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

• Published in: Journal of applied mathematics & computing Vol. 68; no. 5; pp. 3295 - 3316
• Main Authors: Arshad, Sadia, Wali, Mubashara, Huang, Jianfei, Khalid, Sadia, Akbar, Nosheen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2022; Springer Nature B.V
• Subjects: Accuracy; Applied mathematics; Approximation; Boundary conditions; Computational Mathematics and Numerical Analysis; Convergence; Diffusion; Finite difference method; Interpolation; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Theory of Computation; Volterra integral equations; Caputo derivative; 34K28; 65N06; 35R11; Fractional diffusion equation; Stability analysis; Stephenson’s scheme; 65M12; Fractional trapezoid formula; Finite difference method; Convergence analysis
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-021-01635-5

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 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 ( τ 2 + h 4 ) . The numerical examples are presented in accordance with the theoretical results showing the efficiency and accuracy of the presented numerical technique.