A backward parabolic equation with a time-dependent coefficient: Regularization and error estimates

A Regularization of the Backward Problem for Nonlinear Parabolic Equation with Time-Dependent Coefficient

Regularization and Hölder Type Error Estimates for an Initial Inverse Heat Problem with Time-dependent Coefficient

This paper discusses the initial inverse heat problem (backward heat problem) with time-dependent coefficient. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. Two regularization solutions of the backward heat problem will be given by a modified quasi-boundary value method. The Hölder type error estimates between the regularization...

A Nonhomogeneous Backward Heat Problem: Regularization and Error Estimates

We consider the problem of finding the initial temperature, from the final temperature, in the nonhomogeneous heat equation ut − uxx = f(x, t), (x, t) ∈ (0, π)× (0, T ), u(0, t) = u(π, t) = 0, (x, t) ∈ (0, π)× (0, T ). This problem is known as the backward heat problem and is severely ill-posed. Our goal is to present a simple and convenient regularization method, and sharp error estimates for ...

Regularization and New Error Estimates for a Modified Helmholtz Equation

We consider the following Cauchy problem for the Helmholtz equation with Dirichlet boundary conditions at x = 0 and x = π    ∆u− ku = 0, (x, y) ∈ (0, π)× (0, 1) u(0, y) = u(π, y) = 0, y ∈ (0, 1) uy(x, 0) = f(x), (x, y) ∈ (0, π)× (0, 1) u(x, 0) = φ(x), 0 < x < π (1) The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. Using a modifi...

A Cauchy Problem for Helmholtz Equation : Regularization and Error Estimates

In this paper, the Cauchy problem for the Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a new regularization method to solve it based on the solution given by the method of separation of variables. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness ...