A Numerical Approximation On Sideways Heat Equation

In many industrial applications one wishes to determine the temperature in the interior of a body, where the interior itself is inaccessible for measure-ments. It may also be the case that locating a measurement device on the surface of the body to get the temperature, then we can use the measured temperature to inverse the temperature in the interior. This is the so-called sideways heat equation.In this paper, we will consider the heat equation:This is an ill-posed problem:if the solution exists, it does not depend contin-uously on the data. We shall stabilize the problem by imposing a prior bound on u(0,·). suppose that||u(0,·)||≤Mor||u(0,@)||p≤M. Since g is measured, there will be measured errors. We actually have gm to approximate g, such that||g-gm||≤ε. So we will cosider the stabilized problem Numerical procedures for this problem can roughly be divided into two classes. The first class is regularization, such as Fourier methods, Tikhonov regularization and Wavelets method. The second class of numerical procedures is based on the formulation of the problem as a partial differential equation, for example:space-marching difference schemes, hyperbolic approximation and so on.In this paper, we will use to approximate the heat equation and get sharp eatimates about prior bound||·||and||·||p respectively. At the same time we will give the choice of parameters γ and n.The main results in this paper are listed here:定理0.3u is the solution of (0.5) which satisfied the prior bound on L2, v is the solution of (0.7). If we choose then for0≤x<1,we have:定理0.4u is the solution of (0.5) which satisfied the prior bound on Hp, v is the.solution of (0.7). If we choose where e-R(1+R4/4)-p=ε2/M2, then for ε2/M2≤1and0≤x<1, we have the following estimate:...