# A Novel Numerical Scheme for Time-Fractional Partial Integro Differential Equation of Parabolic Type

## DOI:

https://doi.org/10.26713/cma.v15i1.2506## Keywords:

Time-fractional ADE, Variable parameters, Finite difference methods, Convergence analysis, Integro-partial differential equations## Abstract

This work is devoted to study numerical methods for time-fractional integro-differential equations. In order to compute the approximate solutions for highly non-linear or linear forms of various time-fractional integro-differential models, we apply the extended and more generalized finite difference methods. First order and second order spacial derivatives are approximated by the central difference. The integral terms and Capto fractional terms are approximated by the composite trapezoidal rule. Particularly we derive error estimation and stability analysis of the finite difference method for a Volterra type fractional differential equation. Illustrative examples are provided in support of the proposed methods with three distinct problems.

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## References

S. Abbas, M. Benchohra and T. Diagana, Existence and attractivity results for some fractional order partial integro-differential equations with delay, African Diaspora Journal of Mathematics. New Series 15(2) (2013), 87 – 100, DOI: https://projecteuclid.org/journals/african-diaspora-journal-ofmathematics.

D. Benson, R. Schumer, M. Meerschaert and S. Wheatcraft, Fractional dispersion, lévy motion, and the made tracer tests, Transport in Porous Media 42(1) (2001), 211 – 240, DOI: 10.1023/A:1006733002131.

C. Chen, F. Liu, I. Turner and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics 227(2) (2007), 886 – 897, DOI: 10.1016/j.jcp.2007.05.012.

C.-M. Chen, F. Liu and K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation, Applied Mathematics and Computation 198(2) (2008), 754 – 769, DOI: 10.1016/j.amc.2007.09.020.

S. Das, A note on fractional diffusion equations, Chaos, Solitons & Fractals 42(4) (2009), 2074 – 2079, DOI: 10.1016/j.chaos.2009.03.163.

S. Das and P. Gupta, An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy perturbation method, Zeitschrift Für Naturforschung A 65(3) (2010), 182 – 190, DOI: 10.1515/zna-2010-0305.

S. Das, R. Kumar and P. Gupta, An approximate analytical solution of the fractional diffusion equation with external force and different type of absorbent term-revisited, Applications and Applied Mathematics: An International Journal 5(3) (2010), 9, URL: https://digitalcommons.pvamu.edu/aam/vol5/iss3/9.

M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, International Journal of Computer Mathematics 83(1) (2006), 123 – 129, DOI: 10.1080/00207160500069847.

K. Diethelm, The Analysis of Fractional Differential Equations: an Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics series, Springer, Berlin — Heidelberg, viii + 247 pages (2010), DOI: 10.1007/978-3-642-14574-2.

R. Du, W. Cao and Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling 34(10) (2010), 2998 – 3007, DOI: 10.1016/j.apm.2010.01.008.

M. Gordji, H. Baghani and O. Baghani, On existence and uniqueness of solutions of a nonlinear integral equation, Journal of Applied Mathematics 2011 (2011), 743923, 7 pages, DOI: 10.1155/2011/743923.

R. Gorenflo, Fractional calculus: Some numerical methods, in: Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (editors), Vol. 378, pp. 277 – 290, International Centre for Mechanical Sciences, Springer, Vienna, DOI: 10.1007/978-3-7091-2664-6_6.

J. Gracia, E. O’Riordan and M. Stynes, Convergence in positive time for a finite difference method applied to a fractional convection-diffusion problem, Computational Methods in Applied Mathematics 18(1) (2018), 33 – 42, DOI: 10.1515/cmam-2017-0019.

A. Hamoud, K. Ghadle and G. M. Sh. B. Issa, Existence and uniqueness theorems for fractional volterra-fredholm integro-differential equations, International Journal of Applied Mathematics 31(3) (2018), 333 – 348, DOI: 10.12732/ijam.v31i3.3.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo (editors), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies Book series, Vol. 204 (2006), pp. 1 – 523, URL: https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/204/suppl/C.

T. Langlands and B. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, Journal of Computational Physics 205(2) (2005), 719 – 736, DOI: 10.1016/j.jcp.2004.11.025.

E. Lenzi, M. Lenzi, L. Evangelista, L. Malacarne and R. Mendes, Solutions for a fractional nonlinear diffusion equation with external force and absorbent term, Journal of Statistical Mechanics: Theory and Experiment 2009(02) (2009), P02048, DOI: 10.1088/1742-5468/2009/02/P02048.

F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker–Planck equation, Journal of Computational and Applied Mathematics 166(1) (2004), 209 – 219, DOI: 10.1016/j.cam.2003.09.028.

