Analytical and numerical solutions of heat conduction equation

This master's thesis consists of five chapters. In the first chapter, the main purpose of the thesis is briefly mentioned and some information is given about the heat conduction problem. In addition, in this chapter, the historical development and some basic features of the 1-dimensional heat conduction equation, which will be considered in the thesis, are mentioned. In the second chapter, information was given about the standart finite difference method, which is one of the numerical methods to be used in this thesis. Additionally, in this chapter, together with the finite difference method, some information about the von-Neumann stability analysis is given. Additionally, the method of separation of variables used for finding the exact solutions of the ordinary differential equation problems given with initial and boundary conditions are briefly presented. In the third chapter, studies in the literature on the 1-dimensional heat conduction equation are presented. Additionally, in this thesis, two model problems for the 1-dimensional heat conduction equation given with different initial and boundary conditions, of which exact and numerical solutions will be found, are briefly introduced. In the fourth chapter, exact solutions of the model problem given in the second chapter are obtained by separation of variables. Then, numerical schemes of the test problems are found out using the explicit, implicit and Crank-Nicolson finite difference method. Numerical solutions calculated from each scheme are presented in tables and graphics and they are also compared with the results in different studies in the literature along with the available exact solutions. In order to test how accurate and reliable results are produced by those finite difference schemes, in addition to comparisons with analytical and some other solutions, the error norms L₂, L_{∞} and the approximate convergence orders found in these norms are given in tabular form. Finally, in the fifth chapter, a brief evaluation of the results obtained from the applied methods is made and suggestions are made for future studies.