Exponentially fitted methods for solving time fractional nonlinear reaction-diffusion equation. (English) Zbl 1429.65206
Summary: In this article, we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction-diffusion equation with fixed order and distributed order derivatives. These techniques depend on a parameter, which will be used to annihilate the error and increase the order of accuracy. The proposed methods are proved to be unconditionally stable and convergent by Fourier analysis. Also, the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 35K57 | Reaction-diffusion equations |
Keywords:
exponential fitting; parameter selection; distributed order time fractional; time fractional reaction-diffusion equations; stability analysis; convergence analysisReferences:
| [1] | Oldham, K.; Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order, 111 (1974), Elsevier · Zbl 0292.26011 |
| [2] | Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley-Interscience · Zbl 0789.26002 |
| [3] | 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, 297-316 (2013) · Zbl 1312.35179 |
| [4] | Mainardi, F.; Mura, A.; Pagnini, G.; Gorenflo, R., Time-fractional diffusion of distributed order, J. Vib. Control., 14, 1267-1290 (2008) · Zbl 1229.35118 |
| [5] | Diethelm, K., The Analysis of Fractional Differential equations: An application-Oriented Exposition Using Differential Operators of Caputo type (2010), Springer · Zbl 1215.34001 |
| [6] | Angulo, J. M.; Ruiz-Medina, M. D.; Anh, V. V.; Grecksch, W., Fractional diffusion and fractional heat equation, Adv. Appl. Probab., 32, 1077-1099 (2000) · Zbl 0986.60077 |
| [7] | Mainardi, F., The time fractional diffusion-wave equation, Radiophys. Quantum Electron., 38, 13-24 (1995) |
| [8] | Nigmatullin, R. R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solidi., 133, 425-430 (1986) |
| [9] | Richardson, L. F., Atmospheric diffusion shown on a distance-neighbour graph, Proc. R. Soc. Lond. A, 110, 709-737 (1926) |
| [10] | Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004 |
| [11] | Wyss, W., The fractional diffusion equation, J. Math. Phys., 27, 2782-2785 (1986) · Zbl 0632.35031 |
| [12] | I.M. Sokolov, Chechkin, J. Klafter, Distributed-order fractional kinetics, ArXiv Prepr. Cond-Mat/0401146. (2004) 1-18.; I.M. Sokolov, Chechkin, J. Klafter, Distributed-order fractional kinetics, ArXiv Prepr. Cond-Mat/0401146. (2004) 1-18. |
| [13] | Yuste, S. B.; Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42, 1862-1874 (2005) · Zbl 1119.65379 |
| [14] | Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear Dyn., 29, 129-143 (2002) · Zbl 1009.82016 |
| [15] | Murillo, J. Q.; Yuste, S. B., On three explicit difference schemes for fractional diffusion and diffusion-wave equations, Phys. Scr., 2009, 6 (2009) |
| [16] | Liao, H.; Zhang, Y.; Zhao, Y.; Shi, H., Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractional subdiffusion equations, J. Sci. Comput., 61, 629-648 (2014) · Zbl 1339.65150 |
| [17] | 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, 31 (2015) · Zbl 1359.65144 |
| [18] | 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 |
| [19] | 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. with Appl., 73, 1262-1269 (2017) · Zbl 1412.65084 |
| [20] | Zahra, W. K.; Nasr, M. A., Discrete spline solution of time variable order fractional nonlinear parabolic equations, Appl. Math. J., 1-22 (2017), submitted for publication |
| [21] | Ambrosio, R. D.; Paternoster, B., Numerical solution of a diffusion problem by exponentially fitted finite difference methods, SpringerPlus, 3, 425, 7 (2014) |
| [22] | Erdogan, U.; Akarbulut, K.; Tan, N. O., Exponentially fitted finite difference schemes for reaction – diffusion equations, Math. Comput. Appl., 21, 10 (2016) |
| [23] | Owolabi, K. M., Mathematical analysis and numerical simulation of patterns in fractional and classical reaction – diffusion systems, Chaos Solitons Fractals, 93, 89-98 (2016) · Zbl 1372.92091 |
| [24] | Pindza, E.; Owolabi, K. M., Fourier spectral method for higher order space fractional reaction – diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 40, 112-128 (2016) · Zbl 1524.65676 |
