng, W., Jacobian-predictor-corrector approach for fractional differential equations, Adv Comput Math, 40, 1, 137-165 (2014) · Zbl 1322.65079
[320] Alkahtani, B. S.T., Atangana-batogna numerical scheme applied on a linear and non-linear fractional differential equation, Eur Phys J Plus, 133, 3, 111 (2018)
[321] Atangana, A.; Owolabi, K. M., New numerical approach for fractional differential equations, Math Model Natural Phenomena, 13, 1, 3 (2018) · Zbl 1406.65045
[322] Toufik, M.; Atangana, A., New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur Phys J Plus, 132, 10, 444 (2017)
[323] Atangana, A., Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties, Phys A Stat Mech Appl, 505, 688-706 (2018)
[324] Atangana, A.; Aguilar, J. F.G., Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena, Eur Phys J Plus, 133, 1-22 (2018)
[325] Atangana, A.; Aguilar, J. F.G., Fractional derivatives with no-index law property: application to chaos and statistics, Chaos Solit Fract, 114, 516-535 (2018)
[326] Hajipour, M.; Jajarmi, A.; Baleanu, D.; Sun, H., On an accurate discretization of a variable-order fractional reaction-diffusion equation, Commun Nonlinear Sci Numer Simul, 69, 119-133 (2019)
[327] Tavares, D.; Almeida, R.; Torres, D. F., Caputo derivatives of fractional variable order: numerical approximations, Commun Nonlinear Sci Numer Simul, 35, 69-87 (2016)
[328] Chen, C. M.; Liu, F.; Anh, V.; Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J Sci Comput, 32, 4, 1740-1760 (2010) · Zbl 1217.26011
[329] Chen, C. M.; Liu, F.; Anh, V.; Turner, I., Numerical simulation for the variable-order galilei invariant advection diffusion equation with a nonlinear source term, Appl Math Comput, 217, 12, 5729-5742 (2011) · Zbl 1227.65072
[330] Chen, C. M.; Liu, F.; Turner, I.; Anh, V.; Chen, Y., Numerical approximation for a variable-order nonlinear reaction-subdiffusion equation, Numer Algor, 63, 2, 265-290 (2013) · Zbl 1278.65121
[331] Escamilla, A. C.; Aguilar, J. F.G.; Torres, L.; Jimenez, R. F.E., A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel, Phys A Stat Mech Appl, 491, 406-424 (2018)
[332] Chen, S.; Liu, F.; Burrage, K., Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media, Comput Math Appl, 68, 12, 2133-2141 (2014) · Zbl 1369.35105
[333] Abdelkawy, M. A.; Zaky, M. A.; Bhrawy, A. H.; Baleanu, D., Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom Rep Phys, 67, 3, 773-791 (2015)
[334] Heydari, M. H.; Hooshmandasl, M. R.; Cattani, C.; Hariharan, G., An optimization wavelet method for multi variable-order fractional differential equations, Fund Inf, 151, 1-4, 255-273 (2017) · Zbl 1379.65046
[335] Heydari, M. H.; Avazzadeh, Z., Legendre wavelets optimization method for variable-order fractional poisson equation, Chaos Solit Fract, 112, 180-190 (2018) · Zbl 1398.65316
[336] Heydari, M. H.; Hooshmandasl, M. R.; Ghaini, F. M.; Fereidouni, F., Two-dimensional Legendre wavelets for solving fractional poisson equation with dirichlet boundary conditions, Eng Anal Bound Elements, 37, 11, 1331-1338 (2013) · Zbl 1287.65113
[338] Heydari, M. H.; Hooshmandasl, M. R.; Mohammadi, F., Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Appl Math Comput, 234, 267-276 (2014) · Zbl 1298.65181
[339] Gupta, A. K.; Ray, S. S., Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind chebyshev wavelet method, Appl Math Model, 39, 17, 5121-5130 (2015)
[340] Heydari, M. H.; Avazzadeh, Z.; Yang, Y., A computational method for solving variable-order fractional nonlinear diffusion-wave equation, Appl Math Comput, 352, 235-248 (2019)
[341] Heydari, M. H.; Avazzadeh, Z.; Haromi, M. F., A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation, Appl Math Comput, 341, 215-228 (2019)
[342] Hosseininia, M.; Heydari, M. H.; Avazzadeh, Z.; Ghaini, F. M., Two-dimensional Legendre wavelets for solving variable-order fractional nonlinear advection-diffusion equation with variable coefficients, Int J Nonlinear Sci Numer Simul, 19, 7-8, 793-802 (2018) · Zbl 06987921
[343] Lin, X. L.; Ng, M. K.; Sun, H. W., Stability and convergence analysis of finite difference schemes for time-dependent space-fractional diffusion equations with variable diffusion coefficients, J Sci Comput, 75, 2, 1102-1127 (2018) · Zbl 1398.65214
[344] Zhang, Y.; Ding, H., High-order algorithm for the two-dimension Riesz space-fractional diffusion equation, Int J Comput Math, 94, 10, 2063-2073 (2017) · Zbl 1394.65082
[345] Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F., Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains, J Comput Phys, 330, 863-883 (2017) · Zbl 1378.35330
[346] Tian, W.; Zhou, H.; Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math Comput, 84, 294, 1703-1727 (2015) · Zbl 1318.65058
[347] Sousa, E.; Li, C., A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl Numer Math, 90, 22-37 (2015) · Zbl 1326.65111
[349] Lei, S. L.; Huang, Y. C., Fast algorithms for high-order numerical methods for space-fractional diffusion equations, Int J Comput Math, 94, 5, 1062-1078 (2017) · Zbl 1378.65160
[350] Hao, Z. P.; Sun, Z. Z.; Cao, W. R., A fourth-order approximation of fractional derivatives with its applications, J Comput Phys, 281, 787-805 (2015) · Zbl 1352.65238
[351] Lin, X. L.; Ng, M. K.; Sun, H. W., A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations, J Comput Phys, 336, 69-86 (2017) · Zbl 1375.35606
[352] del Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E., Fractional diffusion in plasma turbulence, Phys Plasmas, 11, 8, 3854-3864 (2004)
[353] Celik, C.; Duman, M., Crank-nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J Comput Phys, 231, 4, 1743-1750 (2012) · Zbl 1242.65157
[354] Malesza, W.; Macias, M.; Sierociuk, D., Analytical solution of fractional variable order differential equations, J Comput Appl Math, 348, 214-236 (2019) · Zbl 1409.34014
[355] Sierociuk, D.; Malesza, W.; Macias, M., On the recursive fractional variable-order derivative: equivalent switching strategy, duality, and analog modeling, Circuits Syst Signal Process, 34, 4, 1077-1113 (2015) · Zbl 1342.94132
[356] Mardani, A.; Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C., A meshless method for solving the time fractional advection-diffusion equation with variable coefficients, Comput Math Appl, 75, 1, 122-133 (2018)
[357] Tayebi, A.; Shekari, Y.; Heydari, M. H., A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation, J Comput Phys, 340, 655-669 (2017) · Zbl 1380.65185
[358] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations, Eng Anal Boundary Elements, 50, 412-434 (2015) · Zbl 1403.65082
[359] Zhuang, P.; Gu, Y.; Liu, F.; Turner, I.; Yarlagadda, P. K.D. V., Timeâependent fractional advection-diffusion equations by an implicit MLS meshless method, Int J Numer Methods Eng, 88, 13, 1346-1362 (2011) · Zbl 1242.76262
[360] Morales-Delgado, V. F.; Gomez-Aguilar, J. F.; Taneco-Hernandez, M. A.; Escobar-Jimenez, R. F., A novel fractional derivative with variable-and constant-order applied to a mass-spring-damper system, Eur Phys J Plus, 133, 2, 78 (2018)
[361] Obembe, A. D.; Hossain, M. E.; Abu-Khamsin, S. A., Variable-order derivative time fractional diffusion model for heterogeneous porous media, J Petrol Sci Eng, 152, 391-405 (2017)
[362] Ortigueira, M. D.; Valerio, D.; Machado, J. T., Variable order fractional systems, Commun Nonlinear Sci Numer Simul, 71, 231-243 (2019)
[364] Fang, Z. W.; Sun, H. W.; Wei, H. Q., An approximate inverse pre conditioner for spatial fractional diffusion equations with piecewise continuous coefficients, Int J Comput Math, 1-23 (2019)
