Top

Journal of Applied Mathematics and Computing

Published in:

02-12-2021 | Original Research

A novel approach for solving multi-term time fractional Volterra–Fredholm partial integro-differential equations

Authors: Sudarshan Santra, Abhilipsa Panda, Jugal Mohapatra

Published in: Journal of Applied Mathematics and Computing | Issue 5/2022

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This article deals with an efficient numerical technique to solve a class of multi-term time fractional Volterra–Fredholm partial integro-differential equations of first kind. The fractional derivatives are defined in Caputo sense. The Adomian decomposition method is used to construct the scheme. For simplicity of the analysis, the model problem is converted into a multi-term time fractional Volterra–Fredholm partial integro-differential equation of second kind. In addition, the convergence analysis and the condition for existence and uniqueness of the solution are provided. Several numerical examples are illustrated in support of the theoretical analysis.

previous article Novel algebraic criteria on global Mittag–Leffler synchronization for FOINNs with the Caputo derivative and delay

next article On the 4-adic complexity of the two-prime quaternary generator

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Abbas, S., Benchohra, M., Diagana, T.: Existence and attractivity results for some fractional order partial integro-differential equations with delay. Afr. Diaspora J. Math. 15(2), 87–100 (2013)MathSciNetMATH

Adomian, G.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135(2), 501–544 (1988)MathSciNetMATHCrossRef

Aghazadeh, N., Ravash, E., Rezapour, S.: Existence results and numerical solutions for a multi-term fractional integro-differential equation. Kragujev. J. Math. 43(3), 413–426 (2019)MathSciNetMATH

Alquran, M., Jaradat, I., Sivasundaram, S.: Elegant scheme for solving Caputo-time fractional integro-differential equations. Nonlinear Stud. 25(2), 385–393 (2018)MathSciNetMATH

Assaleh, K., Ahmad, W. M.: Modeling of speech signals using fractional calculus. In: 9th International Symposium on Signal Processing and Its Applications (2007)

Daftardar-Gejji, V., Jafari, H.: Solving a multi-order fractional differential equation using Adomian decomposition. Appl. Math. Comput. 189(1), 541–548 (2007)MathSciNetMATH

Daftardar-Gejji, V., Bhalekar, S.: Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method. Appl. Math. Comput. 202(1), 113–120 (2008)MathSciNetMATH

Darweesh, A., Alquran, M., Aghzawi, K.: New numerical treatment for a family of two-dimensional fractional Fredholm integro-differential equations. Algorithms 13(2), 37 (2020). https://doi.org/10.3390/a13020037MathSciNetCrossRef

Das, P., Rana, S., Ramos, H.: 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 (2020)MathSciNetMATHCrossRef

10.

Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, vol. 2004. Springer, Berlin (2010)MATHCrossRef

11.

Hamoud, A.A., Ghadle, K.P.: Modified Laplace decomposition method for fractional Volterra–Fredholm integro-differential equations. J. Math. Model. 6(1), 91–104 (2018)MathSciNetMATH

12.

Hamoud, A.A., Ghadle, K.P., Issa, M.B.: Giniswamy: existence and uniqueness theorems for fractional Volterra–Fredholm integro-differential equations. Int. J. Appl. Math. 31(3), 333–348 (2018)MathSciNetCrossRef

13.

Jaradat, I., Al-Dolat, M., Al-Zoubi, K., Alquran, M.: Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fract. 108, 107–110 (2018)MathSciNetMATHCrossRef

14.

Kaya, D., Aassila, M.: An application for a generalized KdV equation by the decomposition method. Phys. Lett. A 299(2–3), 201–206 (2002)MathSciNetMATHCrossRef

15.

Kudu, M., Amirali, I., Amiraliyev, G.M.: A fitted second-order difference method for a parameterized problem with integral boundary condition exhibiting initial layer. Mediterr. J. Math. (2021). https://doi.org/10.1007/s00009-021-01758-wMathSciNetMATHCrossRef

16.

Kulish, V.V., Lage, J.L.: Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124(3), 803–806 (2002)CrossRef

17.

Kumar, K., Pramod Chakravarthy, P., Vigo-Aguiar, J.: Numerical solution of time fractional singularly perturbed convection–diffusion problems with a delay in time. Math. Methods Appl. Sci. 44(4), 3080–3097 (2021)MathSciNetMATHCrossRef

18.

Li, C., Zeng, F.: Finite element methods for fractional differential equations. In: Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis, 49–68. World Scientific (2013)

19.

