Numerical and analytical investigations for solving the inverse tempered fractional diffusion equation via interpolating
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Numerical and analytical investigations for solving the inverse tempered fractional diffusion equation via interpolating element‑free Galerkin (IEFG) method Mostafa Abbaszadeh1 · Mehdi Dehghan1 Received: 7 March 2020 / Accepted: 22 July 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract This manuscript is devoted to analysis of a novel meshless numerical procedure for solving the inverse tempered fractional diffusion equation. The employed numerical technique is based on a modification of element-free Galerkin (EFG) method, and the shape functions of interpolating moving least squares approximation are utilized for ingredients of the test and trial functions. At the first stage, the time derivative is discretized by a Crank–Nicolson idea to derive a semi-discrete scheme. In the next stage, the space variable is approximated by the EFG procedure. The convergence rate and stability of the timediscrete formulation are analyzed. Furthermore, the error estimate of the full-discrete plan is discussed in detail. In the end, some numerical experiments are investigated to check the theoretical results and the efficiency of the developed technique. Keywords Inverse tempered fractional diffusion equation · Element-free Galerkin (EFG) method · Interpolating MLS · Error estimation · Convergence · Stability Mathematics Subject Classification 65M70 · 34A34
Introduction Belytschko and his co-workers introduced the element free Galerkin (EFG) method [1–3]. For the first time, IMLS shape functions (IMLSSFs) are introduced in [4–6] and they are employed in EFG method to propose a new meshless weak form procedure that is well known as the interpolating EFG (IEFG) method. A simply way is presented in [5–7] to construct the IMLSSFs for solving 2D potential and elasticity problems based on improved boundary element-free method. Authors of [8, 9] developed an improved IMLS to simulate 2D elasticity and potential problems. Interpolation error of IMLS is studied in [10, 11]. An improved IEFG method based on the nonsingular mass function is proposed in [12] for solving * Mostafa Abbaszadeh [email protected] Mehdi Dehghan [email protected] 1
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran 15914, Iran
elastoplasticity model. Authors of [13] proposed an improved complex variable moving least squares approximation by constructing a new functional with an explicit physical meaning. Authors of [14] presented the boundary integral equation for solving 2D elastodynamic models. Based on the combined field integral equation (CFIE) and the complex variable moving least squares (CVMLS) approximation, a complex variable boundary element-free method is developed in [15] for the exterior Neumann problem for the Helmholtz equation. A complex variable boundary point interpolation method is developed in [16] for solving the nonlinear Signorini problem. Authors of [17] proposed a fundamental solutions-based MLS approximation, named as an augm
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