New Algorithm Of Solving The Coefficient Inverse Problem For Parabolic Equation

As an important class of mathematical physical problems , inverse problems havedeveloped into a popular research direction . Solving an inverse problem is to deter-mine unknown causes based on observation of their effects. Nowadays, inverse prob-lems have been used in many fields, such as inverse medium scattering, computerizedtomography, ect, and the theory and methods are novel and challenging. Two essentialdifficulties appear frequently. One is the observation data possibly does not belong tothe corresponding set to the exact solution, another is that the approximation is notstable. Namely, the little error of initial data due to our measure will lead to the de-viation between approximate solutions and true solutions. Thus inverse problems areoften ill-posed which makes inverse problem more difficult. Also just because of this,inverse problems attracts more and more researchers to study.In this paper, a new algorithm for the solution of an inverse coefficient problem ina parabolic equation in reproducing kernel space is considered. Employing transfor-mation, we transform inverse coefficient p(t) into right-hand inverse coefficient r(t).If r(t) is known, then the exact solution u(t,x) is given in the form of series. Then-term approximation un(t,x) and rn(t) is constructed. They are proved to convergeto the exact solutions and this algorithm is stable. Moreover, the approximate error ofu_n(t,x) is monotone decreasing as n becomes larger. And this algorithm can easily begeneralized to solve multidimensional problems. Finally, numerical experiments arepresented to demonstrate the accuracy and the efficiency of the proposed algorithm.