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A new approach for solving a class of delay fractional partial differential equations. (English) Zbl 1407.65214
Summary: In this article, a new numerical approach has been proposed for solving a class of delay time-fractional partial differential equations. The approximate solutions of these equations are considered as linear combinations of Müntz-Legendre polynomials with unknown coefficients. Operational matrix of fractional differentiation is provided to accelerate computations of the proposed method. Using Padé approximation and two-sided Laplace transformations, the mentioned delay fractional partial differential equations will be transformed to a sequence of fractional partial differential equations without delay. The localization process is based on the space-time collocation in some appropriate points to reduce the fractional partial differential equations into the associated system of algebraic equations which can be solved by some robust iterative solvers. Some numerical examples are also given to confirm the accuracy of the presented numerical scheme. Our results approved decisive preference of the Müntz-Legendre polynomials with respect to the Legendre polynomials.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
44A10 Laplace transform
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
65D32 Numerical quadrature and cubature formulas
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77, (2000) · Zbl 0984.82032
[2] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode processes, J. Electroanal. Chem. Interfacial Electrochem., 33, 253-265, (1971)
[3] Benson, DA; Wheatcraft, SW; Meerschaert, MM, Application of a fractional advection-dispersion equation, Water Resour. Res., 36, 1403-1412, (2000)
[4] Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010) · Zbl 1210.26004
[5] Perestyuk, MO; Chernikova, OS, Some modern aspects of the theory of impulsive differential equations, Ukrain. Math. J., 60, 91, (2008) · Zbl 1164.34300
[6] Rezounenko, AV; Wu, JH, A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, J. Comput. Appl. Math., 190, 99-113, (2006) · Zbl 1082.92039
[7] Wang, H.; Hu, H., Remarks on the perturbation methods in solving the second-order delay differential equations, Nonlinear Dyn., 33, 379-398, (2003) · Zbl 1049.70013
[8] Khasawneh, FA; Barton, DAW; Mann, BP, Periodic solutions of nonlinear delay differential equations using spectral element method, Nonlinear Dyn., 67, 641-658, (2012) · Zbl 1246.65102
[9] Aziz, I.; Amin, R., Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet, Appl. Math. Model., 40, 10286-10299, (2016)
[10] Ghasemi, M.; Fardi, M.; Khoshsiar Ghaziani, R., Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Appl. Math. Comput., 268, 815-831, (2016)
[11] Morgadoa, ML; Fordb, NJ; Limac, PM, Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math., 252, 159-168, (2013) · Zbl 1291.65214
[12] Tumwiine, J.; Luckhaus, S.; Mugisha, JYT; Luboobi, LS, An age-structured mathematical model for the within host dynamics of malaria and the immune system, J. Math. Model. Algorithms, 7, 79-97, (2008) · Zbl 1132.92016
[13] Alvarez-Vázquez, Lino J.; Fernández, FJ; Muũoz-Sola, Rafael, Analysis of a multistate control problem related to food technology, J. Differ. Equ., 245, 130-153, (2008) · Zbl 1147.49003
[14] Cheng, Z.; Lin, YZ, The exact solution of a class of delay parabolic partial differential equation, J. Nat. Sci. Heilongjiang Univ., 25, 155-162, (2008)
[15] Jackiewicz, Z.; Zubik-Kowal, B., Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations, Appl. Numer. Math., 56, 433-443, (2006) · Zbl 1093.65096
[16] Ouyang, Z., Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61, 860-870, (2011) · Zbl 1217.35206
[17] Rihan, FA, Computational methods for delay parabolic and time-fractional partial differential equations, Numer. Methods Partial Differ. Equ., 26, 1556-1571, (2009) · Zbl 1204.65114
[18] Wu, J., A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214, 31-40, (2009) · Zbl 1169.65127
[19] Momani, S.; Odibat, Z., Comparison between the homotopy perturbation method and variational iteration method for a linear partial differential equations, Comput. Math. Appl., 54, 910-919, (2007) · Zbl 1141.65398
[20] Borwein, P.; Erdélyi, T.; Zhang, J., Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Am. Math. Soc., 2, 523-542, (1994) · Zbl 0799.41015
[21] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B. V, Amsterdam (2006)
[22] El-Kady, M., Legendre approximations for solving optimal control problems governed by ordinary differential equations, Int. J. Control Sci. Eng., 4, 54-59, (2012)
[23] Esmaeili, Sh; Shamsi, M.; Luchko, Y., Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials, Comput. Math. Appl., 62, 918-929, (2011) · Zbl 1228.65132
[24] Ejlali, N.; Hosseini, SM, A pseudospectral method for fractional optimal control problems, J. Optim. Theory Appl., 174, 83-107, (2017) · Zbl 1377.49019
[25] Maleki, M.; Hashim, I.; Abbasbandy, S.; Alsaedi, A., Direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method, J. Comput. Appl. Math., 283, 41-57, (2015) · Zbl 1311.65087
[26] Turut, V.; Güzel, N., On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations, Eur. J. Pure Appl. Math., 6, 147-171, (2013) · Zbl 1413.65401
[27] Turut, V.; Güzel, N., Multivariate Padé approximation for solving nonlinear partial differential equations of fractional order, Abstr. Appl. Anal., 2013, article id 746401, (2013) · Zbl 1275.65088
[28] Cuyt, A., How well can the concept of Padé approximant be generalized to the multivariate case?, J. Comput. Appl. Math., 105, 25-50, (1999) · Zbl 0945.41012
[29] Baker, G.A., Graves-Morris, P.R.: Padé Approximants, vol. 59. Cambridge University Press, Cambridge (1996)
[30] Matsuzuka, I.; Nagasawa, K.; Kitahama, A., A proposal for two-sided Laplace transforms and its application to electronic circuits, Appl. Math. Comput., 100, 1-11, (1999) · Zbl 0929.44001
[31] Pol, V.B., Bremmer, H.: Operational Calculus Based on the Two-sided Laplace Integral. Cambridge University Press, London (1955) · Zbl 0066.09101
[32] Fox, W.P.: Mathematical Modeling with Maple. Brooks Cole, Boston (2011)
[33] Sun, Zh; Zhang, Z., A linearized compact difference scheme for a class of nonlinear delay partial differential equations, Appl. Math. Model., 37, 742-752, (2013) · Zbl 1352.65270
[34] Lee, AY, Hereditary optimal control problems: numerical method based upon a padé approximation, J. Optim. Theory Appl., 56, 157-166, (1988) · Zbl 0617.49013
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