Successive approximations of solutions to stochastic functional differential equations

In the present paper, by means of the successive approximations method, the local or global existence and uniqueness theorems for a stochastic functional differential equation of the Ito type are proved.     

On small solutions of nonlinear equations with vector parameter in sectorial neighborhoods

We consider the nonlinear operator equation B(λ)x + R(x, λ) = 0 with parameter λ, which is an element of a linear normed space Λ. The linear operator B(λ) has no bounded inverse for λ = 0. The range of the operator B(0) can be nonclosed. The nonlinear operator R(x, λ) is continuous in a neighborhood of zero and R(0, 0) = 0. We obtain sufficient conditions for the existence of a continuous solution x(λ) → 0 as λ → 0 with maximal order of smallness in an open set S of the space Λ. The zero of the space Λ belongs to the boundary of the set S. The solutions are constructed by the method of successive approximations.     

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form under the very general structural condition . To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P). Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000     

Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.     

Solving nonlinear equations by differentiation with respect to a parameter

Differentiation with respect to a parameter is proposed as a method for solving a system of nonlinear equations with a poorly conditioned Jacobian matrix. Singular and polar decompositions of the Jacobian matrix lead to a system of ordinary differential equations with small parameters multiplying the derivatives. This system is solved by introducing auxiliary linear differential equations. The numerical solution of a poorly conditioned system is reduced in this case to the approximation of an exponential with a large negative argument.     

Finding approximations of continuous solutions to first-kind equations

A method for constructing regularizing families of operators is proposed that is based on combinations of simple operators of the indicated structure.     

Difference approximations to interface curves for nonlinear diffusion equations with absorption

We consider some nonlinear reaction-diffusion equations with extinction phenomena in finite time. For the solution v, there appear interface curves between v > 0 and v = 0. We propose difference approximations to interface curves, and prove the convergence to exact interface curves.     

Diffusive approximations for a class of nonlinear parabolic equations

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.