Abstract
It is presented a way to obtain the closed form for Green’s function related to the nonhomogeneous one-dimensional Helmholtz equation with homogeneous Dirichlet conditions on the boundary of the domain from its Fourier sine series representation. A closed form for the sum of the series ∑k=1∞sinkxsinky/(k2-α2) is found in the process.
Highlights
Green’s function method applied to differential equations is a topic crystallized in mathematical physics textbooks and has appeared in some well-written didactic papers
Green’s function can be obtained from the linearly independent solutions of the corresponding homogeneous differential equation, and in some special circumstances its closed form can be obtained. Another way to obtain the Green function, specially useful when the homogeneous differential equation can not be solved by elementary methods, is by its expansion in a series of orthogonal functions
We seek a closed form for such an infinite summation and show the equivalence with the closed-form Green’s function obtained directly from the homogeneous equation
Summary
Green’s function method applied to differential equations is a topic crystallized in mathematical physics textbooks (see, e.g. [1,2,3,4]) and has appeared in some well-written didactic papers (see, e.g. [5,6,7,8,9]). Green’s function method applied to differential equations is a topic crystallized in mathematical physics textbooks Green’s function method is a tool suitable to solve nonhomogeneous differential equations, or homogeneous differential equations with nonhomogeneous initial or boundaryvalue conditions.
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