Haar wavelet collocation method for variable order fractional integro-differential equations with stability analysis. (English) Zbl 1524.65971
Summary: This paper is focused on a numerical method based on the Haar wavelet collocation method for finding solutions of the variable-order Caputo-Prabhakar fractional integro-differential equations. The method converts the equation under consideration in to a system of linear algebraic equations which is solved by the Gauss elimination technique. Also, we study the existence, uniqueness, convergence and stability analysis of the proposed method. Using the mean square root and maximum absolute errors, the provided numerical experiments confirm that our proposed method is very effective for solving these equations in compared with some other numerical methods.
MSC:
| 65R20 | Numerical methods for integral equations |
| 34K37 | Functional-differential equations with fractional derivatives |
| 45J05 | Integro-ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65T60 | Numerical methods for wavelets |
Keywords:
fractional integro-differential equation; Caputo-Prabhakar fractional derivative; Prabhakar fractional integral; Haar wavelet collocation method; stability analysisSoftware:
MLReferences:
| [1] | Agarwal, P.; El-Sayed, AA; Tariboon, J., Vieta-Fibonacci operational matrices for spectral solutions of variable-order fractional integro-differential equations, J Comput Appl Math (2021) · Zbl 1471.65066 · doi:10.1016/j.cam.2020.113063 |
| [2] | Amin, R.; Shah, K.; Asif, M.; Khan, I.; Ullah, F., An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J Comput Appl Math, 381, 113028 (2021) · Zbl 1451.65230 |
| [3] | Aminikhah, H.; Refahi Sheikhani, AH; Rezazadeh, H., Exact solutions of some nonlinear systems of partial differential equations by using the functional variable method, Mathematica, 56, 79, 103-116 (2014) · Zbl 1389.83002 |
| [4] | Aminikhah, H.; Refahi Sheikhani, AH; Houlari, T.; Rezazadeh, H., Numerical solution of the distributed-order fractional Bagley-Torvik equation, IEEE/CAA J Autom Sin, 6, 3, 760-765 (2019) |
| [5] | Ansari, A.; Sheikhani, AR; Najafi, HS, Solution to system of partial fractional differential equations using the fractional exponential operators, Math Methods Appl Sci, 35, 119-123 (2012) · Zbl 1234.26013 |
| [6] | Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, J Comput Phys, 293, 104-114 (2014) · Zbl 1349.65263 · doi:10.1016/j.jcp.2014.12.043 |
| [7] | Babaei, A.; Jafari, H.; Banihashemi, S., Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, J Comput Appl Math (2020) · Zbl 1451.65231 · doi:10.1016/j.cam.2020.112908 |
| [8] | Babaei, A.; Moghaddam, BP; Banihashemi, S.; Machado, JAT, Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations, Commun Nonlinear Sci Numer Simul (2020) · Zbl 1452.65268 · doi:10.1016/j.cnsns.2019.104985 |
| [9] | Bagherzadeh Tavasania, B.; Refahi Sheikhania, AH; Aminikhah, H., A numerical scheme for solving variable order Caputo-Prabhakar fractional integro-differential equation, Int J Nonlinear Anal Appl, 13, 1, 467-484 (2022) |
| [10] | Bhrawy, AH; Zaky, MA, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn, 80, 101-116 (2015) · Zbl 1345.65060 |
| [11] | Bhrawy, AH; Alofi, AS; Ezz-Eldien, SS, A quadrature tau method for variable coefficients fractional differential equations, Appl Math Lett, 24, 2146-2152 (2011) · Zbl 1269.65068 |
| [12] | Bhrawy, AH; Doha, EH; Baleanu, D.; Ezz-Eldien, SS, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J Comput Phys, 293, 142-156 (2015) · Zbl 1349.65504 |
| [13] | Chen, Y.; Liu, L.; Li, B.; Sun, Y., Numerical solution for the variable order linear cable equation with Bernstein polynomials, Appl Math Comput, 238, 329-341 (2014) · Zbl 1334.65167 |
| [14] | Chen, YM; Wei, YQ; Liu, DY; Yu, H., Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets, Appl Math Lett, 46, 83-88 (2015) · Zbl 1329.65172 |
| [15] | de Oliveira, EC; Mainardi, F.; Vaz, J., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur Phys J Spec Top, 193, 161-171 (2011) |