Matar, M.M.: Existence and uniqueness of solutions to fractional semilinear mixed Volterra–Fredholm integro-differential equations with nonlocal conditions. Electron. J. Differ. Eq. 2009(155), 1–7 (2009)

20.

Momani, S., Noor, M.A.: Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 182(1), 754–760 (2006)MathSciNetMATH

21.

Momani, S., Odibat, Z., Alawneh, A.: Variational iteration method for solving the space-and time-fractional KdV equation. Numer. Methods Part. Differ. Equ. 24(1), 262–271 (2008)MathSciNetMATHCrossRef

22.

Nemati, S., Lima, P.M.: Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions. Appl. Math. Comput. 327, 79–92 (2018)MathSciNetMATH

23.

Panda, A., Santra, S., Mohapatra, J.: Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations. J. Appl. Math. Comput. (2021). https://doi.org/10.1007/s12190-021-01613-xMATHCrossRef

24.

Panda, A., Mohapatra, J., Amirali, I.: A second order post-processing technique for singularly perturbed Volterra integro-differential equations. Mediterr. J. Math. (2021). https://doi.org/10.1007/s00009-021-01873-8MathSciNetMATHCrossRef

25.

Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Vol. 198, Mathematics in Science and Engineering. Academic Press, San Diego (1999)

26.

Rach, R.C.: A new definition of the Adomian polynomials. Kybernetes 6, 66 (2008)MathSciNetMATH

27.

Rahimkhani, P., Ordokhani, Y.: Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions. J. Comput. Appl. Math. (2020). https://doi.org/10.1016/j.cam.2019.112365MathSciNetMATHCrossRef

28.

Rawashdeh, E.A.: Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput. 176(1), 1–6 (2006)MathSciNetMATH

29.

Renardy, M.: Mathematical analysis of viscoelastic ows. Annu. Rev. Fluid Mech. 21(1), 21–34 (1989)CrossRef

30.

Roohollahi, A., Ghazanfari, B., Akhavan, S.: Numerical solution of the mixed Volterra–Fredholm integro-differential multi-term equations of fractional order. J. Comput. Appl. Math. (2020). https://doi.org/10.1016/j.cam.2020.112828MathSciNetMATHCrossRef

31.

Santra, S., Mohapatra, J.: Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity. Math. Methods Appl. Sci. 44(2), 1529–1541 (2020)MathSciNetMATHCrossRef

32.

Santra, S., Mohapatra, J.: Numerical analysis of Volterra integro-differential equations with Caputo fractional derivative. Iran. J. Sci. Technol. Trans. A Sci. 45(5), 1815–1824 (2021)MathSciNetCrossRef

33.

Santra, S., Mohapatra, J.: A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type. J. Comput. Appl. Math. (2021). https://doi.org/10.1016/j.cam.2021.113746MATHCrossRef

34.

Schumer, R., Benson, D., Meerschaert, M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39, 13 (2003)CrossRef

35.

Soczkiewicz, E.: Application of fractional calculus in the theory of viscoelasticity. Mol. Quantum Acoust. 23, 397–404 (2002)

36.

Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)

37.

Wang, Q.: Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. Appl. Math. Comput. 182(2), 1048–1055 (2006)MathSciNetMATH

38.

Zhao, Z., Zheng, Y.: Leapfrog/finite element method for fractional diffusion equation. Sci. World J. (2014). https://doi.org/10.1155/2014/982413CrossRef

39.

Zhou, J., Xu, D.: Alternating direction implicit difference scheme for the multi-term time-fractional integro-differential equation with a weakly singular kernel. Comput. Math. Appl. 79(2), 244–255 (2020)MathSciNetMATHCrossRef

Title: A novel approach for solving multi-term time fractional Volterra–Fredholm partial integro-differential equations
Authors: Sudarshan Santra
Abhilipsa Panda
Jugal Mohapatra
Publication date: 02-12-2021
Publisher: Springer Berlin Heidelberg
Published in: Journal of Applied Mathematics and Computing / Issue 5/2022
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-021-01675-x

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 5/2022

On the structure of 1-generator quasi-polycyclic codes over finite chain rings

Stochastic analysis of a SIRI epidemic model with double saturated rates and relapse

Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation

On the behavior of the solutions of an abstract system of difference equations

Dynamics and control of delayed rumor propagation through social networks

An approximation solution of linear Fredholm integro-differential equation using Collocation and Kantorovich methods

Premium Partner