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 - <p style="text-align: justify;">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 $t$<sub>$k$</sub><sub>−1/2&nbsp;</sub>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.</p>