Abstract
This article deals with a fully discretized numerical scheme for solving fractional order Volterra integro-differential equations involving Caputo fractional derivative. Such problem exhibits a mild singularity at the initial time \(t=0\). To approximate the solution, the classical L1 scheme is introduced on a uniform mesh. For the integral part, the composite trapezoidal approximation is used. It is shown that the approximate solution converges to the exact solution. The error analysis is carried out. Due to presence of weak singularity at the initial time, we obtain the rate of convergence is of order \(O(\tau )\) on any subdomain away from the origin whereas it is of order \(O(\tau ^\alpha )\) over the entire domain. Finally, we present a couple of examples to show the efficiency and the accuracy of the numerical scheme.
Similar content being viewed by others
References
Ahmad B, Nieto JJ (2011) Riemann–Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound Value Probl 36:2011. https://doi.org/10.1186/1687-2770-2011-36
Ali MF, Sharma LN, Mishra Mishra VN (2015) Dirichlet average of generalized Miller–Ross function and fractional derivative. Turk J Anal Number Theory 3(1):30–32. https://doi.org/10.12691/tjant-3-1-7
Alkan S, Hatipoglu V (2017) Approximate solutions of Volterra–Fredholm integro-differential equations of fractional order. Tbilisi Math J 10(2):1–13
Assari P, Dehghan M (2019) A meshless local Galerkin method for solving Volterra integral equations deduced from nonlinear fractional differential equations using the moving least squares technique. Appl Numer Math 143:276–299
Assari P, Adibi H, Dehghan M (2013) A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method. Appl Math Model 37(22):9269–9294
Baglan I, Kanca F, Mishra VN (2018) Determination of an unknown heat source from integral overdetermination condition. Iran J Sci Technol Trans A Sci 42:1373–1382. https://doi.org/10.1007/s40995-017-0454-z
Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210
Das P, Rana S, Ramos H (2019a) Homotopy perturbation method for solving Caputo-type fractional-order Volterra–Fredholm integro-differential equations. Comp Math Methods. https://doi.org/10.1002/cmm4.1047
Das P, Rana S, Ramos H (2019b) A perturbation based approach for solving fractional order Volterra–Fredholm integro differential equations and its convergence analysis. Int J Comput Math 97(10):1994–2014. https://doi.org/10.1080/00207160.2019.1673892
Deepmala V, Mishra N, Marasi HR, Shabanian H, Nosraty M (2017) Solution of Voltra–Fredholm integro-differential equations using Chebyshev collocation method. Glob J Technol Optim 8(1):66. https://doi.org/10.4172/2229-8711.1000210
Diethelm K (2010) The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, vol 2004. Springer, Berlin
Dubey R, Deepmala H, Mishra VN (2020) Higher-order symmetric duality in nondifferentiable multiobjective fractional programming problem over cone constraints. Stat Optim Inf Comput 8(1):187–205. https://doi.org/10.19139/soic-2310-5070-601
Fellah ZEA, Depollier C, Fellah M (2002) Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurements. Acta Acust United Acust 88(1):34–39
Gordji ME, Baghani H, Baghani O (2011) On existence and uniqueness of solutions of a nonlinear integral equation. J Appl Math. https://doi.org/10.1155/2011/743923
Gracia JL, O’Riordan E, Stynes M (2018) Convergence in positive time for a finite difference method applied to a fractional convection–diffusion problem. Comput Methods Appl Math 18(1):33–42
Guo D (2001) Existence of solutions for nth-order integro-differential equations in Banach spaces. Comput Math Appl 41(5–6):597–606
Hamoud AA, Ghadle KP, Issa MB, Giniswamy H (2018) Existence and uniqueness theorems for fractional Volterra–Fredholm integro-differential equations. Int J Appl Math 31(3):333–348
Jhinga A, Daftardar-Gejji V (2019) A new numerical method for solving fractional delay differential equations. Comput Appl Math. https://doi.org/10.1007/s40314-019-0951-0
Kanca F, Mishra VN (2019) Identification problem of a leading coefficient to the time derivative of parabolic equation with nonlocal boundary conditions. Iran J Sci Technol Trans A Scie 43(3):1227–1233. https://doi.org/10.1007/s40995-018-0587-8
Kulish VV, Lage JL (2002) Application of fractional calculus to fluid mechanics. J Fluids Eng 124(3):803–806
Marti TJ (1967) On integro-differential equations in Banach spaces. Pac J Math 20(1):99–108
Mishra VN, Khatri K, Mishra LN, Deepmala S (2013a) Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators. J Inequal Appl 1:1–11. https://doi.org/10.1186/1029-242X-2013-586
Mishra VN, Khan HH, Khatri K, Mishra LN (2013b) Hypergeometric representation for Baskakov–Durrmeyer–Stancu type operators. Bull Math Anal Appl 5(3):18–26
Mittal R, Nigam R (2008) Solution of fractional integro-differential equations by Adomian decomposition method. Int J Appl Math Mech 4(2):87–94
Momani S, Noor MA (2006) Numerical methods for fourth-order fractional integro-differential equations. Appl Math Comput 182(1):754–760
Podlubny I (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of their Applications Mathematics in Science and Engineering, vol 198. Academic Press, San Diego
Rawashdeh EA (2006) Numerical solution of fractional integro-differential equations by collocation method. Appl Math Comput 176(1):1–6
Sales SMSN, Baghani O (2018) On multi-singular integral equations involving n weakly singular kernels. Filomat 32(4):1323–1333
Santra S, Mohapatra J (2020) Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity. Math Methods Appl Sci 44(2):1529–1541. https://doi.org/10.1002/mma.6850
Shahmorad S (2005) Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the Tau method with an error estimation. Appl Math Comput 167(2):1418–1429
Soczkiewicz E (2002) Application of fractional calculus in the theory of viscoelasticity. Mol. Quantum Acoust. 23:397–404
Stynes M, O’Riordan E, Gracia JL (2017) Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J Numer Anal 55(2):1057–1079
Suárez JI, Vinagre BM, Calderón AJ, Monje CA, Chen YQ (2003) Using fractional calculus for lateral and longitudinal control of autonomous vehicles. Lecture notes in computer science. Springer, Berlin, pp 337–348
Wang T, Qin M, Zhang Z (2020) The Puiseux expansion and numerical integration to nonlinear weakly singular Volterra integral equations of the second kind. J Sci Comput 82(2):1–28
Wongyat T, Sintunavarat W (2017) The existence and uniqueness of the solution for nonlinear Fredholm and Volterra integral equations together with nonlinear fractional differential equations via w-distances. Adv Differ Equ 1:211. https://doi.org/10.1186/s13662-017-1267-2
Zhang P, Hao X (2018) Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces. Adv Differ Equ 1:247. https://doi.org/10.1186/s13662-018-1681-0
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Santra, S., Mohapatra, J. Numerical Analysis of Volterra Integro-differential Equations with Caputo Fractional Derivative. Iran J Sci Technol Trans Sci 45, 1815–1824 (2021). https://doi.org/10.1007/s40995-021-01180-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-021-01180-7