Convergence and stability estimates in difference setting for time‐fractional parabolic equations with functional delay

Abstract

A class of one‐dimensional time‐fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work.

Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay

“…The key role in the convergence analysis of the schemes is the fractional Grönwall-type inequations. However, as pointed out in [47][48][49], the similar fractional Grönwall-type inequations can not be directly applied to study the convergence of numerical schemes for the nonlinear timefractional problems with delay.…”

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

“…By invoking the works in [37,43] and due to the nonlocality of time Caputo fractional derivatives which need high computational cost and storage, we can improve our approach in the near future by presenting a high-order scheme based on the sum of exponential functions technique to speed up the evaluation. Furthermore, recalling the methodologies in [37,44] side by side to the numerical analysis in [35] and the appropriate discrete Grönwall inequality in [35,45], the unconditional convergence and stability estimates with out any constraints on time and space steps can be deduced. Assuming that a 0 ≤ a(x 1 , x 2 ) ≤ a 1 and b 0 ≤ b(x 1 , x 2 ) ≤ b 1 , such that a 0 , b 0 and a 1 , b 1 are positive constants, is essential to prove the stability and convergence estimates which will be devoted to a new study in the near future.…”

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

“…Extending the L1 discrete fractional Grönwall inequality to deal with Convergence and stability estimates for time-fractional parabolic equations with functional delay, was done in [12]. Also, the L1 discrete fractional Grönwall inequality was used successfully to prove the convergence and stability estimates for the difference/Galerkin Legendre spectral scheme applied for nonlinear time and space fractional diffusion equation with smooth solutions [27].…”