Ritz–Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions

Abstract

Two types of basis functions are employed to find the approximate solution of the surface heat flux histories and temperature distribution in an inverse heat conduction problem (IHCP). The properties of Bernstein multi-scaling functions are first presented. These properties, together with Galerkin method, are then utilized to reduce the main problem to the solution of nonlinear algebraic equations. The approximation of the problem is based on the Ritz–Galerkin method. Also the B-spline scaling functions are used in the Ritz–Galerkin technique to solve the inverse problem. Both approximations provide greater flexibility in which initial and boundary conditions on rectangular bounded domains are imposed. To keep matters simple, the problem has been considered in one-dimensional case, however the techniques can be employed for two- and three-dimensional cases. Compared with other published methods, it has high accuracy in computations that leads to exact solutions in some cases. Some illustrative examples are included to demonstrate the validity and applicability of the two new techniques.