| [25] | Owolabi, K. M., Efficient numerical simulation of non-integer-order space-fractional reaction – diffusion equation via the Riemann-Liouville operator, Eur. Phys. J. Plus., 133, 98 (2018) |
| [28] | Owolabi, K. M.; Atangana, A., Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative, Eur. Phys. J. Plus., 131, 335 (2016) |
| [29] | Owolabi, K. M., Computational study of noninteger order system of predation, Chaos Interdiscip. J. Nonlinear Sci., 29, 13120 (2019) · Zbl 1406.34014 |
| [30] | Owolabi, K. M., Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus., 133, 15 (2018) |
| [31] | Owolabi, K. M.; Atangana, A., Chaotic behaviour in system of noninteger-order ordinary differential equations, Chaos Solitons Fractals, 115, 362-370 (2018) · Zbl 1416.65180 |
| [32] | Owolabi, K. M., Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44, 304-317 (2017) · Zbl 1465.65108 |
| [33] | Owolabi, K. M.; Hammouch, Z., Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos Interdiscip. J. Nonlinear Sci., 29, 13145 (2019) · Zbl 1406.34091 |
| [34] | Chen, C.; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227, 886-897 (2007) · Zbl 1165.65053 |
| [35] | Chen, C.; Liu, F.; Burrage, K., Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput., 198, 754-769 (2008) · Zbl 1144.65057 |
| [36] | Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228, 7792-7804 (2009) · Zbl 1179.65107 |
| [37] | Gao, G.; Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230, 586-595 (2011) · Zbl 1211.65112 |
| [38] | 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 |
| [39] | Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121 |
| [40] | Liu, F.; Yang, C.; Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231, 160-176 (2009) · Zbl 1170.65107 |
| [41] | Liu, F.; Meerschaert, M.; McGough, R.; Zhuang, P.; Liu, Q., Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal., 16, 9-25 (2013) · Zbl 1312.65138 |
| [42] | Sun, H.; Chen, W.; Sze, K. Y., A semi-discrete finite element method for a class of time-fractional diffusion equations, Phil. Trans. R. Soc. A., 371, 15 (2013) · Zbl 1339.65156 |
| [43] | Jiang, Y.; Ma, J., Moving finite element methods for time fractional partial differential equations, Sci. China Math., 56, 1287-1300 (2013) · Zbl 1290.65091 |
| [44] | Gu, Y.; Zhuang, P., Anomalous sub-diffusion equations by the meshless collocation method, Aust. J. Mech. Eng., 10, 1-8 (2012) |
| [45] | Huang, F., A time-space collocation spectral approximation for a class of time fractional differential equations, Int. J. Differ. Equ., 2012, 19 (2012) · Zbl 1267.35243 |
| [46] | Ferrás, L. L.; Ford, N. J.; Morgado, M. L.; Rebelo, M., A numerical method for the solution of the time-fractional diffusion equation, (Proceedings of the International Conference on Computational Science and Its Application (2014)), 117-131 |
| [47] | Sousa, E., How to approximate the fractional derivative of order 1 < α ≤ 2, Int. J. Bifurc. Chaos., 22, 13 (2012) · Zbl 1258.26006 |
| [48] | Zahra, W. K.; Van Daele, M., Discrete spline methods for solving two point fractional Bagley-Torvik equation, Appl. Math. Comput., 296, 42-56 (2017) · Zbl 1411.65098 |
| [49] | Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77 (2004) · Zbl 1126.76346 |
| [50] | Gao, G. H.; Sun, H. W.; Sun, Z. Z., Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys., 298, 337-359 (2015) · Zbl 1349.65296 |
| [51] | Tian, W.; Zhou, H.; Deng, W., A class of second order differenceapproximations for solving space fractional diffusion equations, Math Comput, 84, 294, 1703-1727 (2015) · Zbl 1318.65058 |
| [52] | Morgado, M. L.; Rebelo, M., Numerical approximation of distributed order reaction – diffusion equations, J. Comput. Appl. Math., 275, 216-227 (2015) · Zbl 1298.35242 |
| [53] | 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 |
| [54] | Ixaru, L. G.; Vanden Berghe, G., Exponential Fitting (2004), Springer Science+Business Media, B.V · Zbl 1060.65622 |
