Singular Solutions Of Nonlinear Equations Research Articles

Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions

This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.

Solving the Two–Dimensional Lane–Emden Type Equations by the Adomian Decomposition Method

In this paper, we introduce the two–dimensional Lane–Emden type equations. We study the linear and the nonlinear forms of these types of equations. We use the systematic Adomian decomposition method to handle these equations with specified initial conditions. The Adomian decomposition method provides an efficient scheme for exact and approximate analytic solutions of singular nonlinear equations. We corroborate this work by studying several two-dimensional Lane-Emden problems that represent numerical examples to show the strength of our approach.

Singular nonlinear elliptic equations in Rn

This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form −Δu + a(x)u = h(x)u−γ in Rn where a, h are given, not necessarily continuous functions, and γ is a positive number. We explore both situations where a, h are radial functions, with a being eventually identically zero, and cases where no symmetry is required from either a or h. Schauder′s fixed point theorem, combined with penalty arguments, is exploited.

Enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations

A solution of a nonlinear equation in Hilbert spaces is said to be a simple singular solution if the Frechet derivative at the solution has one-dimensional kernel and cokernel. In this paper we present the enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations. Once singularities are resolved, we can compute accurately the singular solution by Newton's method. Conditions for which the procedure terminates in finite steps are given. In particular, if the equation defined in ?n is analytic and the simple singular solution is geometrically isolated, the procedure stops in finite steps, and we obtain the enlarged problem with an isolated solution. Numerical examples are given.

Singular solutions of the KdV equation and the inverse scattering method

The paper is devoted to the construction of singular solutions of the KdV equation. The presentation is based on a variant of the inverse scattering method for singular solutions of nonlinear equations developed in previous works of the authors.

Singular solitons: An example of a Sinh-Gordon equation

The application of the inverse scattering transform for a description of singular solutions of nonlinear equations is considered, using an example of a Sinh-Gordon equation. The singular solitons are constructed. The soliton lines of singularity are interpreted as world lines of a Poincare-invariant dynamical system of interacting spin particles.

Singular solutions of nonlinear equations of Ginsburg-Landau type

Singular solutions of nonlinear equations of Ginsburg-Landau type