New method for solving fractional partial integro-differential equations by combination of Laplace transform and resolvent kernel method |
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| Affiliation: | 1. Foundation in Engineering, Faculty of Science and Engineering, University of Nottingham Malaysia, Malaysia;2. Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn Malaysia, Malaysia;3. Faculty of Electrical and Electronic Engineering, Universiti Tun Hussein Onn Malaysia, Malaysia;1. Department of Mathematics, University of Peshawar, Pakistan;2. Department of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan;3. Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, Roma 00186, Italy;4. Department of Mathematics, University of Malakand, Pakistan;5. Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5 Street, Poznan 60-965, Poland;1. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt;2. Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia;1. School of Applied Mathematics, Guangdong University of Technology, Guangdong, Guangzhou 510006, China;2. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Hunan, Changsha 410081, China |
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| Abstract: | In this article, we obtained the approximate solution for a new class of Time-Fractional Partial Integro-Differential Equation (TFPIDE) of the Caputo-Volterra type in which the integral is not limited to the convolution type. This new class of TFPIDE is distinct from the common problem with the convolution integral kernel. The general expression of the analytical solution for this special type of TFPIDE was derived using a combination of Laplace transform and the resolvent kernel method. In the process, Laplace transform will transform the equation into a second kind Volterra integral equation in terms of the transformed function. Two main problems in deriving the approximate analytical solutions were identified as Case I and Case II problems. To obtain the approximate solutions for Case I and Case II problems, numerical methods were designed based on approximation of the resolvent kernel with truncated Neumann series as well as approximation of the Laplace transform based on truncated Taylor series. Several numerical examples are presented to indicate the plausibility, mechanism and performance of the proposed methods. |
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