Numerical solution of the PDE describing unsteady-state heating of the tapered rod by Forward, Backward, Central difference methods and comparison with Analytical solution

The mathematicians, scientists and engineers, generally, describe the real world problem by the partial differential equation, as a result of its mathematical modeling. The mathematical modeling of most problems in science involving rates of change with respect to two or more independent variables, usually representing time, length or angle, leads either to a Partial Differential Equation (PDE) or to a set of such equations. The partial differential equations (PDEs) are obtained by the engineers and scientists in almost all the fields to describe a large number of real world problems. Many of the PDEs which result from engineering problems cannot be readily solved by analytical methods. Only a few of them have analytical solutions. Consequently, knowledge of the methods for obtaining numerical solutions of PDEs is important to the modern engineers. A numerical solution is obtained for the differential equation with specific boundary conditions. These, of course, describe some physical problem. For solving differential equations, the numerical approximation methods such as Finite Difference Methods (FDMs) are frequently used and more universally applicable than any other. The FDMs are most simple, easy and efficient to apply on partial differential equations among all the numerical methods.