Methods of Eigen Function Expansion to Solve Non-homogeneous Boundary Value Problems

Among various methods to solve linear boundary value problems defined on finite domain, method of Eigen function expansion is most popular method. In this method, the solution of given boundary value problem is obtained as a series of Eigen functions, that is, as an expansion of Eigen functions. The general theory of Eigen values, Eigen functions and convergence of Eigen function expansions is most important and deepest part of modern mathematics. For this type of expansion, most probably the method of separation of variables is used. To use the method of separation of variables, the partial differential equation must be linear and homogeneous. Nonhomogeneous boundary value problem means, either the problem is nonhomogeneous or the boundary conditions are nonhomogeneous or both. As for example, a nonhomogeneous heat flow problem means either the problem has source term or it has nonzero boundary conditions or it has a source term and nonzero boundary conditions both. For nonhomogeneous partial differential equations, to get Eigen function expansion, using the method of separation of variables, it is required to convert the given nonhomogeneous conditions into homogeneous boundary conditions. For this purpose, it is required to assume an equilibrium function which is a function of variables x or an appropriate reference function, which is a function of variables x and t for one-dimensional problems [1], (for multidimensional problems, a reference function contains spatial variables and time variable t) in such a way that the nonhomogeneous boundary conditions can be converted into homogeneous boundary conditions. So there arises a question: Do we have any method to solve nonhomogeneous problems without converting nonhomogeneous boundary conditions into homogeneous ones? The answer is yes. Eigen function expansion can be done using Green's formula also and in this method, there is no need to convert nonhomogeneous boundary conditions into homogeneous ones. The only drawback is that the solution obtained by this method will not be valid at boundary points and hence convergence of Eigen function expansion, using Green's formula will be slower compared to that of expansion obtained using method of separation of variables[1].In this paper authors want to discuss theory for both the methods: (1) Eigen function expansion using method of separation of variables and (2) Eigen function expansion using Green's formula. To illustrate the methods, one boundary value problem is also solved using both the methods.