Solutions of Nonlinear Integral Equations and Their Application to Singular Perturbation Problems Thesis By
نویسنده
چکیده
Sufficient conditions for the existence and uniqueness, and estimates, of a continuous vector solution y = y(t) to the integral equation y(t) = u(t) + EL(t) Ef t M(s) f(s,y(s)) ds + K(t,s) f(s,y(s)) ds, t 0 are derived. A successive approximation technique involving a double sequence is used in the proof . This integral equation result is applied to the second order singular perturbation problem with differential equation Ex"+ p(t,e) x ' + q(t,e) x + r(t,e) + Eh(t,x,x',E) = o, and boundary conditions b1(e) x(o,e) + b2(e) x'(o,e) c 1(e) x(l,E) + c2(e) x'(l,e) = £ (e), 0 O 0+ , "'e obtain an asymptotic expansion with leading term w(t,e) for a solution to this problem. 1. We are concerned with proving the existence and uniqueness of solutions for certain classes of ordinary differential equations which depend in a singular manner on a small positive parameter €. We are further concerned with describing these solutions by obtaining for them asymptotic expansions uniformly valid in the whole interval as € -> 0+ More precisely, let P = P(€) be a positive function of € with the property that P(€) -> 0 as € -> O+, and let t belong to the interval 00 I. The formal sum ~ x. ( t, €) is said to be a uniform asymptotic expansion • 1 ~ ~= with scale P(€) for x(t,€) if there exists a function A= A(t,€) such that for € -> 0+ and m = 1,2, ••• , m x(t,€) ~ i=1 uniformly for t in I. It is the nature of most perturbation problems, where the small ( 1.1) parameter multiplies the highest derivative in the differential equation, to exhibit non-uniform convergence, as € -> O+, in the neighborhood of some point or points in the interval. Such problems are usually called singular perturbation problems . Most of the literature on singular perturbation problems has been concerned with the case when the non-uniformity occurs at one of the end points of the interval. It is the custom in this case to call the region near this end point a boundary layer in analogy with certain hydrodynamic phenomena. Most singular perturbation problems with linear differential equations exhibit boundary layers, and in this case theoretical means exist for determining when a particular end point is part of a boundary layer (see Wasow [1]). Usually, different analytic expressions are developed for the boundary layers and the rest of the interval (see, e.g., Levin and Levinson [2]).
منابع مشابه
Numerical solutions of two-dimensional linear and nonlinear Volterra integral equations: Homotopy perturbation method and differential transform method
متن کامل
A finite difference method for the smooth solution of linear Volterra integral equations
The present paper proposes a fast numerical method for the linear Volterra integral equations withregular and weakly singular kernels having smooth solutions. This method is based on the approx-imation of the kernel, to simplify the integral operator and then discretization of the simpliedoperator using a forward dierence formula. To analyze and verify the accuracy of the method, weexamine samp...
متن کاملModified homotopy method to solve non-linear integral equations
In this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. Almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. But we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as Legendre ...
متن کاملNumerical Solution of Weakly Singular Ito-Volterra Integral Equations via Operational Matrix Method based on Euler Polynomials
Introduction Many problems which appear in different sciences such as physics, engineering, biology, applied mathematics and different branches can be modeled by using deterministic integral equations. Weakly singular integral equation is one of the principle type of integral equations which was introduced by Abel for the first time. These problems are often dependent on a noise source which a...
متن کاملSolving infinite system of nonlinear integral equations by using F-generalized Meir-Keeler condensing operators, measure of noncompactness and modified homotopy perturbation.
In this article to prove existence of solution of infinite system of nonlinear integral equations, we consider the space of solution containing all convergence sequences with a finite limit, as with a suitable norm is a Banach space. By creating a generalization of Meir-Keeler condensing operators which is named as F-generalized Meir-Keeler condensing operators and measure of noncompactness, we...
متن کاملSolving singular integral equations by using orthogonal polynomials
In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...
متن کامل