Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

Abstract

In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.

Homotopy perturbation method for solving Caputo‐type fractional‐order Volterra‐Fredholm integro‐differential equations

“…We have provided sufficient conditions under which the present model problem has a unique solution. In addition, we have considered a semi-analytical approach that can be considered as an alternative to other approaches presented in several works, [9][10][11][12][13] based on the homotopy perturbation technique to approximate the solutions of the model equations.…”

“…Here, we are interested in obtaining the solution of fractional nonlinear integro-differential equations by using semi-analytical homotopy-based perturbation methods. [6][7][8] This approach makes the approximation simple for complicated models and can be considered as an alternative way to avoid the numerical discretizations provided in other works [9][10][11][12][13][14][15] for perturbation problems. Furthermore, the present method is not only an effective tool for approximation of the solution but it can also provide the exact solution of certain problems.…”

Using epidemic simulators for monitoring an ongoing epidemic

“…Mathematical treatment of epidemics has its origins more than a century ago 1,2 . The recent interest in modeling epidemics was rekindled with the onset of the severe acute respiratory syndrome (SARS) epidemics during the early part of the century 3 .…”

Optimal Control Approach to Image Registration

“…Note: Algorithms have been produced lately to solve ODEs and PDEs by moving grids, the latest papers which show its effectiveness can be read in [19], [20], [21], [22], [23], [24], [32] and [33]. These references provide the technique that show the moving mesh algorithm and the discrete problem can be patched up to find the solution on adaptive mesh.…”