On the iterative method for solution of direct and inverse problems for parabolic equations

The paper is devoted to approximate methods for solution of direct and inverse problems for parabolic equations. An approximate method for the solution of the initial problem for multidimensional nonlinear parabolic equation is proposed. The method is based on the reduction of the initial problem to a nonlinear multidimensional intergral Fredholm equation of the second kind which is approximated by a system of nonlinear algebraic equations with the help of the method of mechanical quadratures. For constructing the computational scheme we use the nodes of the local splines which realize order-optimal approximation of the functional class that contains solutions of parabolic equations. For implementation of the computational scheme we use the generalization of the continuous method for solution of nonlinear operator equations that is described in the paper. We also analyse the inverse problem for parabolic equation with fractional order derivative with respect to the time variable. The approximate methods for defining the fractional order of the time derivative and the coeffcient at spatial derivative are proposed.