Coupling of Two Partial Differential Equations and its Application

Abstract

The paper considers the following two partial differential equations

in the complex domain, and gives an answer to the following question: when can we say that the two equations (A) and (B) are equivalent? or when can we transform (A) into (B) (or (B) into (A))? The discussion is done by considering the coupling of two equations (A) and (B), and by solving their coupling equation. The most important fact is that the coupling equation has infinitely many variables and so the meaning of the solution is not so trivial. The result is applied to the problem of analytic continuation of the solution.