An Analytical Solution of Linear/Nonlinear Fractional-Order Partial Differential Equations and with New Existence and Un
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RESEARCH ARTICLE
An Analytical Solution of Linear/Nonlinear Fractional-Order Partial Differential Equations and with New Existence and Uniqueness Conditions Pratibha Verma1 • Manoj Kumar1
Received: 28 August 2019 / Revised: 15 August 2020 / Accepted: 22 October 2020 Ó The National Academy of Sciences, India 2020
Abstract In this paper, we have adopted the two-step Adomian decomposition method (TSADM) for solving linear and nonlinear partial differential equations of fractional order and compared with the variational iteration method (VIM) and Adomian decomposition method (ADM). It is successfully applied to linear and nonlinear partial differential equations involving variable coefficients. It is shown that the TSADM is a more effective and promising method and provides an exact solution with one iteration of the problems without discretization and linearization. Keywords Adomian decomposition method Two-step adomian decomposition method (TSADM) Fractional partial differential equation Caputo fractional derivative
Significance Statement The concept TSADM has been extended to provide the analytical solutions for initial value problems of linear and nonlinear partial differential equations of fractional order and compared with existing numerical methods. & Pratibha Verma [email protected] Manoj Kumar [email protected] 1
Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh 211004, India
1 Introduction Ordinary and partial differential equations of integral/ derivative type are studied in different cases because they appear in various fields of science and engineering [1]. In general, the exact solution is not available for the solution of nonlinear fractional differential equations; for this reason, numerical methods [2–5] help to solve such problems. Because of non-local character in fractional derivatives, the geometrical interpretation is not evident. The analytical results based on the existence and uniqueness of solutions of differential equations of fractional order had been investigated by different authors [1, 2, 6, 9–11]. In the existing literature, many analytical and numerical methods have been developed to solve ordinary/partial fractional differential equations, such as the homotopy analysis method, the homotopy perturbation method, spectral methods, and variational iteration method, Adomian decomposition method, and several others [6–8]. Zaid and Shaher [2] implemented methods like Adomian decomposition method and variational iteration method for solving nonlinear partial differential equations involving fractional derivatives. The Adomian decomposition method (ADM) [9] has been proven to be an effective method and successfully implemented on various problems of fractional orders. Using the ADM, we obtain a series solution, but in practice, we approximate the solution by a truncated series. The series sometimes coincides with the Taylor expansion of the exact solution in the neighborhood about the initial point for initial value
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