Numerical solution of variable-order time fractional weakly singular partial integro-differential equations with error estimation
Abstract
In this paper, we apply Legendre-Laguerre functions (LLFs) and collocation method to obtain the approximate solution of variable-order time-fractional partial integro-differential equations (VO-TF-PIDEs) with the weakly singular kernel. For this purpose, we derive the pseudo-operational matrices with the use of the transformation matrix. The collocation method and pseudo-operational matrices transfer the problem to a system of algebraic equations. Also, the error analysis of the proposed method is given. We consider several examples to illustrate the proposed method is accurate.
Keyword : variable-order fractional partial integro-differential equations, weakly singular kernel, Legendre-Laguerre functions, pseudo-operational matrix
How to Cite
Dehestani, H., Ordokhani, Y., & Razzaghi, M. (2020). Numerical solution of variable-order time fractional weakly singular partial integro-differential equations with error estimation. Mathematical Modelling and Analysis, 25(4), 680-701. https://doi.org/10.3846/mma.2020.11692
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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H. Dehestani, Y. Ordokhani and M. Razzaghi. Fractional-order Legendre– Laguerre functions and their applications in fractional partial differential equations. Applied Mathematics and Computation, 336:433–453, 2018. https://doi.org/10.1016/j.amc.2018.05.017
H. Dehestani, Y. Ordokhani and M. Razzaghi. Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations. Mathematical Methods in the Applied Sciences, 42(18):7296–7313, 2019. https://doi.org/10.1002/mma.5840
H. Dehestani, Y. Ordokhani and M. Razzaghi. Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(5):e2606, 2019. https://doi.org/10.1002/jnm.2606
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G. Diaz and C.F.M Coimbra. Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dynamics, 56(1):145–157, 2009. https://doi.org/10.1007/s11071-008-9385-8
G. Fairweather. Spline collocation methods for a class of hyperbolic partial integro-differential equations. SIAM journal on numerical analysis, 31(2):444– 460, 1994. https://doi.org/10.1137/0731024
L. Hörmander. The analysis of linear partial differential operators I: Distribution theory and Fourier analysis. Springer, 2015. https://doi.org/10.1007/978-3-642-61497-2
D. Ingman and J. Suzdalnitsky. Control of damping oscillations by fractional differential operator with time-dependent order. Computer Methods in Applied Mechanics and Engineering, 193(52):5585–5595, 2004. https://doi.org/10.1016/j.cma.2004.06.029
W. Jiang and N. Liu. A numerical method for solving the time variable fractional order mobile–immobile advection–dispersion model. Applied Numerical Mathematics, 119:18–32, 2017. https://doi.org/10.1016/j.apnum.2017.03.014
J.B. Keller and W.E. Olmstead. Temperature of a nonlinearly radiating semi-infinite solid. Quarterly of Applied Mathematics, 29(4):559–566, 1972. https://doi.org/10.1090/qam/403430
R. Lin, F. Liu, V. Anh and I. Turner. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Applied Mathematics and Computation, 212(2):435–445, 2009. https://doi.org/10.1016/j.amc.2009.02.047
P. Linz. Analytical and numerical methods for Volterra equations. Society for Industrial and Applied Mathematics, 1985. https://doi.org/10.1137/1.9781611970852
W. McLean, I.H. Sloan and V. Thomée. Time discretization via Laplace transformation of an integro-differential equation of parabolic type. Numerische Mathematik, 102(3):497–522, 2006. https://doi.org/10.1007/s00211-005-0657-7