[365] Xu, T.; Lu, S.; Chen, W.; Chen, H., Finite difference scheme for multi-term variable-order fractional diffusion equation, Adv Differ Equ, 2018, 1, 103 (2018) · Zbl 1445.65041
[366] Zaky, M. A.; Baleanu, D.; Alzaidy, J. F.; Hashemizadeh, E., Operational matrix approach for solving the variable-order nonlinear galilei invariant advection-diffusion equation, Adv Differ Equ, 2018, 1, 102 (2018) · Zbl 1445.65042
[367] Wei, S.; Chen, W.; Zhang, Y.; Wei, H.; Garrard, R. M., A local radial basis function collocation method to solve the variable-order time fractional diffusion equation in a two-dimensional irregular domain, Numer Methods Part Differ Equ, 34, 4, 1209-1223 (2018) · Zbl 1407.65231
[368] Ramirez, L. E.; Coimbra, C. F., On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Phys D Nonlinear Phen, 240, 13, 1111-1118 (2011) · Zbl 1219.76054
[370] Aissani, K.; Benchohra, M., Fractional integro-differential equations with state-dependent delay, Adv Dyn Syst Appl, 9, 1, 17-30 (2014)
[371] Benchohra, M.; Berhoun, F., Impulsive fractional differential equations with state-dependent delay, Commun Appl Anal, 14, 2, 213 (2010) · Zbl 1203.26007
[372] Benchohra, M.; Litimein, S.; NGuerekata, G., On fractional integro-differential inclusions with state-dependent delay in banach spaces, Appl Anal, 92, 2, 335-350 (2013) · Zbl 1269.34083
[373] Benchohra, M.; Litimein, S.; Trujillo, J. J.; Velasco, M. P., Abstract fractional integro-differential equations with state-dependent delay, Int J Evol Equat, 6, 2, 25-38 (2012)
[374] dos Santos, J. P.C.; Arjunan, M. M.; Cuevas, C., Existence results for fractional neutral integro-differential equations with state-dependent delay, Comput Math Appl, 62, 3, 1275-1283 (2011) · Zbl 1228.45014
[375] Zhou, Y.; Suganya, S.; Mallika Arjunan, M.; Ahmad, B., Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in hilbert spaces, IMA J Math Control Inf, 00, 1-20 (2018)
[376] Ganesh, R.; Sakthivel, R.; Ren, Y.; Anthoni, S. M.; Mahmudov, N. I., Controllability of neutral fractional functional equations with impulses and infinite delay, Abst Appl Anal Hindawi, 2013 (2013) · Zbl 1266.93019
[377] Ganesh, R.; Sakthivel, R.; Mahmudov, N. I., Approximate controllability of fractional functional equations with infinite delay, Topolog Methods Nonlinear Anal, 43, 2, 345-364 (2014) · Zbl 1360.93097
[378] Kavitha, V.; Wang, P. Z.; Murugesu, R., Existence results for neutral functional fractional differential equations with state dependent-delay, Malaya J Matematik, 5061, 1, 1 (2012)
[379] Rathinasamy, S.; Yong, R., Approximate controllability of fractional differential equations with state-dependent delay, Results Math, 63, 3-4, 949-963 (2013) · Zbl 1272.34105
[380] Wang, J.; Ibrahim, A. G.; Feckan, M., Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on banach spaces, Appl Math Comput, 257, 103-118 (2015) · Zbl 1338.34027
[381] Kothari, K.; Mehta, U.; Vanualailai, J., A novel approach of fractional-order time delay system modeling based on haar wavelet, ISA Trans, 80, 371-380 (2018)
[382] Peng, C.; Li, W.; Wang, Y., Frequency domain identification of fractional order time delay systems, Proceedings of the Chinese Control and Decision Conference IEEE, 2635-2638 (2010)
[383] Das, S.; Molla, N. U.; Pan, I.; Pakhira, A.; Gupta, A., Online identification of fractional order models with time delay: An experimental study, Proceedings of the International Conference on Communication and Industrial Application IEEE, 1-5 (2011)
[384] Narang, A.; Shah, S. L.; Chen, T., Continuous-time model identification of fractional-order models with time delays, IET Control Theory Appl, 5, 7, 900-912 (2011)
