On solutions of PDEs by using algebras
AbstractThe 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 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. 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. In the case of PDEs of the form Au xx + Bu xy + Cu yy + F u = 0, sine, cosine, hyperbolic sine, and hyperbolic cosine functions can be used instead of the exponential function. Also, solutions for two dependent variables 3 th order PDEs and a 4 th order PDE are constructed.