On solutions of PDEs by using algebras

Abstract

The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form Auxx+Bux𝑦+Cu𝑦𝑦 = 0, and for each affine planar vector field 𝜑, we give an algebra A with unit e=e1, with respect to which the components of all functions of the form L◦𝜑 are all the solutions for this PDE, where L is differentiable in the sense of Lorch with respect to A. Solutions are also constructed for the following equations: Auxx+Bux𝑦+Cu𝑦𝑦+Dux+Eu𝑦+Fu = 0, 3rd-order PDEs, and 4th-order PDEs; among these are the bi-harmonic, the bi-wave, and the bi-telegraph equations.