A higher order unconditionally stable numerical technique for multi-term time-fractional diffusion and advection-diffusion equations

Constructing a higher order collocation framework for solving the Caputo multi-term time-fractional advection-diffusion and diffusion-type problems is the primary objective of this work, which has influenced the field of scientific disciplines. Advection-diffusion and reaction-diffusion equations were developed by modeling scientific phenomena in fluid flow issues, solid oxide fuel cells, and solvent diffusion into heavy oils. As a result, numerical solutions to these problems have garnered significant attention. The L1-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L1-2$$\end{document} approximation approach approximates the fractional derivatives of orders eta,eta i is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upeta ,\, \upeta _i \in (0,1)$$\end{document} that are present in the considered problem. This approach provides a higher accuracy of O(k3-max{eta,eta i})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(k<^>{3-\max \{\upeta ,\upeta _i\}})$$\end{document} in time direction. Fourth-order convergence in space is achieved by employing a spline collocation technique with trigonometric quintic splines. Results from applying the suggested computational approach to four test examples have demonstrated its superiority and validity.