Development the Numerical Method to Solve the Inverse Initial Value Problem for the Thermal Conductivity Equation of Composite Materials

Abstract In this paper, the heat conduction equation for composite materials posed and solved. This problem is known as an inverse initial value problem for the heat conduction equation. In order to solve and formulate this inverse problem, the function spaces must be defined and represented. By studying and solving the direct problem for the heat equation in composite materials, it is possible to determine the function spaces and solve the inverse initial value problem. Scientific methods used: the separation of variables method used to solve the direct problem for the heat equation. It found that method separation of variables does not completely lead to the solution of the inverse initial value problem, since this method leads to a divergent series of solutions and rather massive errors. The heat conduction problem can be formulated as Fredholm integral first kind equations. The discretization algorithm applied to reformulated the problem as linear operator problem as matrix and vectors form. Then, Tikhonov’s regularization inversion method has been used to find an approximation solution. Finally, as shown in the numerical example the regularized approximate solutions obtained.