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Numerical approach for a class of distributed order time fractional partial differential equations. (English) Zbl 1407.65122

Summary: The numerical solution of distributed order time fractional partial differential equations based on the midpoint quadrature rule and linear B-spline interpolation is studied. The proposed discretization algorithm follows the Du Fort-Frankel method. The unconditional stability and the convergence of the scheme are analyzed. Simulations illustrate the application of the new algorithm.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65D07 Numerical computation using splines
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35R09 Integro-partial differential equations
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[1] Al-Shibani, F.; Ismail, A., Compact Crank-Nicolson and Du Fort-Frankel method for the solution of the time fractional diffusion equation, Int. J. Comput. Methods, 12, 06, (2015) · Zbl 1359.65144
[2] Atanackovic, T. M.; Pilipovic, S.; Zorica, D., Existence and calculation of the solution to the time distributed order diffusion equation, Phys. Scr. T, 136, (2009)
[3] Biswas, K.; Bohannan, G.; Caponetto, R.; Lopes, A. M.; Machado, J. A.T., Fractional-order models of vegetable tissues, (Fractional-Order Devices, (2017), Springer International Publishing), 73-92
[4] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M., Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66, 4, (2002)
[5] Chen, H.; Lü, S.; Chen, W., Finite difference/spectral approximations for the distributed order time fractional reaction-diffusion equation on an unbounded domain, J. Comput. Phys., 315, 84-97, (2016) · Zbl 1349.65507
[6] Dabiri, A.; Butcher, E. A., Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods, Appl. Math. Model., 56, 424-448, (2018)
[7] Dabiri, A.; Butcher, E. A.; Nazari, M., Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation, J. Sound Vib., 388, 230-244, (2017)
[8] Dabiri, A.; Moghaddam, B. P.; Machado, J. T., Optimal variable-order fractional PID controllers for dynamical systems, J. Comput. Appl. Math., 339, 40-48, (2018) · Zbl 1392.49033
[9] Ford, N. J.; Morgado, M. L.; Rebelo, M., An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44, 289-305, (2015) · Zbl 1330.65130
[10] Gao, G. H.; Sun, Z. Z., Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations, J. Sci. Comput., 66, 3, 1281-1312, (2015) · Zbl 1373.65055
[11] Gao, G. H.; Sun, Z. Z., Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations, Numer. Methods Partial Differ. Equ., 32, 2, 591-615, (2015) · Zbl 1339.65115
[12] Gorenflo, R.; Luchko, Y.; Stojanović, M., Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density, Fract. Calc. Appl. Anal., 16, 2, 297-316, (2013) · Zbl 1312.35179
[13] 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. Algorithms, 72, 2, 393-407, (2015) · Zbl 1343.65110
[14] Hu, Y.; Li, C.; Li, H., The finite difference method for Caputo-type parabolic equation with fractional Laplacian: one-dimension case, Chaos Solitons Fractals, 102, 319-326, (2017) · Zbl 1422.65157
[15] Jiao, Z.; Chen, Y.; Podlubny, I., Distributed-order linear time-invariant system (DOLTIS) and its stability analysis, (Distributed-Order Dynamic Systems, (2012), Springer: Springer London), 11-28
[16] Jin, B.; Lazarov, R.; Sheen, D.; Zhou, Z., Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal., 19, 1, (2016) · Zbl 1333.65111
[17] Khaliq, A. Q.M.; Twizell, E. H., The extrapolation of stable finite difference schemes for first order hyperbolic equations, Int. J. Comput. Math., 11, 2, 155-167, (1982) · Zbl 0485.65065
[18] Kleefeld, B.; Khaliq, A. Q.M.; Wade, B., An ETD Crank-Nicolson method for reaction-diffusion systems, Numer. Methods Partial Differ. Equ., 28, 4, 1309-1335, (2011) · Zbl 1253.65128
[19] Li, C.; Zeng, F., Numerical Methods for Fractional Calculus, (2015), Chapman and Hall/CRC · Zbl 1326.65033
[20] Li, C.; Chen, A.; Ye, J., Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys., 230, 9, 3352-3368, (2011) · Zbl 1218.65070
[21] 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
[22] Liao, H. L.; Zhang, Y. N.; Zhao, Y.; Shi, H. S., Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractional subdiffusion equations, J. Sci. Comput., 61, 3, 629-648, (2014) · Zbl 1339.65150
[23] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dyn., 29, 1-4, 57-98, (2002) · Zbl 1018.93007
[24] Luchko, Y., Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12, 4, 409-422, (2009) · Zbl 1198.26012
[25] Luo, W.-H.; Li, C.; Huang, T.-Z.; Gu, X.-M.; Wu, G.-C., A high-order accurate numerical scheme for the Caputo derivative with applications to fractional diffusion problems, Numer. Funct. Anal. Optim., 39, 5, 600-622, (2017)
[26] 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)
[27] Mainardi, F.; Pagnini, G., The role of the Fox-Wright functions in fractional sub-diffusion of distributed order, J. Comput. Appl. Math., 207, 2, 245-257, (2007) · Zbl 1120.35002
[28] Meerschaert, M. M.; Nane, E.; Vellaisamy, P., Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379, 1, 216-228, (2011) · Zbl 1222.35204
[29] 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. Calc. Appl. Anal., 20, 4, (2017)
[30] Moghaddam, B. P.; Machado, J. A.T., SM-algorithms for approximating the variable-order fractional derivative of high order, Fundam. Inform., 151, 1-4, 293-311, (2017) · Zbl 1377.65031
[31] 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)
[32] Moghaddam, B. P.; Machado, J.; Behforooz, H., An integro quadratic spline approach for a class of variable-order fractional initial value problems, Chaos Solitons Fractals, 102, 354-360, (2017) · Zbl 1422.65131
[33] Morgado, M. L.; Rebelo, M., Numerical approximation of distributed order reaction-diffusion equations, J. Comput. Appl. Math., 275, 216-227, (2015) · Zbl 1298.35242
[34] Morgado, M. L.; Rebelo, M.; Ferrás, L. L.; Ford, N. J., Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method, Appl. Numer. Math., 114, 108-123, (2017) · Zbl 1357.65198
[35] Sokolov, I. M.; Chechkin, A. V.; Klafter, J., Distributed-order fractional kinetics, Acta Phys. Pol. B, 35, 1323-1341, (2004)
[36] Wang, X.; Liu, F.; Chen, X., Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv. Math. Phys., 2015, 1-14, (2015) · Zbl 1380.65188
[37] Ye, H.; Liu, F.; Anh, V.; Turner, I., Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math., 80, 3, 825-838, (2014) · Zbl 1337.65120
[38] Ye, H.; Liu, F.; Anh, V., Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, J. Comput. Phys., 298, 652-660, (2015) · Zbl 1349.65353
[39] Zaky, M. A., A Legendre collocation method for distributed-order fractional optimal control problems, Nonlinear Dyn., 91, 4, 2667-2681, (2018) · Zbl 1392.35331
[40] Zaky, M.; Machado, J. T., On the formulation and numerical simulation of distributed-order fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 52, 177-189, (2017)
[41] Zeng, F.; Li, C., A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation, Appl. Numer. Math., 121, 82-95, (2017) · Zbl 1372.65276
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