| [55] | Vanden Berghe, G.; Van Daele, M., Exponentially-fitted numerov methods, J. Comput. Appl. Math., 200, 140-153 (2007) · Zbl 1110.65071 |
| [56] | Coleman, J. P.; Ixaru, L. G., Truncation errors in exponential fitting for oscillatory problems, SIAM J. Numer. Anal., 44, 1441-1465 (2006) · Zbl 1123.65016 |
| [57] | Ixaru, L., Approximation formulae generated by exponential fitting, Ann. Acad. Rom. Sci. Ser. Math. Appl., 3, 164-187 (2011) · Zbl 1284.65042 |
| [58] | Hollevoet, D.; Van Daele, M.; Vanden Berghe, G., Exponentially fitted methods applied to fourth-order boundary value problems, J. Comput. Appl. Math., 235, 5380-5393 (2011) · Zbl 1225.65075 |
| [59] | Hollevoet, D.; Van Daele, M.; Vanden Berghe, G., The optimal exponentially-fitted Numerov method for solving two-point boundary value problems, J. Comput. Appl. Math., 230, 260-269 (2009) · Zbl 1167.65406 |
| [60] | Zahra, W. K.; Nasr, M. A., Exponentially fitted methods for solving two-dimensional time fractional damped Klein-Gordon equation with nonlinear source term, Commun. Nonlinear Sci. Numer. Simul., 73, 177-194 (2019) · Zbl 1464.65095 |
| [61] | Simos, T. E., Explicit exponentially fitted methods for the numerical solution of the Schrödinger equation, Appl. Math. Comput., 98, 185-198 (1999) · Zbl 0934.65085 |
| [62] | Simos, T. E.; Vigo-Aguiar, J., A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems, Comput. Phys. Commun., 152, 274-294 (2003) · Zbl 1196.65123 |
| [63] | Anastassi, Z. A.; Simos, T. E., A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation, J. Math. Chem., 41, 79-100 (2007) · Zbl 1125.81017 |
| [64] | Berg, D. B.; Simos, T. E., An efficient six-step method for the solution of the Schrödinger equation, J. Math. Chem., 55, 1521-1547 (2017) · Zbl 1422.65112 |
| [65] | Monovasilis, T.; Kalogiratou, Z.; Simos, T. E., Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation, J. Math. Chem., 37, 263-270 (2005) · Zbl 1073.65071 |
| [66] | Monovasilis, T.; Kalogiratou, Z.; Simos, T. E., Exponentially fitted symplectic Runge-Kutta-Nyström methods derived by partitioned Runge-Kutta methods, AIP Conf. Proc., 1558, 1181-1185 (2013) |
| [67] | Konguetsof, A.; Simos, T. E., An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems, Comput. Math. with Appl., 45, 547-554 (2003) · Zbl 1035.65071 |
| [68] | Akram, G.; Tariq, H., An exponential spline technique for solving fractional boundary value problem, Calcolo, 53, 545-558 (2016) · Zbl 1377.65093 |
| [69] | Ixaru, L. G.; Vanden Berghe, G., Exponential Fitting. Exponential Fitting, Series on mathematics and its applications, 568 (2004), Kluwer Academic Publishers · Zbl 1105.65082 |
| [70] | Li, X.; Wong, P. J.Y., A higher order non-polynomial spline method for fractional sub-diffusion problems, J. Comput. Phys., 328, 46-65 (2017) · Zbl 1406.65097 |
| [71] | Hosseini, S. M.; Ghaffari, R., Polynomial and nonpolynomial spline methods for fractional sub-diffusion equations, Appl. Math. Model., 38, 3554-3566 (2014) · Zbl 1427.65168 |
| [72] | Lin, B., Non-polynomial splines method for numerical solutions of the regularized long wave equation, Int. J. Comput. Math., 92, 1591-1607 (2015) · Zbl 1317.65054 |
| [73] | Zahra, W. K., A smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems, Appl. Math. Comput., 217, 8447-8457 (2011) · Zbl 1218.65075 |
| [74] | Chen, C.; Liu, F.; Anh, V.; Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32, 1740-1760 (2010) · Zbl 1217.26011 |
| [75] | Zahra, W. K.; Ouf, W. A.; El-Azab, M. S., Cubic B - spline collocation algorithm for the numerical solution of Newell Whitehead Segel type equations, Electron. J. Math. Anal. Appl., 2, 81-100 (2014) · Zbl 1463.65330 |
| [76] | Ucar, Y.; Yagmurlu, N. M.; Tasbozan, O.; Esen, A., Numerical solution of some fractional partial differential equations using collocation finite element method, Prog. Fract. Differ. Appl., 1, 157-164 (2015) · Zbl 1342.80013 |
| [77] | Zahra, W. K., Trigonometric B-spline collocation method for solving PHI-Four and Allen-Cahn equations, Mediterr. J. Math., 14, 19 (2017) · Zbl 1377.65138 |
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