Nonlinear Second-order Boundary Value Problems Research Articles

Justification of perturbation algorithm for weakly nonlinear elliptic second-order boundary value problem

Article Justification of perturbation algorithm for weakly nonlinear elliptic second-order boundary value problem was published on January 1, 1991 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 6, issue 3).

Approximation of solutions for nonlinear second-order boundary value problems

Abstract. The purpose of this paper is to study two classes of nonlinear boundary value problems of second order when the nonlinearity is a Carathéodory function. We develop the monotone method in order to approximate the extremal solutions between the upper and lower solutions.

Compactness method in the finite element theory of nonlinear elliptic problems

Finite element approximation of a nonlinear elliptic pseudomonotone second-order boundary value problem in a bounded nonpolygonal domain Ω with mixed Dirichlet-Neumann boundary conditions is studied. In the discretization we approximate the domain Ω by a polygonal one, use linear conforming triangular elements and evaluate integrals by numerical quadratures. We prove the solvability of the discrete problem and on the basis of compactness properties of the corresponding operator (which is not monotone in general) we prove the convergence of approximate solutions to an exact weak solutionu∈H1 Ω). No additional assumption on the regularity of the exact solution is needed.

Multiderivative methods for non-linear second-order boundary value problems

Second-, fourth- and sixth-order methods are developed and analysed for the numerical solution of non-linear second order boundary value problems. The methods arise from a two-step recurrence relation involving exponential terms, these being replaced by Padé approximants. The methods are tested on two problems from the literature.

Nonlinear second-order boundary value problems: Intervals of existence, uniqueness and continuous dependence

Nonlinear second-order boundary value problems: Intervals of existence, uniqueness and continuous dependence

Discrete convergence of the approximations obtained by the multi-step method for the solutions of non-linear second-order boundary value problems

THE DISCRETE convergence of the approximate solutions of non-linear boundary value problems, obtained by a multi-step method, is discussed. Some theoretical aspects are examined, a convergence theorem is stated, and its proof is illustrated by a specific method. It is shown that the order of convergence is equal to the order of stability.

Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems

The purpose of this paper is to study the solvability of a nonlinear functional equation f E (A + B)u, where A is a maximal monotone (possibly multivalued) mapping from a reflexive Banach space X to its dual X* and B: X-+ X* satisfies some hind of sign condition. The main feature of the present method is that no asymptotic hypothesis (such as coerciveness, semicoerciveness, asymptotic oddness or homogeneity) is imposed on A + B. In Section 1 we state and prove the abstract result (Theorem 1). In Section 2 we apply Theorem 1 to the discussion of solvability of nonlinear second-order boundary value problems of Neumann type. Such problems were previously considered by Schatzman [6] and Hess [5] by different methods. This research was stimulated by some recent results on the range of the sum of maximal monotone operators announced by BrCzis [3].

A second-order nonlinear boundary value problem

A second-order nonlinear boundary value problem

A priori bounds and upper and lower solutions for nonlinear second-order boundary-value problems

y” = f(h y, r’), (1.1) r(a) kL(Y’(a)) = 0, (1.2) Y(b) i92(3”(b)) = 0, (1.3) where we assume throughout that f(t, y, y’) is continuous on [a, b] x Rg, and g,( y’) and g,( y’) are continuous and nondecreasing in (--CO, co). Our method originates from the observation that if y(t) is a solution to (l.l)--(1.3) with ~rlax[,,,~y(t) = y(t,) M > 0 and a --I#( y, y’) on an appropriate set and (A) The function $((x, x’) is positive, continuous, and satisfies a locai f,ipschitz condition on H”. 291 Copyright