B.P. Moghaddam and J.A.T. Machado. A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fractional Calculus and Applied Analysis, 20(4):1023– 1042, 2017. https://doi.org/10.1515/fca-2017-0053
B.P. Moghaddam and J.A.T. Machado. Time analysis of forced variable-order fractional Van der Pol oscillator. The European Physical Journal Special Topics, 226(16):3803–3810, 2017. https://doi.org/10.1140/epjst/e2018-00019-7
B.P. Moghaddam, J.A.T. Machado and H. Behforooz. An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos, Solitons & Fractals, 102:354–360, 2017. https://doi.org/10.1016/j.chaos.2017.03.065
B.P. Moghaddam and J.T. Machado. A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Computers & Mathematics with Applications, 73(6):1262– 1269, 2017. https://doi.org/10.1016/j.camwa.2016.07.010
P. Muthukumar and B. Ganesh Priya. Numerical solution of fractional delay differential equation by shifted Jacobi polynomials. International Journal of Computer Mathematics, 94(3):471–492, 2017. https://doi.org/10.1080/00207160.2015.1114610
S. Nemati, P.M. Lima and Y. Ordokhani. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials. Journal of Computational and Applied Mathematics, 242:53–69, 2013. https://doi.org/10.1016/j.cam.2012.10.021
V.K. Patel, S. Singh, V.K. Singh and E. Tohidi. Two dimensional wavelets collocation scheme for linear and nonlinear Volterra weakly singular partial integrodifferential equations. International Journal of Applied and Computational Mathematics, 4(5):132, 2018. https://doi.org/10.1007/s40819-018-0560-4
L.E.S. Ramirez and C.F.M. Coimbra. On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D: Nonlinear Phenomena, 240(13):1111–1118, 2011. https://doi.org/10.1016/j.physd.2011.04.001
R. Schumer, D.A. Benson, M.M. Meerschaert and B. Baeumer. Fractal mobile/immobile solute transport. Water Resources Research, 39(10), 2003. https://doi.org/10.1029/2003WR002141
H. Sheng, H. Sun, Y.Q. Chen and T.S. Qiu. Synthesis of multifractional Gaussian noises based on variable-order fractional operators. Signal Processing, 91(7):1645–1650, 2011. https://doi.org/10.1016/j.sigpro.2011.01.010
S. Singh, V.K. Patel and V.K. Singh. Operational matrix approach for the solution of partial integro-differential equation. Applied Mathematics and Computation, 283:195–207, 2016. https://doi.org/10.1016/j.amc.2016.02.036
S. Singh, V.K. Patel and V.K. Singh. Convergence rate of collocation method based on wavelet for nonlinear weakly singular partial integro-differential equation arising from viscoelasticity. Numerical Methods for Partial Differential Equations, 34(5):1781–1798, 2018. https://doi.org/10.1002/num.22245
S. Singh, V.K. Patel, V.K. Singh and E. Tohidi. Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices. Applied Mathematics and Computation, 298:310–321, 2017. https://doi.org/10.1016/j.amc.2016.11.012
C.M. Soon, C.F.M. Coimbra and M.H. Kobayashi. The variable viscoelasticity oscillator. Annalen der Physik, 14(6):378–389, 2005. https://doi.org/10.1002/andp.200410140
T. Tang. A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Applied Numerical Mathematics, 11(4):309–319, 1993. https://doi.org/10.1016/0168-9274(93)90012-G
A.-M. Wazwaz. A reliable treatment for mixed Volterra–Fredholm integral equations. Applied Mathematics and Computation, 127(2-3):405–414, 2002. https://doi.org/10.1016/S0096-3003(01)00020-0
H. Zhang, F. Liu, M.S. Phanikumar and M.M. Meerschaert. A novel numerical method for the time variable fractional order mobile–immobile advection– dispersion model. Computers & Mathematics with Applications, 66(5):693–701, 2013. https://doi.org/10.1016/j.camwa.2013.01.031
Y. Zhang, D.A. Benson and D.M. Reeves. Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources, 32(4):561–581, 2009. https://doi.org/10.1016/j.advwatres.2009.01.008
