Some applications of generalized exponentials to partial differential equations

Abstract:

Using what may be considered as a natural generalization of the exponential function, some of the formalism of the theory of ordinary linear differential equations is extended to a class of linear partial differential equations among which are some important equations of mathematical physics. In ${\text {\S }}1$ we give the definitions of the generalized exponentials and derive expressions for them. $\S 2$ is devoted to the study of some of the properties of the exponential in two independent variables. In $\S 3$ we derive the general solutions of some key partial differential equations using the method of recursion. The last section is devoted to extending the formalism of the method of variation of parameters to a class of linear partial differential equations.