A higher order stable numerical approximation for time‐fractional non‐linear Kuramoto–Sivashinsky equation based on quintic B‐‐spline

Abstract

This article deals with designing and analyzing a higher order stable numerical analysis for the time‐fractional Kuramoto–Sivashinsky (K‐S) equation, which is a fourth‐order non‐linear equation. The fractional derivative of order present in the considered problem is taken into Caputo sense and approximated using the scheme. In space direction, the discretization process uses quintic ‐spline functions to approximate the derivatives and the solution of the problem. The present approach is unconditionally stable and is convergent with rate of accuracy , where and denote the space and time step sizes, respectively. We have also noted that the linearized version of the K‐S equation leads the rate of accuracy to . The present approach is also highly effective for the time‐fractional Burgers' equation. We have shown that the present approach provides better accuracy than the scheme with the same computational cost for several linear/non‐linear problems, with classical as well as fractional time derivatives.

Adaptive mesh based efficient approximations for Darcy scale precipitation–dissolution models in porous media

“…We want to highlight that the usual time uniform 25,26 or time graded meshes 27 and adaptive moving mesh strategy based on equidistribution principle in space-time governed by a moving mesh PDE will not work for the present Darcy scale precipitation-dissolution model as the required adaptive time steps based on Matlab inbuilt ODE solver, goes below the machine precision for small values of the diffusion parameter. Hence we need to modify the algorithm accordingly to ensure there is no mesh crossing.…”