A new approach to solving partial differential equations using tensor product theory of banach spaces

In this paper, an alternative and systematic solution method is developed for first and second order partial differential equations (PDEs) that are not directly separable into their variables, inhomogeneous and linear in nature. The basic approach is to assume the solution in separable form, i.e. u(x,y)=P(x)Q(y), and to rewrite this assumption in tensor product form by applying it to the corresponding differential equations. Thanks to the tensor product structure, each term can be treated as the product of functions depending on only one variable and thus the solution process can be simplified and analyzed. The method presented in this thesis is based on the idea of reducing the obtained equations to atomic solutions. In this context, two basic solution scenarios (Case i and Case ii) obtained by fixing the components depending on both x and y are systematically analyzed. The ordinary differential equations obtained from each case are solved by classical solution methods and the existence of atomic solutions is evaluated. As a result of the study, it is shown that the developed approach can provide effective solutions even in cases where classical separation methods are insufficient. Moreover, the consistency of the obtained solution structures with the initial conditions is analyzed in detail and it is shown that atomic solution is possible only for some cases. The developed method is shown to be a powerful alternative for solving partial differential equations from both theoretical and applied point of view.