Many physical processes appear to exhibit fractional order behavior that may vary with time or space.The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a ...Many physical processes appear to exhibit fractional order behavior that may vary with time or space.The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable.Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive.This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain.It is worth mentioning that here we use the Coimbradefinition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems.An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated.Finally,numerical examples are provided to show that the implicit Euler approximation is computationally efficient.展开更多
基金supported by the National NSF of China 11201077 and 11001090the Fujian Provincial Department of Education Fund Class A JA11034,China+1 种基金the Talent Foundation of Fuzhou University XRC-0811,Chinathe Natural Science Foundation of Fujian province 2013J05005.
文摘Many physical processes appear to exhibit fractional order behavior that may vary with time or space.The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable.Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive.This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain.It is worth mentioning that here we use the Coimbradefinition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems.An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated.Finally,numerical examples are provided to show that the implicit Euler approximation is computationally efficient.
