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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 = {Jianfei Huang, Sadia Arshad, Yandong Jiao and YifaTang}, 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 AU - Jianfei Huang, Sadia Arshad, Yandong Jiao & YifaTang 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 and 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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