[385] Nie, Z.; Wang, Q.; Liu, R.; Lan, Y., Identification and PID control for a class of delay fractional-order systems, IEEE/CAA J Autom Sin, 3, 4, 463-476 (2016)
[386] Tang, Y.; Li, N.; Liu, M.; Lu, Y.; Wang, W., Identification of fractional-order systems with time delays using block pulse functions, Mech Syst Signal Process, 91, 382-394 (2017)
[387] Baleanu, D.; Agheli, B.; Darzi, R., An optimal method for approximating the delay differential equations of noninteger order, Adv Differ Equ, 2018, 1, 284 (2018) · Zbl 1446.35012
[388] Sakar, M. G.; Uludag, F.; Erdogan, F., Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method, Appl Math Model, 40, 13-14, 6639-6649 (2016)
[389] Baleanu, D.; Magin, R. L.; Bhalekar, S.; Daftardar-Gejji, V., Chaos in the fractional order nonlinear Bloch equation with delay, Commun Nonlinear Sci Numer Simul, 25, 1-3, 41-49 (2015)
[390] Benchohra, M.; Bennihi, O.; Ezzinbi, K., Existence results for some neutral partial functional differential equations of fractional order with state-dependent delay, Cubo (Temuco), 16, 3, 37-53 (2014) · Zbl 1333.34114
[391] Bu, S.; Cai, G., Well-posedness of fractional integro-differential equations in vector-valued functional spaces, Math Nachr, 292, 5, 969-982 (2019) · Zbl 1417.34191
[392] Dehghan, M.; Abbaszadeh, M., A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed-order fractional damped diffusion-wave equation, Math Methods Appl Sci, 41, 9, 3476-3494 (2018) · Zbl 1395.65098
[393] Cermak, J.; Dosla, Z.; Kisela, T., Fractional differential equations with a constant delay: stability and asymptotics of solutions, Appl Math Comput, 298, 336-350 (2017) · Zbl 1411.34099
[394] Pimenov, V. G.; Hendy, A. S.; De Staelen, R. H., On a class of non-linear delay distributed order fractional diffusion equations, J Comput Appl Math, 318, 433-443 (2017) · Zbl 1357.65127
[395] Zhu, B.; Liu, L.; Wu, Y., Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl Math Lett, 61, 73-79 (2016) · Zbl 1355.35196
[396] Hao, Z.; Fan, K.; Cao, W.; Sun, Z., A finite difference scheme for semilinear space-fractional diffusion equations with time delay, Appl Math Comput, 275, 238-254 (2016) · Zbl 1410.65310
[397] Chalishajar, DN; Karthikeyan, K., Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces, Acta Math. Sci., Ser. B, 33, 3, 1-16 (2013) · Zbl 1299.34059
[398] Liu, S.; Zhou, X. F.; Li, X.; Jiang, W., Asymptotical stability of riemannliouville fractional singular systems with multiple time-varying delays, Appl Math Lett, 65, 32-39 (2017)
[399] Brzdk, J.; Eghbali, N., On approximate solutions of some delayed fractional differential equations, Appl Math Lett, 54, 31-35 (2016) · Zbl 1381.34103
[400] Xu, M. Q.; Lin, Y. Z., Simplified reproducing kernel method for fractional differential equations with delay, Appl Math Lett, 52, 156-161 (2016) · Zbl 1330.65102
[401] Cermak, J.; Hornicek, J.; Kisela, T., Stability regions for fractional differential systems with a time delay, Commun Nonlinear Sci Numer Simul, 31, 1-3, 108-123 (2016)
[402] Hristova, S.; Tunc, C., Stability of nonlinear volterra integro-differential equations with Caputo fractional derivative and bounded delays, Electron J Differ Equ, 2019, 30, 1-11 (2019) · Zbl 07037988
[403] Nguyen, D., Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays, Commun Nonlinear Sci Numer Simul, 19, 1, 1-7 (2014) · Zbl 1344.34086
[404] Wang, F.; Chen, D.; Xu, B.; Zhang, H., Nonlinear dynamics of a novel fractional-order francis hydro-turbine governing system with time delay, Chaos Solit Fract, 91, 329-338 (2016) · Zbl 1372.93150
[405] Bolat, Y., On the oscillation of fractional-order delay differential equations with constant coefficients, Commun Nonlinear Sci Numer Simul, 19, 11, 3988-3993 (2014)
