Approximation of a Function by Solutions of an Underdetermined Nonhomogeneous Heat Equation

The best (in the L2 sense) approximation of a given function by solutions of an underdetermined nonhomogeneous heat equation is considered. The problem is reduced to a well-posed boundary value problem for a fourth-order partial differential equation. The eigenfunction expansion of the solution of this well-posed problem is obtained in a one-dimensional spatial case. The found best approximation and partial sums of its eigenfunction expansion provide an enhanced approximation of any unknown temperature distribution