Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo–Fabrizio fractional derivative

Abstract

AbstractWe consider distributed-order partial differential equations with time fractional derivative proposed by Caputo and Fabrizio in a one-dimensional space. Two finite difference schemes are established via Grünwald formula. We show that these two schemes are unconditionally stable with convergence rates $O(\tau ^{2}+h^{2}+ \Delta \alpha ^{2})$O(τ2+h2+Δα2) and $O(\tau ^{2}+h^{4}+\Delta \alpha ^{4})$O(τ2+h4+Δα4) in discrete $L^{2}$L2, respectively, where Δα, h, and τ are step sizes for distributed-order, space, and time variables, respectively. Finally, the performance of difference schemes is illustrated via numerical examples.