# An algebraic approach for solving fourth-order partial differential equations

## TL;DR

This paper extends the classical link between harmonic functions and holomorphic functions to fourth-order hyperbolic and elliptic PDEs using hypercomplex algebra, providing a new algebraic solution framework.

## Contribution

It introduces an algebraic method for solving fourth-order PDEs by leveraging hypercomplex algebras, extending known complex analysis techniques.

## Key findings

- Solutions derived from hypercomplex algebra components satisfy the PDEs.
- The approach generalizes classical harmonic function methods to higher-order PDEs.
- Provides a new algebraic perspective for solving complex PDEs.

## 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.