| [16] | Dehestani, H.; Ordokhani, Y.; Razzaghi, M., Application of fractional Gegenbauer functions in variable-order fractional delay-type equations with non-singular kernel derivatives, Chaos Solitons Fractals (2020) · Zbl 1524.65645 · doi:10.1016/j.chaos.2020.110111 |
| [17] | Dehghan, M.; Safarpoor, M.; Abbaszadeh, M., Two high-order numerical algorithms for solving the multi- term time fractional diffusion-wave equations, J Comput Appl Math, 290, 174-195 (2015) · Zbl 1321.65129 |
| [18] | Derakhshan, MH; Aminataei, A., A new approach for solving multi-variable orders differential equations with Prabhakar function, J Math Model, 8, 139-155 (2020) · Zbl 1488.65200 |
| [19] | Doha, EH; Abdelkawy, MA; Amin, AZ; Lopes, AM, On spectral methods for solving variable-order fractional integro-differential equations, Comput Appl Math, 37, 3937-3950 (2018) · Zbl 1404.65192 |
| [20] | D’Ovidio, M.; Polito, F., Fractional diffusion-telegraph equations and their associated stochastic solutions, Theory Probab Appl, 62, 552-574 (2018) · Zbl 1405.60086 |
| [21] | Eshaghi, S.; Ansari, A., Lyapunov inequality for fractional differential equations with Prabhakar derivative, Math Inequal Appl, 19, 1, 349-358 (2016) · Zbl 1337.34011 |
| [22] | Eshaghi, S.; Ansari, A., Finite fractional Sturm-Liouville transforms for generalized fractional derivatives, Iran J Sci Technol, 41, 4, 931-937 (2017) · Zbl 1391.34010 |
| [23] | Eshaghi, S.; Khoshsiar Ghaziani, R.; Ansari, A., Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay, Math Methods Appl Sci, 42, 7, 2302-2323 (2019) · Zbl 1418.34009 |
| [24] | Farid, G.; Pecaric, J.; Tomovski, Z., Opial-type inequalities for fractional integral operator involving Mittag-Leffler function, Fract Differ Calc, 5, 93-106 (2015) · Zbl 1412.26047 |
| [25] | Garra, R.; Garrappa, R., The Prabhakar or three parameter Mittag-Leffler function: theory and application, Commun Nonlinear Sci Numer Simul, 56, 314-329 (2018) · Zbl 1524.33083 |
| [26] | Garra, R.; Gorenflo, R.; Polito, F.; Tomovski, Ž., Hilfer-Prabhakar derivatives and some applications, Appl Math Comput, 242, 576-589 (2014) · Zbl 1334.26008 |
| [27] | Garrappa, R., Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun Nonlinear Sci Numer Simul, 38, 178-191 (2016) · Zbl 1471.47032 · doi:10.1016/j.cnsns.2016.02.015 |
| [28] | Garrappa, R.; Mainardi, F.; Guido, M., Models of dielectric relaxation based on completely monotone functions, Fract Calc Appl Anal, 19, 1105-1160 (2016) · Zbl 1499.78010 |
| [29] | Giusti, A.; Colombaro, I., Prabhakar-like fractional viscoelasticity, Commun Nonlinear Sci Numer Simul, 56, 138-43 (2018) · Zbl 1510.74015 |
| [30] | Gorenflo, R.; Kilbas, AA; Mainardi, F.; Rogosin, SV, Mittag-Leffler functions, related topics and applications (2014), Berlin: Springer, Berlin · Zbl 1309.33001 |
| [31] | Heydari, MH, A new direct method based on the Chebyshev cardinal functions for variable-order fractional optimal control problems, J Franklin Inst, 355, 4970-4995 (2018) · Zbl 1395.49025 |
| [32] | Heydari, MH; Avazzadeh, Z., Legendre wavelets optimization method for variable-order fractional Poisson equation, Chaos Solitons Fractals, 112, 180-190 (2018) · Zbl 1398.65316 |
| [33] | Hsiao, CH; Wang, WJ, Haar wavelet approach to nonlinear stiff systems, Math Comput Simul, 57, 347-353 (2001) · Zbl 0986.65062 |
| [34] | Katugampola, UN, New approach to a generalized fractional integral, Appl Math Comput, 218, 860-865 (2011) · Zbl 1231.26008 |
| [35] | Keshi, FK; Moghaddam, BP; Aghili, A., A numerical approach for solving a class of variable-order fractional functional integral equations, Comput Appl Math, 37, 4821-4834 (2018) · Zbl 1432.65106 |
| [36] | Kilbas, AA; Saigo, M.; Saxena, RK, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transform Spec Funct, 15, 31-49 (2004) · Zbl 1047.33011 |
| [37] | Lepik, U., Solving fractional integral equations by the Haar wavelet method, Appl Math Comput, 214, 468-478 (2009) · Zbl 1170.65106 |
| [38] | Li, C.; Zhang, F., Fractional-order system identification based on continuous order-distributions, Signal Process, 83, 2287-2300 (2003) · Zbl 1145.93433 |
| [39] | Li, X.; Li, H.; Wu, B., A new numerical method for variable order fractional functional differential equations, Appl Math Lett, 68, 80-86 (2017) · Zbl 1361.65044 |
