Numerical Solution of Singularly Perturbed Two-Point Boundary Value Problem via Liouville-Green Transform

In this paper, authors describe a Liouville-Green transform to solve a singularly perturbed two-point boundary value problem with right end boundary layer in the interval [0, 1]. They reply Liouville-Green transform into original given problem and finds the numerical solution. Then they implemented this method on two linear examples with right end boundary layer which nicely approximate the exact solution.

A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations.Then,a Legendre-based spectral collocation method is developed for solving the transformed system.Therefore,we can make good use of the advantages of the Gauss quadrature rule.We present the construction and analysis of the collocation method.These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations.Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

Optimal L_∞ Estimates for Galerkin Methods for Nonlinear Singular Two-point Boundary Value Problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.

Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid

Matrix methods, now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineers for which the existing methodologies do not give reliable results, these methods are solving them competitively. In this work, a matrix methods is presented for approximate solution of the second-order singularly-perturbed delay differential equations. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given.