[406] Cheng, Y.; Gao, S.; Wu, Y., Exact controllability of fractional order evolution equations in banach spaces, Adv Differ Equ, 2018, 1, 332 (2018) · Zbl 1448.93024
[407] Suganya, S.; Mallika Arjunan, M., Existence of mild solutions for impulsive fractional integro-differential inclusions with state-dependent delay, Mathematics, 5, 1, 9 (2017) · Zbl 1365.34020
[408] Wang, L.; Wu, Y.; Ren, Y.; Chen, X., Two analytical methods for fractional partial differential equations with proportional delay, Int J Appl Math, 49, 1 (2019)
[409] Singh, B. K.; Kumar, P., Extended fractional reduced differential transform for solving fractional partial differential equations with proportional delay, Int J Appl Comput Math, 3, 1, 631-649 (2017)
[410] Singh, B. K.; Kumar, P., Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay, SeMA J, 75, 1, 111-125 (2018) · Zbl 06859008
[411] Sherif, M. N., Numerical solution of system of fractional delay differential equations using polynomial spline functions, Appl Math, 7, 06, 518 (2016)
[412] Zayernouri, M.; Cao, W.; Zhang, Z.; Karniadakis, G. E., Spectral and discontinuous spectral element methods for fractional delay equations, SIAM J Scient Comput, 36, 6, B904-B929 (2014) · Zbl 1314.34159
[413] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., Numerical algorithm for solving multi-pantograph delay equations on the half-line using Jacobi rational functions with convergence analysis, Acta Math Appl Sin Engl Ser, 33, 2, 297-310 (2017) · Zbl 1371.65063
[414] Wang, J.; Zhang, Y., Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63, 8, 1181-1190 (2014) · Zbl 1296.34034
[415] Mophou, G. M.; NGuerekata, G. M., Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl Math Comput, 216, 1, 61-69 (2010) · Zbl 1191.34098
[416] Siracusa, G.; Henriquez, H. R.; Cuevas, C., Existence results for fractional integro-differential inclusions with state-dependent delay, Nonautonomous Dyn Syst, 4, 1, 62-77 (2017) · Zbl 1377.34099
[417] Nouri, K.; Nazari, M.; Keramati, B., Existence results for a coupled system of fractional integro-differential equations with time-dependent delay, J Fixed Point Theory Appl, 19, 4, 2927-2943 (2017) · Zbl 1376.34065
[418] Kalamani, P.; Baleanu, D.; Selvarasu, S.; Arjunan, M. M., On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv Differ Equ, 2016, 1, 163 (2016) · Zbl 1422.34221
[419] Shah, K.; Khalil, H.; Khan, R. A., Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Chaos Solit Fract, 77, 240-246 (2015) · Zbl 1353.34028
[420] Ahmad, B.; Ntouyas, S. K.; Alsaedi, A., On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Solitons Fract, 83, 234-241 (2016) · Zbl 1355.34012
[421] Rahimkhani, P.; Ordokhani, Y.; Babolian, E., A new operational matrix based on bernoulli wavelets for solving fractional delay differential equations, Numer Algor, 74, 1, 223-245 (2017) · Zbl 1358.65043
[422] Ravichandran, C.; Baleanu, D., Existence results for fractional neutral functional integro-differential evolution equations with infinite delay in banach spaces, Adv Differ Equ, 2013, 1, 215 (2013) · Zbl 1379.34073
[423] Saeed, U.; Rehman, M. U., Hermite wavelet method for fractional delay differential equations, J Differ Equ, 1-8 (2014)
[424] Suganya, S.; Baleanu, D.; Kalamani, P.; Arjunan, M. M., On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses, Adv Differ Equ, 2015, 1, 372 (2015) · Zbl 1422.34224
[425] Suganya, S.; Kalamani, P.; Arjunan, M. M., Existence of a class of fractional neutral integro-differential systems with state-dependent delay in banach spaces, Comput Math Appl (2016) · Zbl 1348.34130
[426] Suganya, S.; Baleanu, D.; Selvarasu, S.; Arjunan, M. M., About the existence results of fractional neutral integro-differential inclusions with state-dependent delay in Frchet spaces, J Funct Spaces (2016)
