We consider the problem of finding, from the final data u(x, T ) = ϕ(x), the temperature function u(x, t), x ∈ (0, π), t ∈ [0, T ] satisfies the following nonlinear systemThe nonlinear problem is severely ill-posed. We shall improve the quasi-boundary value method to regularize the problem and to get some error estimates. The approximation solution is calculated by the contraction principle. A numerical experiment is given.

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Let Q be a heat conduction body and let ' = '(t) be given. We consider the problem of nding a two-dimensional heat source having the form '(t)f(x; y) in Q. The problem is ill-posed. Assuming @Q is insulated and ' 6 0, we show that the heat source is de ned uniquely by the temperature history on @Q and the temperature distribution in Q at the initial time t = 0 and at the nal time t = 1. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate.

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We consider the problem of finding u ∈ L 2 (I), I = (0, 1), satisfyingis a sequence of distinct real numbers greater than −1/2, and µ = (µ kl ) is a given bounded sequence of real numbers. This is an ill-posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given.