Upper and lower solutions in quasilinear parabolic boundary value problems

The authors study a class of initial boundary value problems associated with parabolic quasilinear equations: by introducing special auxiliary functions, upper and lower solutions are obtained, which turn out to be sharp in the sense that they coincide with the solution in particular situations.     

Upper and lower solutions in quasilinear parabolic boundary value problems

The authors study a class of initial boundary value problems associated with parabolic quasilinear equations: by introducing special auxiliary functions, upper and lower solutions are obtained, which turn out to be sharp in the sense that they coincide with the solution in particular situations. To Larry Payne on the occasion of his 80th birthday. Received: February 3, 2004; revised: April 26, 2004 Partially supported by University of Cagliari     

Uniqueness of solutions for third-order two-point boundary value problems

This paper is devoted to study the uniqueness of solutions for the third-order boundary value problems. Differential inequalities and fixed point theorem are used. In particular, nonlinearity term is a L ^p -Carathédory function, p≥1.     

Multiple solutions of two-point nonlinear boundary value problems

A quasi-linear boundary value problem has a solution with properties induced by oscillatory properties of the linear part of an equation. This result is proved for two dimensional systems. Consequences for Φ-Laplacian equations and problems with resonant linear parts are discussed.     

Explicit solutions of two-point boundary value operator problems

This work has been partially supported by a grant from the Conselleria de Cultura, Educació i Ciència de la Generalitat de Valencia     

Nonlinear two-point boundary value problems with multiple solutions

In the first part of this paper we study the convergence of finite difference methods to approximate the maximal solution of problems of the form:u+f (x, u)=0, with boundary conditions eitheru(0)=u(b)=0 oru(0)=u (b)=0, 0<b1. The functionf(x, u) satisfies several conditions that are explicitly given in § 1. This work extends earlier results of Parter (see references at the end).Since this problem has in general more than one solution we develop in the second part two algorithms to approximate solutions characterized by the number of their zeros in (0, 1). We include in the last Section numerical results and some additional comments on the implementation of the algorithms on a digital computer.The research of this author was supported by the Office of Naval Research under contract N^o N-0014-67-A-0128-004.Part of this research was carried on while this author was visiting the Mathematics Research Center, University of Wisconsin, during the summer of 1969.     

Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method

Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.     

Existence of multiple positive solutions for fourth order two-point boundary value problems

The present article tackles two-point boundary value problems for fourth-order differential equations as follows <artwork name="GAPA31044eu1"> Several existence theorems on multiple positive solutions to the problems are obtained, and some examples are given to show the validity of these results.     

Relationship between solutions of families of two-point boundary value problems and Cauchy problems

We carry out a qualitative analysis and suggest a method for the solution of the two-point boundary value problems A _l υ = g, x ? [0, l], l ? (0, L) = B ? R ₊; $$ \alpha v'_x \left| {_{x = + 0} = F_1 (v,l)} \right|_{x = 0} , \beta v'_x \left| {_{x = l - 0} = F_2 (v,l)} \right|_{x = l} , $$ , where α, β ? R, υ is the unknown function, g = g(x) is a given real function, F ₁(y, l) and F ₂(y, l) are known real functions defined on the sets Y ₁ × B and Y ₂ × B, respectively, where Y ₁ ∪ Y ₂ ? R, and A _l is the restriction of A corresponding to the embedding parameter l. (Here A is an operator taking an arbitrary function in the set of function classes defined in the paper to C(B ₁), where B ₁ = [0, L).) The study takes into account the dependence of solutions of various versions of these two-point boundary value problems on the parameter l. We construct algorithms for the reduction of these families of two-point boundary value problems to Cauchy problems for ordinary differential equations and integro-differential equations that contain only first derivatives of the unknown functions with respect to the parameter l.     

Positive solutions of two-point boundary value problems for fractional singular differential equations

For nonlinear fractional differential equations with singularities in the phase variable, we establish tests for the existence of several positive solutions of two-point boundary value problems.     

Newton-like methods for two-point boundary value problems

A class of Newton-like methods for discrete two-point boundary value problems is constructed from the sum equation formulation of the problem. Each step of the Newton-like method can be described as first solving a system of linear algebraic equations. The solution vector of this system gives boundary values to a number of discrete boundary value problems which can be solved explicitly.     

Galerkin-wavelet methods for two-point boundary value problems

Summary Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.This work was supported by National Science Foundation     

Upper and lower solutions for singular problems with nonlinear boundary data

We obtain via Schauder's fixed point theorem new results for singular boundary value problems when our nonlinear term f is allowed to change sign. Our boundary data include the Sturm Liouville condition , and the Stefan condition .     

Bivariational bounds on solutions of two-point boundary-value problems

Bivariational principles for a linear equation in a Hilbert space are used to derive complementary upper and lower bounds on solutions of two-point boundary-value problems. The functional dependence of the bounds is exhibited, and various simplified versions of them are discussed. Illustrative examples are presented, showing encouraging accuracy with simple trial vectors.