Abstract

The components of complex differentiable 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 Au xx +Bu xy +Cu yy = 0 and for each affine planar vector field ϕ, we give an associative and commutative 2D algebra with unit A, with respect to which the components of all functions of the form L • ϕ define solutions for this PDE, where L is differentiable in the sense of Lorch with respect to A. By using the generalized Cauchy-Riemann equations associated with ϕA-differentiability we show that each solution of these PDEs is a component of a ϕA-differentiable function. In the same way, for each PDE of the form Au xx + Bu xy + Cu yy + Du x + Eu y + F u = 0, the components of the exponential function e ϕ defined with respect to A, define solutions for this PDE. Also, solutions for two independent variables 3 th order PDEs and 4 th order PDE are constructed; among these are the bi-harmonic, bi-wave, and bi-telegraph equations.