A Higher-Order Uniformly Convergent Numerical Method for a Singularly Perturbed Nonlinear Reaction–Diffusion Equation

This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We will prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.

<p>In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.</p>

In this work, a new access for solving fourth-order singular boundary value problem (SBVP) by using sextic B-spline is presented. The SBVP yields in mathematical modelling of several real-world development essentially electro-hydrodynamics, chemical reactions, thermal explosions, fluid dynamics, aerodynamics and atomic nuclear reactions. It must be indicated that the original non-linear boundary value problem (BVP) is first remodelled into a linear BVP by quasilinearization technique. Convergence analysis of the method is explored. Three trial evidences are examined to determine the accuracy and suitability of the described technique and to validate the analytical results. The proposed method is simple, robust and straightforward for solving the non-linear SBVP.

A numerical method for solving linear two-point boundary-value problems with time varying coefficients is presented in this paper. The method is based upon constructing the Nth degree polynomial interpolation using Legendre-Gauss-Lobatto nodes to approximate the solution of linear two-point value problems with time-varying coefficients. The method can be applied to obtain the optimal control of linear time-varying systems subject to a quadratic cost criteria. The two-point value problem is thereby transformed into a linear programming problem. The method is efficient and yields very accurate results. Illustrative examples are included to demonstrate the accuracy of the proposed method.

In this paper, a collocation method based on Laguerre wavelets is proposed for the numerical solutions of linear and nonlinear singular boundary value problems. Laguerre wavelet expansions together with operational matrix of integration are used to convert the problems into systems of algebraic equations which can be efficiently solved by suitable solvers. Illustrative examples are given to demonstrate the validity and applicability of this technique, and the results have been compared with the exact solutions.

The main objective of this paper is to develop new spectral solutions for linear and nonlinear fifth-order boundary value problems. For this aim, a new operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The key idea for obtaining the suggested spectral solutions is to convert the fifth-order linear and nonlinear boundary value problems into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. A theoretical error bound is derived and the convergence analysis of the proposed expansion is proved. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the suggested algorithms.

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π .

In this study, we applied the homotopy perturbation (HP) method for solving linear and nonlinear fourth-order boundary value problems. The analytical results of the boundary value problems have been obtained in terms of a convergent series with easily computable components. Comparisons between the results of the HP method and the analytical solution showed that this method gives very precise results with a few terms. In the implied HP method, some unknown parameters in the initial guess are introduced, which are identified after applying boundary conditions. This improvement results in higher accuracy.

Linear two-point boundary value problems defined on an infinite domain are converted to initial-value problems using invariant imbedding. The resulting Riccati equations are integrated numerically until the desired accuracy is obtained. Several criteria for determining the appropriate length of integration are presented. Several example problems are presented.

Questions of existence, uniqueness, and continuous dependence for weak solutions of linear hyperbolic boundary value problems are considered. The differential equations have the form utt + Au=f, where A is elliptic in the spatial variables, and the boundary conditions are homogeneous in both space and time. Resolution of these questions depends on the relationship of the eigenvalues of A and those of an associated scalar problem in time.

A numerical algorithm for solving a class of linear nonlocal boundary value problems

A computer program is described which has been developed for obtaining approximate solutions to linear initial and boundary value problems involving differential equations. The program can be used for ordinary curve fitting as well. For each problem, input to the program includes:1. The equations (in symbolic form) to be satisfied - the differential equations, equations describing auxiliary conditions such as boundary conditions, etc.2. A numerical description of the regions in which each of the equations are to be satisfied.3. Sets of functions (in symbolic form) to be used in linear combinations to approximate the solution functions.Given the above input, the program generates an approximation to the solutuon of the specified problem in terms of the specified functions which is optimum in the least squares sense. A number of examples illustrating the use of the program are included in the paper.

An invariant imbedding algorithm for the solution of inhomogeneous linear two-point boundary value problems

Factorization of linear elliptic boundary value problems in non-cylindrical domains

4 - Boundary Value Problems