Shehu Adomian Decomposition Method for Solving Fractional Integro-Differential Equation
Abstract
In this paper, we apply Shehu transform and Adomian decomposition method to find the approximate solution of nonlinear fractional Volterra integro-differential and fractional Volterra-Fredholm integro-differential equation. The fractional derivative is described in Caputo sense. Finally, we provide some applications to validate the efficiency and the high accuracy of this technique. Keywords : Shehu transform ; Adomian polynomial ; integro-differential equation ; fractional derivativeReferences
Belgacem, R., Baleanu, D., & Bokhari, A. (2019). Shehu transform and applications to Caputo-fractional
differential equations. International Journal of Analysis and Applications, 17(6), 917-927. DOI:10.28924/2291-8639-17-2019-917
Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency in dependent-II, Geophys. J. Roy.
Astron. Soc., 13, 529-539.
Das, P., Rana S., & Ramos, H. (2019). Homotopy perturbation method for solving Caputo-type fractional-order
Volterra-Fredholm integro-differential equations. Computational and Mathematical Methods, 1(5), 1-9. DOI:10.1002/cmm4.1047
Duan, J.-S., Rach, R., Baleanu, D., & Wazwaz, A.-M. (2012). A review of Adomian decomposition method and its
applications to fractional di®erential equations. Commun. Frac. Calc., 3(2), 73-99.
Hamoud, A. A., & Ghadle, K. P. (2018). Modifed Laplace decomposition method for fractional Volterra-Fredholm
integro-diferential equations. Journal of Mathematical Modeling, 6(1), 91-104.
Hamoud, A. A., Abdo M. S., & Ghadle K. P. (2018). Existence and uniqueness results for Caputo fractional
integro-differential equation. J. Ksiam, 12(3), 163-177.
Hamoud, A. A., Ghadle K. P., Issa M.Sh. B., & Giniswamy. (2018). Existence and uniqueness theorems for
fractional Volterra-Fredholm integro-differential equations. International Journal of Applied Mathematics, 31(3), 333-348.
Khan. H., Farooq, U., Shah, R., Baleanu. D., Kumam, P., & Arif M. (2020). Analytical Solutions of (2+Time
Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method. Appl. Sci, 10(122), 1-20. DOI:10.3390/app10010122
Maitama, S., & Zhao, W. (2019). New homotopy analysis transform method for solving multidimensional
fractional diffusion equations. Arab Journal of Basic and Applied Sciences, 27(1), 27-44. DOI: 10.1080/
25765299.2019.1706234
Maitama, S., & Zhao, W. (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace
transform for solving differential equations. International Journal of Analysis and Applications, 7(2),
167-190. DOI:10.28924/2291-8639-17-2019-167
Noor, M. A. (2010). Iterative Methods for Nonlinear Equations Using Homotopy Perturbation Technique. Applied
Mathematics & Information Sciences, 4(2), 227-235.
Qureshi, S., & Kumar P. (2019). Using Shehu integral transform to solve fractional order Caputo type initial value
problems. Journal of Applied Mathematics and Computational Mechanics, 18(2), 75-83. DOI:10.17512/
jamcm.2019.2.07
Rawashdeh, E.A. (2011). Legendre Wavelets Method for Fractional Integro-Differential Equations. Applied
Mathematical Sciences, 5, 2467-2474.
Wazwaz, A.-M. (1999). A reliable modification of Adomian decomposition method. Applied Mathematics and
Computation, 102, 77-86.
Yang, C., & Hou, J. (2013). Numerical solution of integro-differential equations of fractional order by Laplace
decomposition method. Wseas Transactions on Mathematics, 12(12), 1173-1183.
differential equations. International Journal of Analysis and Applications, 17(6), 917-927. DOI:10.28924/2291-8639-17-2019-917
Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency in dependent-II, Geophys. J. Roy.
Astron. Soc., 13, 529-539.
Das, P., Rana S., & Ramos, H. (2019). Homotopy perturbation method for solving Caputo-type fractional-order
Volterra-Fredholm integro-differential equations. Computational and Mathematical Methods, 1(5), 1-9. DOI:10.1002/cmm4.1047
Duan, J.-S., Rach, R., Baleanu, D., & Wazwaz, A.-M. (2012). A review of Adomian decomposition method and its
applications to fractional di®erential equations. Commun. Frac. Calc., 3(2), 73-99.
Hamoud, A. A., & Ghadle, K. P. (2018). Modifed Laplace decomposition method for fractional Volterra-Fredholm
integro-diferential equations. Journal of Mathematical Modeling, 6(1), 91-104.
Hamoud, A. A., Abdo M. S., & Ghadle K. P. (2018). Existence and uniqueness results for Caputo fractional
integro-differential equation. J. Ksiam, 12(3), 163-177.
Hamoud, A. A., Ghadle K. P., Issa M.Sh. B., & Giniswamy. (2018). Existence and uniqueness theorems for
fractional Volterra-Fredholm integro-differential equations. International Journal of Applied Mathematics, 31(3), 333-348.
Khan. H., Farooq, U., Shah, R., Baleanu. D., Kumam, P., & Arif M. (2020). Analytical Solutions of (2+Time
Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method. Appl. Sci, 10(122), 1-20. DOI:10.3390/app10010122
Maitama, S., & Zhao, W. (2019). New homotopy analysis transform method for solving multidimensional
fractional diffusion equations. Arab Journal of Basic and Applied Sciences, 27(1), 27-44. DOI: 10.1080/
25765299.2019.1706234
Maitama, S., & Zhao, W. (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace
transform for solving differential equations. International Journal of Analysis and Applications, 7(2),
167-190. DOI:10.28924/2291-8639-17-2019-167
Noor, M. A. (2010). Iterative Methods for Nonlinear Equations Using Homotopy Perturbation Technique. Applied
Mathematics & Information Sciences, 4(2), 227-235.
Qureshi, S., & Kumar P. (2019). Using Shehu integral transform to solve fractional order Caputo type initial value
problems. Journal of Applied Mathematics and Computational Mechanics, 18(2), 75-83. DOI:10.17512/
jamcm.2019.2.07
Rawashdeh, E.A. (2011). Legendre Wavelets Method for Fractional Integro-Differential Equations. Applied
Mathematical Sciences, 5, 2467-2474.
Wazwaz, A.-M. (1999). A reliable modification of Adomian decomposition method. Applied Mathematics and
Computation, 102, 77-86.
Yang, C., & Hou, J. (2013). Numerical solution of integro-differential equations of fractional order by Laplace
decomposition method. Wseas Transactions on Mathematics, 12(12), 1173-1183.
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2021-01-05
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