Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients—Forward collocation with finite integration method

We investigate a sideways problem of reconstructing an inaccessible boundary value for parabolic equation with variable coefficients. Formulating the sideways problem into a sequence of well-posed direct problems (DP) and a system of Ordinary Differential Equations (ODE), we combine the recently developed finite integration method (FIM) with radial basis functions (RBF) to iteratively obtain the solution of each DP by solving an ill-posed linear system. The use of numerical integration instead of finite quotient formula in FIM completely avoids the well known roundoff-discretization errors problem in finite difference method and the use of RBF as forward collocation method (FCM) gives a truly meshless computational scheme. For tackling the ill-posedness of the sideways problem, we adapt the traditional Tikhonov regularization technique to obtain stable solution to the system of ODEs. Convergence analysis is then derived and error estimate shows that the error tends to zero when perturbation δ→0δ→0. We can then obtain highly accurate and stable solution under some assumptions. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.