A. Babaei, B.P. Moghaddam, S. Banihashemi and J.A.T. Machado. Numerical solution of variable-order fractional integro-partial differential equations via sinc collocation method based on single and double exponential transformations. Communications in Nonlinear Science and Numerical Simulation, 82:104985, 2020. https://doi.org/10.1016/j.cnsns.2019.104985
D. Baleanu, R. Darzi and B. Agheli. New study of weakly singular kernel fractional fourth-order partial integro-differential equations based on the optimum q-homotopic analysis method. Journal of Computational and Applied Mathematics, 320:193–201, 2017. https://doi.org/10.1016/j.cam.2017.01.032
Sh.S. Behzadi. The use of iterative methods to solve two-dimensional nonlinear Volterra-Fredholm integro-differential equations. Communications in Numerical Analysis, 2012:1–20, 2012. https://doi.org/10.5899/2012/cna-00108
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang. Spectral methods: fundamentals in single domains. Springer Science & Business Media, 2007. https://doi.org/10.1007/978-3-540-30726-6
C.F.M Coimbra. Mechanics with variable-order differential operators. Annalen der Physik, 12(11-12):692–703, 2003. https://doi.org/10.1002/andp.200310032
H. Dehestani, Y. Ordokhani and M. Razzaghi. Fractional-order Legendre– Laguerre functions and their applications in fractional partial differential equations. Applied Mathematics and Computation, 336:433–453, 2018. https://doi.org/10.1016/j.amc.2018.05.017
H. Dehestani, Y. Ordokhani and M. Razzaghi. Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations. Mathematical Methods in the Applied Sciences, 42(18):7296–7313, 2019. https://doi.org/10.1002/mma.5840
H. Dehestani, Y. Ordokhani and M. Razzaghi. Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(5):e2606, 2019. https://doi.org/10.1002/jnm.2606
M. Dehghan. Solution of a partial integro-differential equation arising from viscoelasticity. International Journal of Computer Mathematics, 83(1):123–129, 2006. https://doi.org/10.1080/00207160500069847
G. Diaz and C.F.M Coimbra. Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dynamics, 56(1):145–157, 2009. https://doi.org/10.1007/s11071-008-9385-8
G. Fairweather. Spline collocation methods for a class of hyperbolic partial integro-differential equations. SIAM journal on numerical analysis, 31(2):444– 460, 1994. https://doi.org/10.1137/0731024
L. Hörmander. The analysis of linear partial differential operators I: Distribution theory and Fourier analysis. Springer, 2015. https://doi.org/10.1007/978-3-642-61497-2
D. Ingman and J. Suzdalnitsky. Control of damping oscillations by fractional differential operator with time-dependent order. Computer Methods in Applied Mechanics and Engineering, 193(52):5585–5595, 2004. https://doi.org/10.1016/j.cma.2004.06.029
W. Jiang and N. Liu. A numerical method for solving the time variable fractional order mobile–immobile advection–dispersion model. Applied Numerical Mathematics, 119:18–32, 2017. https://doi.org/10.1016/j.apnum.2017.03.014
J.B. Keller and W.E. Olmstead. Temperature of a nonlinearly radiating semi-infinite solid. Quarterly of Applied Mathematics, 29(4):559–566, 1972. https://doi.org/10.1090/qam/403430
R. Lin, F. Liu, V. Anh and I. Turner. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Applied Mathematics and Computation, 212(2):435–445, 2009. https://doi.org/10.1016/j.amc.2009.02.047
P. Linz. Analytical and numerical methods for Volterra equations. Society for Industrial and Applied Mathematics, 1985. https://doi.org/10.1137/1.9781611970852
W. McLean, I.H. Sloan and V. Thomée. Time discretization via Laplace transformation of an integro-differential equation of parabolic type. Numerische Mathematik, 102(3):497–522, 2006. https://doi.org/10.1007/s00211-005-0657-7
