Existence and uniqueness results and analytical solution of the multi-dimensional Riesz space distributed-order advectio
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ORIGINAL ARTICLE
Existence and uniqueness results and analytical solution of the multi‑dimensional Riesz space distributed‑order advection–diffusion equation via two‑step Adomian decomposition method Pratibha Verma1 · Manoj Kumar1 Received: 28 July 2020 / Accepted: 6 October 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this article, we introduced for the first time the two-step Adomian decomposition method (TSADM) for solving the multi-dimensional Riesz space distributed-order advection–diffusion (RSDOAD) equation. The TSADM was successfully applied to obtain the analytical solution of the multi-dimensional (RSDOAD) equation. The analytical solution has been obtained without approximation/discretization of the Riesz fractional operator. Furthermore, new results for the existence are obtained with the help of some fixed point theorems, while the uniqueness of the solution was investigated employing the Banach contraction principle. Finally, we included a generalized example to demonstrate the validity and application of the proposed method. The obtained results conclude that the proposed method is powerful and efficient for the considered problem compared to the other existing methods. Keywords Fractional derivatives · Riesz space distributed-order advection–diffusion equation · Riesz derivative, Two-step Adomian decomposition method · Fixed point theorem Mathematics Subject Classification 26A33 · 35R11 · 47H10
1 Introduction The theory of fractional calculus has been more focused and getting attention of the researches in the different areas of science and engineering [1–3]. Fractional calculus is an updated version of the standard calculus, in which the the operators (derivative/integral) are of fractional order. Nowadays, studies of fractional calculus have received considerable attention [4–8]. The authors proposed a unified method to solve time fractional Burgers’ equation with the Caputo derivative. Using numerical methods, the authors obtained
* Manoj Kumar [email protected] Pratibha Verma [email protected] 1
Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh 211004, India
approximate solution and compared the obtained results with the exact [9, 10]. Furthermore, the analytical solution of fractional differential equations are difficult to find and has not been the focus of much attention. Therefore, a numerical approach is needed for solving fractional differential equations (FDEs) [11, 12]. The most developed methods for the numerical approximation of FDEs are spectral methods, spectral collocation method, Adomian decompostion method (ADM), improved collection method, Sinc collocation methods, and so on. Many works have done on the existence and uniqueness of the FDEs. The most urgent problems listed are how to confirm the existence and uniqueness of the FDEs for the solution and find the existence of the method which provides the analytical solution of the FDEs [13–15]. For the first time, Caputo i
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