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Volume 9, Issue 3
Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations

Jianfei Huang, Sadia Arshad, Yandong Jiao & YifaTang

DOI: 10.4208/eajam.230718.131018

East Asian J. Appl. Math., 9 (2019), pp. 538-557.

Published online: 2019-06

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  • Abstract

Two second-order convolution quadrature methods for fractional nonlinear diffusion-wave equations with Caputo derivative in time and Riesz derivative in space are constructed. To improve the numerical stability, the fractional diffusion-wave equations are firstly transformed into equivalent partial integro-differential equations. Then, a second-order convolution quadrature is applied to approximate the Riemann-Liouville integral. This deduced convolution quadrature method can handle solutions with low regularity in time. In addition, another second-order convolution quadrature method based on a new second-order approximation for discretising the Riemann-Liouville integral at time $t$$k$−1/2 is constructed. This method reduces computational complexity if Crank-Nicolson technique is used. The stability and convergence of the methods are rigorously proved. Numerical experiments support the theoretical results.

  • Keywords

Fractional diffusion-wave equation, nonlinear source, convolution quadrature, generating function, stability and convergence.

  • AMS Subject Headings

65M06, 65M12, 35R1

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-9-538, author = {}, title = {Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2019}, volume = {9}, number = {3}, pages = {538--557}, abstract = {

Two second-order convolution quadrature methods for fractional nonlinear diffusion-wave equations with Caputo derivative in time and Riesz derivative in space are constructed. To improve the numerical stability, the fractional diffusion-wave equations are firstly transformed into equivalent partial integro-differential equations. Then, a second-order convolution quadrature is applied to approximate the Riemann-Liouville integral. This deduced convolution quadrature method can handle solutions with low regularity in time. In addition, another second-order convolution quadrature method based on a new second-order approximation for discretising the Riemann-Liouville integral at time $t$$k$−1/2 is constructed. This method reduces computational complexity if Crank-Nicolson technique is used. The stability and convergence of the methods are rigorously proved. Numerical experiments support the theoretical results.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.230718.131018 }, url = {http://global-sci.org/intro/article_detail/eajam/13166.html} }
TY - JOUR T1 - Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations JO - East Asian Journal on Applied Mathematics VL - 3 SP - 538 EP - 557 PY - 2019 DA - 2019/06 SN - 9 DO - http://doi.org/10.4208/eajam.230718.131018 UR - https://global-sci.org/intro/article_detail/eajam/13166.html KW - Fractional diffusion-wave equation, nonlinear source, convolution quadrature, generating function, stability and convergence. AB -

Two second-order convolution quadrature methods for fractional nonlinear diffusion-wave equations with Caputo derivative in time and Riesz derivative in space are constructed. To improve the numerical stability, the fractional diffusion-wave equations are firstly transformed into equivalent partial integro-differential equations. Then, a second-order convolution quadrature is applied to approximate the Riemann-Liouville integral. This deduced convolution quadrature method can handle solutions with low regularity in time. In addition, another second-order convolution quadrature method based on a new second-order approximation for discretising the Riemann-Liouville integral at time $t$$k$−1/2 is constructed. This method reduces computational complexity if Crank-Nicolson technique is used. The stability and convergence of the methods are rigorously proved. Numerical experiments support the theoretical results.

Jianfei Huang, Sadia Arshad, Yandong Jiao & YifaTang. (2019). Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations. East Asian Journal on Applied Mathematics. 9 (3). 538-557. doi:10.4208/eajam.230718.131018
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