Research On Inverse Problem Of Acoustics And Heat Equation

The field of inverse problem is a relatively new area of Mathematical physics research. The inverse acoustic problem is a typical inverse problem, which takes the answer of scattering function as its starting point to find the scattering field and the other conditions (for example the coefficient). It is realized that inverse scattering problems were ill-posed and the Tikhonov method is used to solve it. In this paper, the inverse scattering problems and the inverse problem of initial condition of the heat equation are studied.For the inverse scattering problems with impedance boundary condition, firstly, the combined single and double layer potentials are used to approach the scattered waves in order to determine the impedance. The numerical results are better and the convergence is proven. Secondly, the shape is determined and the convergence is proven, too. The numerical results show that this method is both accurate and simple to use. Thirdly, the single layer potential is used to find both the shape and the impedance. The proof and numerical examples are given and the results are better. finally, the combined Hankel function is used to determine the shape and the impedance. The numerical result shows that the combined Hankel function method is more simple than the combined single and double layer potentials method but the latter is more accurate.For the inverse scattering problems with impedance boundary condition of crack, the combined single and double layer potential are used to approach the scattered waves in order to determine the shape and the impedance. The Tikhonov method is used to convert this problem to the minimize problem and the imitate Newton method is used to solve it. This method does not require the solution to u and (?)u/(?)v at each iteration step. So it is simple to use.The inverse problem of the second boundary condition of heat equation considered in this paper is to determine the initial condition if the temperature distribution inside a domain are known at some time. Based on the existence and uniqueness proof of solution, Tikhonov regularization method is used to convert the problem with Neuman boundary conditions into a nonlinear optimization problem. And integral operator is discretized by trapezoidal rule for an approximation solution. Numerical examples show that this method is both accurate and simple to use.