Numerical Method to Modify the Fractional-Order Diffusion Equation

Abstract

Time or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equations, and some achievements have been obtained. However, to our knowledge of the literature, up to date, all the proposed numerical methods to modify FODE have achieved at most a second-order time accuracy. In this study, we focus mainly on the numerical methods based on numerical integration in order to modify the fractional-order diffusion equation: $\left(1+\left(1/12\right){\delta }_x^2\right)p_j^k-\left(1+\left(1/12\right){\delta }_x^2\right)p_j^{k-1}={\mu }_{\alpha }{\displaystyle {\sum }_{l=0}^k{\lambda }_{\alpha }^{\left(l\right)}{\delta }_x^2{p}_j^{k-l}}+\hspace{0.17em}{\mu }_{\beta }{\displaystyle {\sum }_{l=0}^k{\lambda }_{\beta }^{\left(l\right)}{\delta }_x^2{p}_j^{k-l}}+\left(\tau /2\right)\left(1+\left(1/12\right){\delta }_x^2\right)\left(f_j^{k-1}+f_j^k\right)$ , $k=\mathrm{1,2},\dots ,K;j=\mathrm{1,2},\dots ,j-1,p_j^0={\omega }_j,j=\mathrm{0,1},\dots ,J$ , $p_0^k=\phi \left(t_k\right),p_j^k=\psi \left(t_k\right),k=\mathrm{0,1},\dots ,K,f_j^l=f\left(x_j,t_l\right),{\omega }_j=\omega \left(x_j\right)$ . Accordingly, numerical methods can be built to modify FODE with second-order time accuracy and fourth-order spatial accuracy in $\left(\partial p\left(x,t\right)/\partial t\right)=\left(\left({\partial }^{1-\alpha }/\partial t^{1-\alpha }\right)+B\left({\partial }^{1-\beta }/\partial t^{1-\beta }\right)\right)\left({\partial }^{2}p\left(x,t\right)/\partial x^2\right)+f\left(x,t\right),0<t\le \mathrm{1,0}<x<1,p\left(x,0\right)=\mathrm{0,0}\le x\le 1$ . $p\left(0,t\right)=t^2,p\left(1,t\right)=e{t}^2,0\le t\le 1$ . Our suggested method can improve the time precision with a certain value.