B.P. Moghaddam and J.A.T. Machado. A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fractional Calculus and Applied Analysis, 20(4):1023– 1042, 2017. https://doi.org/10.1515/fca-2017-0053
B.P. Moghaddam and J.A.T. Machado. Time analysis of forced variable-order fractional Van der Pol oscillator. The European Physical Journal Special Topics, 226(16):3803–3810, 2017. https://doi.org/10.1140/epjst/e2018-00019-7
B.P. Moghaddam, J.A.T. Machado and H. Behforooz. An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos, Solitons & Fractals, 102:354–360, 2017. https://doi.org/10.1016/j.chaos.2017.03.065
B.P. Moghaddam and J.T. Machado. A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Computers & Mathematics with Applications, 73(6):1262– 1269, 2017. https://doi.org/10.1016/j.camwa.2016.07.010
P. Muthukumar and B. Ganesh Priya. Numerical solution of fractional delay differential equation by shifted Jacobi polynomials. International Journal of Computer Mathematics, 94(3):471–492, 2017. https://doi.org/10.1080/00207160.2015.1114610
S. Nemati, P.M. Lima and Y. Ordokhani. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials. Journal of Computational and Applied Mathematics, 242:53–69, 2013. https://doi.org/10.1016/j.cam.2012.10.021
V.K. Patel, S. Singh, V.K. Singh and E. Tohidi. Two dimensional wavelets collocation scheme for linear and nonlinear Volterra weakly singular partial integrodifferential equations. International Journal of Applied and Computational Mathematics, 4(5):132, 2018. https://doi.org/10.1007/s40819-018-0560-4
L.E.S. Ramirez and C.F.M. Coimbra. On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D: Nonlinear Phenomena, 240(13):1111–1118, 2011. https://doi.org/10.1016/j.physd.2011.04.001
R. Schumer, D.A. Benson, M.M. Meerschaert and B. Baeumer. Fractal mobile/immobile solute transport. Water Resources Research, 39(10), 2003. https://doi.org/10.1029/2003WR002141
H. Sheng, H. Sun, Y.Q. Chen and T.S. Qiu. Synthesis of multifractional Gaussian noises based on variable-order fractional operators. Signal Processing, 91(7):1645–1650, 2011. https://doi.org/10.1016/j.sigpro.2011.01.010
S. Singh, V.K. Patel and V.K. Singh. Operational matrix approach for the solution of partial integro-differential equation. Applied Mathematics and Computation, 283:195–207, 2016. https://doi.org/10.1016/j.amc.2016.02.036
S. Singh, V.K. Patel and V.K. Singh. Convergence rate of collocation method based on wavelet for nonlinear weakly singular partial integro-differential equation arising from viscoelasticity. Numerical Methods for Partial Differential Equations, 34(5):1781–1798, 2018. https://doi.org/10.1002/num.22245
S. Singh, V.K. Patel, V.K. Singh and E. Tohidi. Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices. Applied Mathematics and Computation, 298:310–321, 2017. https://doi.org/10.1016/j.amc.2016.11.012
C.M. Soon, C.F.M. Coimbra and M.H. Kobayashi. The variable viscoelasticity oscillator. Annalen der Physik, 14(6):378–389, 2005. https://doi.org/10.1002/andp.200410140
T. Tang. A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Applied Numerical Mathematics, 11(4):309–319, 1993. https://doi.org/10.1016/0168-9274(93)90012-G
A.-M. Wazwaz. A reliable treatment for mixed Volterra–Fredholm integral equations. Applied Mathematics and Computation, 127(2-3):405–414, 2002. https://doi.org/10.1016/S0096-3003(01)00020-0
H. Zhang, F. Liu, M.S. Phanikumar and M.M. Meerschaert. A novel numerical method for the time variable fractional order mobile–immobile advection– dispersion model. Computers & Mathematics with Applications, 66(5):693–701, 2013. https://doi.org/10.1016/j.camwa.2013.01.031
Y. Zhang, D.A. Benson and D.M. Reeves. Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources, 32(4):561–581, 2009. https://doi.org/10.1016/j.advwatres.2009.01.008