Abstract
In this paper we develop a way of obtaining Green's functions of partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat transfer in a conducting plate which is bent in half.
Highlights
The study of differential equations with involutions dates back to the work of Silberstein [10] who, in 1940, obtained the solution of the equation f (x) = f (1/x)
In the field of differential equations there has been quite a number of publications but most of them relate to ordinary differential equations (ODEs)
There has been some work in partial differential equations (PDEs), for instance [11] or [2], where they study a PDE with reflection
Summary
In what Green’s functions for equations with involutions is concerned, we find in [3] the first Green’s function for ODEs with reflection and in [4] we have a framework that allows the reduction of any differential equation with reflection and constant coefficients. This setting is established in a general way, so it can be used as well for other operators (the Hilbert transform, for instance) or in other yet unexplored problems, like PDEs [8]. We study the problem for different kinds of boundary conditions and a general heat source
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