scholarly journals Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

2021 ◽  
Vol 94 ◽  
pp. 136-154
Author(s):  
Huan-Yan Jian ◽  
Ting-Zhu Huang ◽  
Xian-Ming Gu ◽  
Xi-Le Zhao ◽  
Yong-Liang Zhao
Keyword(s):  
Wave Equations ◽  
Fractional Diffusion ◽  
Second Order ◽  
Riesz Space ◽  
Difference Schemes ◽  
Diffusion Wave ◽  
Implicit Difference Schemes ◽  
Distributed Order

scholarly journals Subordination in a class of generalized time-fractional diffusion-wave equations

2018 ◽  
Vol 21 (4) ◽  
pp. 869-900 ◽  
Author(s):  
Bazhlekova Emilia
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Bernstein Function ◽  
Present Survey ◽  
Diffusion Wave ◽  
Survey Paper ◽  
Completely Monotone ◽  
Generalized Time ◽  
Distributed Order

Abstract Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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