An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

“…The primary objective of our paper is to evaluate the performance, validity, efficiency, and accuracy of the developed techniques. We confirm this by comparing the computed results with existing numerical solutions [13,42,43,59] and analytical solutions [40,41,44]. To assess the convergence and accuracy of the developed methods, we use the error computation method outlined in [13,42,43,59]:…”

“…Tables 2 and 3 investigate the effect of values on the accuracy of DSCDQM-RSK and DSCDQM-DLK such as regularized Shannon factor (σ), computational parameter (τ), and step size (∆). Table 2 explains that DSCDQM-RSK is more accurate than DSCDQM-DLK for computing the solute concentration υ(x, t), compared with earlier numerical [13,42,43,59] and exact [40,41,44] solutions. Also, the bandwidth (2M + 1 = 7) and (σ = 1.45 × ∆) are the most suitable choices for numerical results for problem (4.1), which achieve more efficient results.…”

“…In addition, Tables 2 and 3 exhibit that increasing the time levels (L) and decreasing the step size (δt) for each block (δt = 0.1, L = 12) lead to more efficiency at a lower bandwidth. Exact [40,41,44] 4.5929…”

“…Additionally, we design a MATLAB code for each approach to obtain a numerical solution for three problems to be examined. When compared to earlier analytical [40,41] and numerical [13,42,43] methodologies, the derived numerical results attain excellent efficiency and accuracy. Furthermore, we present some parametric studies to highlight the reliability of our methods with the influence of fractional order derivative, the velocity, and positive diffusion parameters on the results.…”

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

“…To resolve this issue, Zeng et al theoretically and numerically shown that the accuracy of numerical solution can be efficiently improved with only a few correction terms [19]. Since then, a variety of numerical schemes based on the addition of correction terms have emerged for fractional problems with nonsmooth solutions, see [3,20,21]. To the best of our knowledge, it seems that the method of adding correction terms with DST for solving equation (1) has not been considered in the existing literatures yet.…”