Multidimensional Laplace transforms over Cayley-Dickson algebras and partial differential equations

TL;DR

This paper develops multidimensional noncommutative Laplace transforms over Cayley-Dickson algebras, proves fundamental theorems, and applies them to solve various classes of partial differential equations, including higher-order and variable coefficient types.

Contribution

It introduces new theoretical results on Laplace transforms over Cayley-Dickson algebras and demonstrates their application to complex PDEs with diverse coefficients.

Findings

Established direct and inverse Laplace transform theorems over Cayley-Dickson algebras

Applied transforms to solve elliptic, parabolic, and hyperbolic PDEs

Extended methods to higher-order PDEs with variable coefficients

Abstract

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.