Boundary value problem in the free boundary domain

In this thesis, the exact and numerical solutions of the nonlinear heat equation in a free boundary domain is investigated.In the first section, the Green?s function for ordinary differential equations is analyzed.In the second section, using the Green?s function the solution of the linear heat equation in domain with given and moving boundary is obtained.In the last section, the nonlinear heat equation with zero initial condition in right half space is investigated. The solution in the form traveling wave is found and the differentiable properties of the problem is also analyzed. It is shown that the solution of this problem on the moving boundary which is unknown beforehand is degenerated.In order to find both the weak solution and unknown front, the auxiliary problem which has some advantages over the main problem is introduced. The weak solution of the problem is obtained by referring to the auxiliary problem. Using the advantages of this auxiliary problem the essential algorithms for obtaining the numerical solution are also suggested.