Yapman, Ömer; Amiraliyev, Gabil M.

In this paper, we deal with the second-order accurate homogeneous (non-hybrid) type difference scheme for solving a singularly perturbed first-order Volterra integro-differential equation. It is shown that the method displays uniform convergence of O(N−2 ln N) on a special non-uniform mesh, where N is the mesh parameter. Numerical results are included to verify the theoretical estimates.

Cakir, Musa; Cimen, Erkan; Amirali, Ilhame; Amiraliyev, Gabil M.

The paper deals with the singularly perturbed quasilinear initial value problem exhibiting initial layer. First the nature of solution of differential problem before presenting method for its numerical solution is discussed. The numerical solution of the problem is performed with the use of a finite-fitted difference scheme on an appropriate piecewise uniform mesh (Shishkin-type mesh). An error analysis shows that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Finally, numerical results supporting the theory are presented.