An algebraic approach for solving fourth-order partial differential equations

Abstract

It is well-known that any solution of the Laplace equation is a real or imaginary part of a complex holomorphic function. In this paper, in some sense, we extend this property into four order hyperbolic and elliptic type PDEs. To be more specific, the extension is for a $c$-biwave PDE with constant coefficients, and we show that the components of a differentiable function on the associated hypercomplex algebras provide solutions for the equation.