Numerical solution of the nonlinear conformable space-time fractional partial differential equations. (English) Zbl 07423837

Summary: In this paper, a numerical approach for solving the nonlinear space-time fractional partial differential equations with variable coefficients is proposed. The fractional derivatives are described in the conformable sense. The numerical approach is based on shifted Chebyshev polynomials of the second kind and finite difference method. The proposed scheme reduces the main problem to a system of nonlinear algebraic equations. The validity and the applicability of the proposed technique are shown by numerical examples.


65-XX Numerical analysis
35G31 Initial-boundary value problems for nonlinear higher-order PDEs
35R11 Fractional partial differential equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Bagley, RL; Torvik, PJ, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51, 294-298 (1984) · Zbl 1203.74022
[2] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus. App. Anal., 5, 367-386 (2002) · Zbl 1042.26003
[3] Podlubny, I., Fractional differential equations (1999), New York: Academic press, New York · Zbl 0924.34008
[4] Mainardi, F.; Carpinteri, A.; Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (1997), New York: Springer-Verlag, New York · Zbl 0917.73004
[5] Thomas, MD; Bamforth, PB, Modelling chloride diffusion in concrete: Effect of fly ash and slag, Cem. Concr. Res., 29, 4, 487-495 (1999)
[6] Khitab, A.; Lorente, S.; Ollivier, JP, Predictive model for chloride penetration through concrete, Mag. Concr. Res., 57, 9, 511-520 (2005)
[7] Firoozjaee, MA; Yousefi, SA, A numerical approach for fractional partial differential equations by using Ritz approximation, Appl. Math. Comput., 338, 711-721 (2018) · Zbl 1427.65245
[8] Dehestani, H.; Ordokhani, Y.; Razzaghi, M., Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations, Appl. Math. Comput., 336, 433-453 (2018) · Zbl 1427.35314
[9] 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, 85-95 (2008) · Zbl 1148.65099
[10] Saeed, U.; Rehman, M., Haar wavelet Picard method for fractional nonlinear partial differential equations, Appl. Math. Comput., 264, 310-322 (2015) · Zbl 1410.65401
[11] Biala, TA; Khaliq, AQM, Parallel algorithms for nonlinear time-space fractional parabolic PDEs, J. Comput. Phys., 375, 135-154 (2018) · Zbl 1416.65303
[12] Khader, MM; Saad, KM, A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method, Chaos Soliton Fract., 110, 169-177 (2018) · Zbl 1448.65185
[13] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Appl. Math. Model., 32, 28-39 (2008) · Zbl 1133.65116
[14] Mohamed, Z.; Elzaki, TM, Applications of new integral transform for linear and nonlinear fractional partial differential equations (2018), Sci: J. King Saud Univ, Sci
[15] Zhang, S.; Hong, S., Variable separation method for a nonlinear time fractional partial differential equation with forcing term, J. Comput. Appl. Math., 339, 297-305 (2018) · Zbl 06867160
[16] Demir, A.; Bayrak, MA; Ozbilge, E., An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method, Math. Probl. Eng., 2018, 1-8 (2018) · Zbl 1427.35315
[17] Nagy, AM, Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collocation method, Appl. Math. Comput., 310, 139-148 (2017) · Zbl 1427.65294
[18] Yusuf, A.; Inc, M.; Aliyu, AI; Baleanu, D., Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations, Chaos Soliton Fract., 116, 220-226 (2018) · Zbl 1442.35533
[19] Prakash, P.; Harikrishnan, S.; Benchohra, M., Oscillation of certain nonlinear fractional partial differential equation with damping term, Appl. Math. Lett., 43, 72-79 (2015) · Zbl 1406.35475
[20] Khalil, R.; Horani, MA; Yousef, A.; Sababheh, M., A new defnition of fractional derivative, J. Comput. Appl. Math., 264, 65-70 (2014) · Zbl 1297.26013
[21] Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279, 57-66 (2015) · Zbl 1304.26004
[22] Feng, Q., A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation, Chinese J. Phys., 56, 2817-2828 (2018)
[23] Chen, C.; Jiang, YL, Simplest equation method for some time-fractional partial differential equations with conformable derivative, Comput. Math. Appl., 75, 2978-2988 (2018) · Zbl 1415.35275
[24] Hosseini, K.; Ansari, R., New exact solutions of nonlinear conformable time- fractional Boussinesq equations using the modified Kudryashov method, Wave Random Complex., 27, 628-636 (2017)
[25] Abdelsalam, UM, Exact travelling solutions of two coupled (2 + 1)-Dimensional Equations, J. Egyptian Math. Soc., 25, 125-128 (2017) · Zbl 1372.35065
[26] Thabet, H.; Kendre, S., Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform, Chaos Soliton Fract., 109, 238-245 (2018) · Zbl 1390.35406
[27] Yang, Y., Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method, AM, 4, 113-118 (2013)
[28] Mohammadi, F.; Cattani, C., A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations, J. Comput. Appl. Math., 339, 306-316 (2018) · Zbl 1464.65079
[29] Doha, EH; Bhrawy, AH; Ezz-Eldien, SS, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35, 5662-5672 (2011) · Zbl 1228.65126
[30] Sweilam, NH; Nagy, AM; El-Sayed, AA, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos Soliton Fract., 73, 141-147 (2015) · Zbl 1352.65401
[31] Sweilam, NH; Nagy, AM; El-Sayed, AA, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind, J. King Saud Univ. Sci., 28, 41-47 (2016)
[32] Sweilam, NH; Nagy, AM; El-Sayed, AA, Numerical approach for solving space fractional order diffusion equations using shifted Chebyshev polynomials of the fourth kind, Turk J. Math., 40, 1283-1297 (2016) · Zbl 1438.35442
[33] Khader, MM, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16, 2535-42 (2011) · Zbl 1221.65263
[34] H. Azizi, G.B. Loghmani. Numerical approximation for space fractional diffusion equations via Chebyshev finite difference method. J. Fract. Ca.l Appl. 4 (2013) 303-311.
[35] Azizi, H.; Loghmani, GB, A numerical method for space fractional diffusion equations using a semi-disrete scheme and Chebyshev collocation method, J. Math. Comput. Sci., 8, 226-235 (2014)
[36] Saadatmandia, A.; Dehghan, M., A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl., 62, 1135-1142 (2011) · Zbl 1228.65203
[37] Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 247, 2108-2131 (2009) · Zbl 1193.35243
[38] Piret, C.; Hanert, E., A radial basis functions method for fractional diffusion equations, J. Comput. Phys., 14, 71-81 (2012) · Zbl 1286.65135
[39] Khader, MM, An efficient approximate method for solving linear fractional Klein-Gordon equation based on the generalized Laguerre polynomials, Int. J. Comput. Math., 90, 1853-1864 (2013) · Zbl 1291.65185
[40] Yaslan, HC, Numerical solution of the conformable space-time fractional wave equation, Chinese J. Phys., 56, 2916-2925 (2018)
[41] Mason, JC; Handscomb, DC, Chebyshev Polynomials (2003), CRC, New York, NY, Boca Raton: Chapman and Hall, CRC, New York, NY, Boca Raton · Zbl 1015.33001
[42] Wazwaz, AM; Gorguis, A., An analytic study of Fishers equation by using Adomian decomposition method, Appl. Math. Comput., 154, 609-20 (2004) · Zbl 1054.65107
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.