Two unconditionally stable and convergent difference schemes with the extrapolation method for the one‐dimensional distributed‐order differential equations

Abstract

The Grünwald formula is used to solve the one-dimensional distributed-order differential equations. Two difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(τ + h 2 + α 2 ) and O(τ + h 4 + α 4 ) in maximum norm, respectively, where τ , h and α are step sizes in time, space and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders O(τ 2 + h 2 + α 2 ) and O(τ 2 + h 4 + α 4 ), respectively. An illustrative numerical example is given to confirm the theoretical results.

Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L2 Regularization

“…The authors of [20][21][22] also proved the existence, uniqueness, and regularity properties for the weak solutions of DOTFDEs. In addition, research on numerical calculations of DOTFDE can be found in [23][24][25][26][27][28][29][30].…”

A kernel‐based pseudo‐spectral method for multi‐term and distributed order time‐fractional diffusion equations

“…The have appeared in the physical modeling of some problems of engineering and science, such as in describing dielectric induction, diffusion dielectric induction and diffusion [20], in dynamic systems [21], in modeling the ultra-slow diffusion/accelerating super diffusion which particles spread in a logarithmic rate [22][23][24], in describing transport phenomena in complex heterogeneous media with multi-fractal properties [25], and so on. A variety of numerical methods have been proposed for distributed-order TFDEs [26][27][28][29][30][31][32][33][34][35].…”

Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations

“…Very recently, we are devoted to the investigation of numerical solutions to the time distributedorder fractional partial differential equations. In [21], for the one-dimensional time distributed-order fractional diffusion equations, the composite trapezoid rule was used to approximate the distributed integral and then the first-order Grünwald formula was applied to discretize the involved time-fractional derivatives together with the second-order approximation for the spatial second-order derivatives. A difference scheme with the second-order accuracy in all variables was derived, and the stability along with the convergence was proved by the maximum principle.…”