Minimal structured perturbations are constructed such that an approximate eigenpair of a nonlinear eigenvalue problem in homogeneous form is an exact eigenpair of an appropriately perturbed nonlinear matrix function. Structured and unstructured backward errors are compared. These results extend previous results for (structured) matrix polynomials to more general functions. Structured and unstru...

We analyze fully implicit and linearly implicit backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of the coefficient functions. We combine maximal parabolic regularity and energy estimates to derive optimal-order error bounds for the time-discrete approximation to the solution and its gradient in the maximum norm...

Existence, uniqueness, and a priori estimates for solutions are studied for stochastic parabolic Ito equations. An analog of the second fundamental inequality and the related existence theorem are obtained for backward stochastic parabolic Ito equation. AMS 1991 subject classification: Primary 60J55, 60J60, 60H10. Secondary 34F05, 34G10.

In order to solve backward parabolic problems F. John [Comm. Pure. Appl. Math. (1960)] introduced the two constraints “‖u(T )‖ ≤ M” and ‖u(0) − g‖ ≤ δ where u(t) satisfies the backward heat equation for t ∈ (0, T ) with the initial data u(0). The slow-evolution-from-the-continuation-boundary (SECB) constraint has been introduced by A. Carasso in [SIAM J. Numer. Anal. (1994)] to attain continuou...

We rigorously prove that the semidiscrete schemes of a Perona-Malik type equation converge, in a long time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formal asymptotic expansion argument, and on a careful construction of discrete comparison functions. Despite the equation has a region where it is backward parab...

In order to solve backward parabolic problems F. John [Comm. Pure. Appl. Math. (1960)] introduced the two constraints “‖u(T )‖ ≤ M” and ‖u(0) − g‖ ≤ δ where u(t) satisfies the backward heat equation for t ∈ (0, T ) with the initial data u(0). The slow-evolution-from-the-continuation-boundary (SECB) constraint has been introduced by A. Carasso in [SIAM J. Numer. Anal. (1994)] to attain continuou...

The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div (a(x, t, u)|u|�(x, t)|�u|p(x, t)-2 with given variable exponents �(x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type. DOI: https...

A class of q-nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities q ∈ (1, 2), q = 2, q > 2, are analyzed.

We determine new, more favourable, and in a sense optimal, multipliers for the threeand five-step backward difference formula (BDF) methods. We apply the new multipliers to establish stability of these methods as well as of their implicit–explicit counterparts for parabolic equations by energy techniques, under milder conditions than the ones recently imposed in [4, 1].

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequency-domain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable fo...