[427] Yan, Z.; Jia, X., On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls, Stochastics, 88, 8, 1115-1146 (2016) · Zbl 1354.34133
[428] Yan, Z.; Lu, F., Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay, Appl Math Comput, 292, 425-447 (2017) · Zbl 1410.93026
[429] Yan, Z.; Lu, F., Complete controllability of fractional impulsive multivalued stochastic partial integro-differential equations with state-dependent delay, Int J Nonlinear Sci Numer Simul, 18, 3-4, 197-220 (2017) · Zbl 1401.93040
[430] Yan, Z.; Yan, X., The optimal behavior of solutions to fractional impulsive stochastic integro-differential equations and its control problems, J Fixed Point Theory Appl, 21, 1, 12 (2019) · Zbl 1409.34069
[431] Gomez-Aguilar, J. F.; Atangana, A., New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications, Eur Phys J Plus, 132, 1, 13 (2017)
[432] Atangana, A.; Gomez-Aguilar, J. F., A new derivative with normal distribution kernel: theory, methods and applications, Phys A Stat Mech Appl, 476, 1-14 (2017)
[433] Atangana, A.; Gomez-Aguilar, J. F., Fractional derivatives with no-index law property: application to chaos and statistics, Chaos Solit Fract, 114, 516-535 (2018)
[434] Yepez-Martinez, H.; Gomez-Aguilar, F.; Sosa, I. O.; Reyes, J. M.; Torres-Jimenez, J., The Feng’s first integral method applied to the nonlinear MKDV space-time fractional partial differential equation, Revista Mexicana de fsica, 62, 4, 310-316 (2016)
[435] Gomez-Aguilar, J. F.; Yepez-Martinez, H.; Torres-Jimenez, J.; Cordova-Fraga, T.; Escobar-Jimenez, R. F.; Olivares-Peregrino, V. H., Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Adv Differ Equ, 2017, 1, 68 (2017) · Zbl 1422.35165
[436] Gomez-Aguilar, J. F., Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Phys A Stat Mech Appl, 494, 52-75 (2018)
[437] Saad, K. M.; Khader, M. M.; Gomez-Aguilar, J. F.; Baleanu, D., Numerical solutions of the fractional fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos Interdisci J Nonlinear Sci, 29, 2, 023116 (2019) · Zbl 1409.35225
[438] Morales-Delgado, V. F.; Gomez-Aguilar, J. F.; Escobar-Jimenez, R. F.; Taneco-Hernandez, M. A., Fractional conformable derivatives of Liouville-Caputo type with low-fractionality, Phys A Stat Mech Appl, 503, 424-438 (2018)
[439] Atangana, A.; Gomez-Aguilar, J. F., Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solit Fract, 102, 285-294 (2017) · Zbl 1374.34296
[440] Yepez-Martinez, H.; Gomez-Aguilar, J. F., Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and mittag-leffler kernel, Math Model Natural Phenomena, 13, 1, 13 (2018) · Zbl 07007614
[441] Atangana, A.; Gomez-Aguilar, J. F., Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu, Numer Methods Part Differ Equ, 34, 5, 1502-1523 (2018) · Zbl 1417.65113
[442] Gomez-Aguilar, J. F.; Atangana, A., Fractional hunter-saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense, Eur Phys J Plus, 132, 2, 100 (2017)
[443] Morales-Delgado, V. F.; Gomez-Aguilar, J. F.; Yepez-Martinez, H.; Baleanu, D.; Escobar-Jimenez, R. F.; Olivares-Peregrino, V. H., Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv Differ Equ, 2016, 1, 164 (2016) · Zbl 1419.35220
[444] Gomez-Aguilar, J. F.; Yepez-Martinez, H.; Escobar-Jimenez, R. F.; Olivares-Peregrino, V. H.; Reyes, J. M.; Sosa, I. O., Series solution for the time-fractional coupled MKDV equation using the homotopy analysis method, Math Probl Eng, 2016, 7845874 (2016) · Zbl 1400.35066
[445] Zeid, S. S.; Kamyad, A. V.; Effati, S., Measurable functions approach for approximate solutions of Linear space-time-fractional diffusion problems, Iran J. Nume. Anal Optim., 8, 2, 1-24 (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.