Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations
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Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations Jalil Rashidinia1
· Elham Mohmedi1
Received: 9 February 2020 / Revised: 11 June 2020 / Accepted: 28 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract We develop a numerical scheme for finding the approximate solution for one- and twodimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey. Keywords Multi-term time fractional diffusion and diffusion-wave equations · Caputo derivative · Legendre collocation method · Convergence analysis Mathematics Subject Classification 65M10 · 78A48
1 Introduction Differential equations are usually drawn by so many applied real-world problems in science and engineering challenges. Many scientists, especially including mathematicians and engineering scientists are very interested in fractional-based calculus. Fractional calculus is conventionally used in many applications of science and engineering, for instance it can be utilized in modeling phenomena in fluid flows, rheology, relaxation, oscillation, anomalous diffusion, reaction–diffusion turbulence, diffusive transport akin to diffusion, electric
Communicated by José Tenreiro Machado.
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Jalil Rashidinia [email protected] Elham Mohmedi [email protected]
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School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran 0123456789().: V,-vol
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networks, polymer physics, chemical physics, electrochemistry of corrosion, relaxation processes in complex non-linear systems, propagation of seismic waves, and dynamic processes. To describe the mentioned important aspects of such physical phenomena, fractional partial differential equations have been effectively applied (Uchaikin 2013; Richard 2014; Povstenko 2015). Though, studies of the multi-term time fractional diffusion-wave equations are still under progress and development (Liu et al. 2013). Multi-term FDEs have been commonly studied in rheology, mechanical models as well as many related areas (Schiessel et al. 1995; Srivastava and Rai 2010; Ren and Liu 2019; Li et al. 2018a). Most fractional differential equations do not have exact analytical solutions, so for tackling such approximation and numerical schemes
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