Partial Differential Equations

John M. Neuberger ()
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John M. Neuberger: Northern Arizona University, Department of Mathematics and Statistics

Chapter 4 in Difference Matrices for ODE and PDE, 2023, pp 93-179 from Springer

Abstract: Abstract We first build a second difference block matrix corresponding to the Laplacian on the square. We use this Laplacian matrix with various enforced boundary conditions to extend the ideas developed in Chap. 3 to partial differential equations. In particular, for the square domain we investigate eigenvalues of the Laplacian, solutions to semilinear elliptic boundary value problems, Laplace’s equation with nonhomogeneous boundary conditions, the heat equation, and the wave equation. We introduce techniques for other domains, including the Laplacian on the cube and in polar coordinates, and a fairly simplistic method for constructing a Laplacian on an arbitrarily bounded two-dimensional domain. We include a section on Tricomi’s equation, and a brief tutorial on solving first-order PDE numerically via the method of characteristics. We conclude with an overview of the method of separation of variables as applied to obtaining theoretical solutions to the fundamental PDE covered in this chapter, with examples.

Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-12000-8_4

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DOI: 10.1007/978-3-031-12000-8_4

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