V. Lynch, B. Carreras, D. del-Castillo-Negrete, K. Ferreira-Mejias and H. Hicks, Numerical methods for the solution of partial differential equations of fractional order, Journal of Computational Physics 192(2) (2003), 406 – 421, DOI: 10.1016/j.jcp.2003.07.008.

F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas, Fractional calculus and continuous-time finance II: The waiting-time distribution, Physica A: Statistical Mechanics and its Applications 287(3-4) (2000), 468–481.

M. Matar, Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions, Electronic Journal of Differential Equations 2009 (2009), Paper number: 155, 7 pages, https://eudml.org/doc/231434.

M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, Journal of Computational and Applied Mathematics 172(1) (2004), 65 – 77, DOI: 10.1016/j.cam.2004.01.033.

I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edition, Vol. 198, Elsevier, 340 pages (1998).

A. Prakash and H. Kaur, Numerical solution for fractional model of Fokker-Planck equation by using q-hatm, Chaos, Solitons & Fractals 105 (2017), 99 – 110, DOI: 10.1016/j.chaos.2017.10.003.

S. Ray and R. Bera, Analytical solution of the bagley torvik equation by adomian decomposition method, Applied Mathematics and Computation 168(1) (2005), 398 – 410, DOI: 10.1016/j.amc.2004.09.006.

K. Saad, H. Srivastava and J. Gómez-Aguilar, A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos, Solitons & Fractals 132 (2020), 109557, DOI: 10.1016/j.chaos.2019.109557.

S. Santra and J. Mohapatra, A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type, Journal of Computational and Applied Mathematics 400 (2022), 113746, DOI: 10.1016/j.cam.2021.113746.

A. Schot, M. Lenzi, L. Evangelista, L. Malacarne, R. Mendes and E. Lenzi, Fractional diffusion equation with an absorbent term and a linear external force: Exact solution, Physics Letters A 366(4-5) (2007), 346 – 350, DOI: 10.1016/j.physleta.2007.02.056.

S. Shahmorad, Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the tau method with an error estimation, Applied Mathematics and Computation 167(2) (2005), 1418 – 1429, DOI: 10.1016/j.amc.2004.08.045.

S. Shen and F. Liu, Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, Anziam Journal 46 (2004), C871 – C887, DOI: 10.21914/anziamj.v46i0.995.

A. Silva, E. Lenzi, L. Evangelista, M. Lenzi and L. da Silva, Fractional nonlinear diffusion equation, solutions and anomalous diffusion, Physica A: Statistical Mechanics and its Applications 375(1) (2007), 65 – 71, DOI: 10.1016/j.physa.2006.09.001.

E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics 228(11) (2009), 4038 – 4054, DOI: 10.1016/j.jcp.2009.02.011.

A. Tari and S. Shahmorad, A computational method for solving two-dimensional linear Fredholm integral equations of the second kind, The ANZIAM Journal 49(4) (2008), 543 – 549, DOI: 10.1017/S1446181108000126.

J. Thorwe and S. Bhalekar, Solving partial integro-differential equations using Laplace transform method, American Journal of Computational and Applied Mathematics 2(3) (2012), 101 – 104, DOI: 10.5923/j.ajcam.20120203.06.

T. Wang, M. Qin and Z. Zhang, The Puiseux expansion and numerical integration to nonlinear weakly singular Volterra integral equations of the second kind, Journal of Scientific Computing 82(3) (2020), 1 – 28, DOI: 10.1007/s10915-020-01167-3.

T. Wongyat and W. Sintunavarat, The existence and uniqueness of the solution for nonlinear Fredholm and Volterra integral equations together with nonlinear fractional differential equations via w-distances, Advances in Difference Equations 2017(1) (2017), 1 – 15, DOI: 10.1186/s13662-017-1267-2.

S. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM Journal on Numerical Analysis 42(5) (2005), 1862 – 1874, DOI: 10.1137/030602666.

M. Zahran, On the derivation of fractional diffusion equation with an absorbent term and a linear external force, Applied Mathematical Modelling 33(7) (2009), 3088 – 3092, DOI: 10.1016/j.apm.2008.10.013.

P. Zhang and X. Hao, Existence and uniqueness of solutions for a class of nonlinear integrodifferential equations on unbounded domains in Banach spaces, Advances in Difference Equations 2018(1) (2018), 1 – 7, DOI: 10.1186/s13662-018-1681-0.

P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, Journal of Applied Mathematics and Computing 22(3) (2006), 87 – 99, DOI: 10.1007/BF02832039.

P. Zhuang, F. Liu, V. Anh and I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM Journal on Numerical Analysis 46(2) (2008), 1079 – 1095, DOI: 10.1137/060673114.

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*Communications in Mathematics and Applications*,

*15*(1), 463–482. https://doi.org/10.26713/cma.v15i1.2506

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