*Mathematical Methods in the Applied Sciences*On solutions of PDEs by using algebras

The components of complex differentiable functions define solutions for
the Laplace’s equation. In this paper we generalize this result; for
each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}=0$ we give an
affine planar vector field $\varphi$ and an associative
and commutative 2D algebra with unit $\mathbb A$, with
respect to which the components of all functions of the form
$\mathcal L\circ\varphi$
define solutions for this PDE, where $\mathcal L$ is
differentiable in the sense of Lorch with respect to
$\mathbb A$. In the same way, for each PDE of the form
$Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu=0$, the components of
the exponential function $e^{\varphi}$ defined
with respect to $\mathbb A$, define solutions for this
PDE. In the case of PDEs of the form
$Au_{xx}+Bu_{xy}+Cu_{yy}+Fu=0$, sine, cosine, hyperbolic
sine, and hyperbolic cosine functions can be used instead of the
exponential function. Also, solutions for two dependent variables
$3^{\text{th}}$ order PDEs and a
$4^{\text{th}}$ order PDE are constructed.

26 Mar 2021

27 Mar 2021

27 Mar 2021

03 Apr 2021