A comparative study on numerical methods for Singularly Perturbed Boundary Turning Point Problems (SPBTPPs) featuring discontinuous source terms are examined. The study involves developing and analyzing two specific numerical techniques: the finite difference method and a hybrid difference method incorporating a Shishkin-type mesh. This approach demonstrates notable capabilities, exhibiting almost first-order and second-order convergence for the finite difference and hybrid difference methods, respectively. Numerical results are given  to support the theoretical findings.

A uniformly convergent numerical scheme is developed for solving a singularly perturbed parabolic turning point problem. The properties of continuous solutions and the bounds of the derivatives are discussed. Due to the presence of a small parameter as a multiple of the diffusion coefficient, it causes computational difficulty when applying classical numerical methods. As a result, the scheme is formulated using the Crank-Nicolson method in the temporal discretization and an exponentially fitted finite difference method in the space on a uniform mesh. The existence of a unique discrete solution is guaranteed by the comparison principle. The stability and convergence analysis of the method are investigated. Two numerical examples are considered to validate the applicability of the scheme. The numerical results are displayed in tables and graphs to support the theoretical findings. The scheme converges uniformly with order one in space and two in time.

This paper presents a numerical method for solving singularly perturbed parabolic convection-diffusion problems with boundary turning points. As the perturbation parameter $\varepsilon$ approaches zero, the solution shows rapid changes on the left side of the spatial domain, forming a small boundary layer. The classical finite difference methods on uniform meshes fail to capture these oscillations without using a large number of mesh points. To solve this, we use the implicit Euler method for time discretization and a non-standard finite difference method in space. The method satisfies stability, the discrete minimum principle, and $\varepsilon$-uniform convergence. Error estimates show that the proposed method is first-order convergence in time and space. The order of convergence is improved by applying the Richardson extrapolation method. Two model examples are provided to show the scheme's applicability. It demonstrates that the numerical results are in agreement with the theoretical findings.

A three point variable mesh finite difference method of third order, have been derived to solve the singular two-point boundary value problems. The method reduces to a method of order four for the uniform mesh case and may be considered as a modification of the well known Numerov"s method. The methods are self starting and are exact for y=1/x. The convergence of the fourth order method has also been discussed. The various measures of error for solution of two problems, using the methods and the fourth order method in [1] are listed. The numerical results are also compared with five well known classical methods to show the self starting and accuracy of the present methods. Subject Classifications: AMS 64L10, CR: G1.7

We consider a class of singularly perturbed semilinear three-point boundary value problems. An accelerated uniformly convergent numerical method is constructed via the exponential fitted operator method using Richardson extrapolation techniques to solve the problem. To treat the semilinear term, we use quasi-linearization techniques. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε, where the classical numerical methods fail to give a good result. It also improves the results of the methods existing in the literature. The method is shown to be second-order convergent independent of perturbation parameter ε.

In this paper, we investigate the canonical property of solutions of a system of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with a turning point. Using the asymptotic estimates for a special fundamental system of solutions of Sturm-Liouville equation, we study the infinite product representation of solutions of the system and investigate the uniqueness of the solution for the dual equations of the Sturm- Liouville equation. Then, we transform the Sturm-Liouville equation with a turning point to the equation with a singularity, and study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

We consider finite difference method to find best approximation of nonlinear singularly perturbed problem which contains multi-point boundary conditions. We surveyed the asymptotic estimates of the corresponding problem that needs to be solved with maximum principle. We constructed a finite difference scheme by using Bakhvalov mesh. Based on the error estimation, we proved that this method was first-order, uniformly convergent method with in the discrete maximum norm. Finally, we conducted a numerical experiment in order to check the theoretical results