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 \(L1-2\) $L1-2$ approximation approach approximates the fractional derivatives of orders \(\upeta ,\, \upeta _i \in (0,1)\) $\mathrm{\eta },{\mathrm{\eta }}_i\in (0,1)$ that are present in the considered problem. This approach provides a higher accuracy of \(O(k^{3-\max \{\upeta ,\upeta _i\}})\) $O(k^{3-max\{\mathrm{\eta },{\mathrm{\eta }}_i\}})$ 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.