Abstract
The components of complex analytic 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 , and for each affine planar vector field φ, we give an algebra with unit e = e1, with respect to which the components of all functions of the form are all the solutions for this PDE, where is differentiable in the sense of Lorch with respect to . Solutions are also constructed for the following equations: , ‐order PDEs, and ‐order PDEs; among these are the bi‐harmonic, the bi‐wave, and the bi‐telegraph equations.
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