Numerical Calculation To Some Kinds Of Direct And Inverse Problems Of Parabolic Equation

The direct and inverse problem of parabolic equation is a kind of classical mathematical physics problems. Both in terms of theoretical analysis and in practical applications, it has a wide range of research value. Finite difference methods are given respectively to the parabolic equation with direct and inverse problems. And the numerical results are gotten in Matlab software. Both analytical solution and numerical solution are given in order to make the results more convincing.Based on Darcy’s seepage law and heat conduction rule in porous media, this article deduces the governing equation and finite difference scheme of water head values, and presents the governing equation and finite difference scheme of 3D coupling model of seepage field and temperature field. It finally computes the water temperature values at some location inside the body after the control equations, the boundary condition (include the water head and temperature value), and the initial temperature condition are known in porous medium. Numerical results show that the numerical format of 3D coupled model is very efficient.The paper considers the problem of finding u(X,t) and p(t) which satisfy ui=Δu+p(t)u+(?)(X,t) with u(X,0)=Φ(X), X∈Q, u(X,t)=G(X,t), (X,t)∈(?)Q×[0,L] and u(X0,t)= E(t), Xo in Q,0< t< L, where Q (?)R2 is a bounded domain with smooth boundary (?)Q, X represents a vector, A is a Laplace operator,f,Φ, G and E are known functions. The inverse problem can be changed into a direct problem by introducing a transformation, then the direct problem can be calculated, finally we can find the solution of the inverse problem in accordance with the transformation. Two explicit and implicit finite difference methods and ADI algorithm are used to solve above equations, so five numerical techniques are presented. The accuracy and efficiency of the procedures are discussed and the results of numerical simulation are presented.