Numerical framework for the Caputo time-fractional diffusion equation with fourth order derivative in space

A finite difference scheme along with a fourth-order approximation is examined in this article for finding the solution of time-fractional diffusion equation with fourth-order derivative in space subject to homogeneous and non-homogeneous boundary conditions. Caputo fractional derivative is used to describe the time derivative. The time-fractional diffusion equation of order 0<γ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< \gamma < 1$$\end{document} is transformed into Volterra integral equation which is then approximated by linear interpolation. A novel numerical scheme based on fractional trapezoid formula for time discretization followed by Stephenson’s scheme to discretize the fourth order space derivative is developed for the linear diffusion equation. Afterward, convergence and stability are investigated thoroughly showing that the proposed scheme is unconditionally stable and hold convergence accuracy of order O(τ2+h4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\tau ^{2}+h^{4})$$\end{document}. The numerical examples are presented in accordance with the theoretical results showing the efficiency and accuracy of the presented numerical technique.