| [40] | Liang, H.; Stynes, M., Collocation methods for general Caputo two-point boundary value problems, J Sci Comput, 76, 390-425 (2018) · Zbl 1398.65180 |
| [41] | Liang, H.; Stynes, M., Collocation methods for general Riemann-Liouville two-point boundary value problems, Adv Comput Math, 45, 897-928 (2019) · Zbl 1415.65168 · doi:10.1007/s10444-018-9645-1 |
| [42] | 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, 435-445 (2009) · Zbl 1171.65101 |
| [43] | Mainardi, F.; Garrappa, R., On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics, J Comput Phys, 293, 70-80 (2015) · Zbl 1349.65085 |
| [44] | Mehrdoust, F.; Refahi Sheikhani, AH; Mashoof, M.; Hasanzadeh, S., Block-pulse operational matrix method for solving fractional Black-Scholes equation, J Econ Stud, 44, 3, 489-502 (2017) |
| [45] | Moghaddam, BP; Machado, JA, 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, 1023-1042 (2017) · Zbl 1376.65159 |
| [46] | Odzijewicz T, Malinowska AB, Torres DFM (2013) Fractional variational calculus of variable order. In: Advances in harmonic analysis and operator theory, operator theory: advances and applications, vol 229. Birkhauser/Springer Basel AG, Basel, pp 291-301 · Zbl 1275.26016 |
| [47] | Pandey, SC, The Lorenzo-Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics, Comput Appl Math, 37, 2648-2666 (2018) · Zbl 1401.26015 |
| [48] | Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1998), San Diego: Elsevier, Academic Press, San Diego · Zbl 0924.34008 |
| [49] | Polito F, Tomovski Ž (2016) Some properties of Prabhakar-type fractional calculus operators. Fract Differ Calc 6:73-94. arXiv:1508.03224 · Zbl 1424.26017 |
| [50] | Prabhakar, TR, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J., 19, 7-15 (1971) · Zbl 0221.45003 |
| [51] | Rahimkhani, P.; Ordokhani, Y., Approximate solution of non-linear fractional integro-differential equations using fractional alternative Legendre functions, J Comput Appl, 365, 112365-112379 (2020) · Zbl 1524.65979 |
| [52] | Saberi Najafi, H.; Refahi Sheikhani, AH, New restarting method in the Lanczos algorithm for generalized eigenvalue problem, Appl Math Comput, 184, 2, 421-428 (2007) · Zbl 1116.65045 |
| [53] | Samko, SG, Fractional integration and differentiation of variable order, Anal Math, 21, 213-236 (1995) · Zbl 0838.26006 |
| [54] | Samko, SG, Fractional integration and differentiation of variable order: an overview, Nonlinear Dyn, 71, 653-662 (2013) · Zbl 1268.34025 |
| [55] | Samko, SG; Ross, B., Integration and differentiation to a variable fractional order, Integral Transforms Spec Funct, 1, 277-300 (1993) · Zbl 0820.26003 |
| [56] | Soradi-Zeid, S.; Jahanshahi, H.; Yousefpour, A.; Bekiros, S., King algorithm: a novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems, Chaos Solitons Fractals (2020) · Zbl 1434.65084 · doi:10.1016/j.chaos.2019.109569 |
| [57] | Sun, K.; Zhu, M., Numerical algorithm to solve a class of variable order fractional integral-differential equation based on Chebyshev polynomials, Math Probl Eng (2015) · Zbl 1394.65113 · doi:10.1155/2015/902161 |
| [58] | Sun, H.; Chen, W.; Wei, H.; Chen, Y., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur Phys J Spec Top, 193, 185-192 (2011) |
| [59] | Valério, D.; Costa, J., Variable-order fractional derivatives and their numerical approximations, Signal Process, 91, 470-483 (2011) · Zbl 1203.94060 |
| [60] | Xu, J., Time-fractional particle deposition in porous media, J Phys A Math Theor, 50, 195002 (2017) · Zbl 1369.82034 |
| [61] | Xu, Y.; Ertürk, VS, A finite difference technique for solving variable-order fractional integro-differential equation, Bull Iran Math Soc, 40, 699-712 (2014) · Zbl 1305.65249 |
| [62] | Yan, R.; Han, M.; Ma, Q.; Ding, X., A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative, Comput Appl Math, 38, 66-90 (2019) · Zbl 1438.65164 |
| [63] | Yi, M.; Huang, J.; Wang, L., Operational matrix method for solving variable order fractional integro-differential equations, CMES Comput Model Eng Sci, 96, 361-377 (2013) · Zbl 1356.65205 |
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