Elementary Boundary Value Problems
This textbook elucidates the role of BVPs as models of scientific phenomena, describes traditional methods of solution and summarizes the ideas that come from the solution techniques, centering on the concept of orthonormal sets of functions as generalizations of the trigonometric functions. To reinforce important concepts, the book contains exercises that range in difficulty from routine applications of the material just covered to extensions of that material.;Emphasizing the unifying nature of the material, this book: constructs physical models for both bounded and unbounded domains using rectangular and other co-ordinate systems; develops methods of characteristics, eigenfunction expansions, and transform procedures using the traditional fourier series, D'Alembert's method , and fourier integral transforms; makes explicit connections with linear algebra, analysis, complex variables, set theory, and topology in response to the need to solve BVP's employing Sturm-Liouville ststems as the primary vehicle; and presents illustrative examples in science and engineering, such as versions of the wave, diffusion equations and Laplace's equations.;Providing fundamental definitions for students with no prior experience in this topic other than differential equations, this text is intended as a resource for upper-level undergraduates in mathematics, physics and engineering, and students on courses on boundary value problems.
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Boundary Value Problems As Models
The Method of Characteristics
Linear Algebra and SturmLiouville Systems
A Fourier Series Theorem
2L-periodic extension absolutely integrable arbitrary assume Assumption auxiliary conditions Bessel's boundary conditions called Cauchy sequence Chapter complete conclude consider constant continuous on a,b converges uniformly coordinates cosine series course curve d2u d2u defined derivatives differentiable function differential equation diffusion equation Dirichlet domain dx2 dx dy dz dw eigenfunctions eigenfunctions belonging eigenvalues example Exercise formula Fourier series Fourier sine series Fourier transform Hankel transform Hint homogeneous infinite initial conditions inner product space integral transform interval a,b Laplace transform Laplace's equation Lemma linear algebra linear operator linearly independent membrane method metric models nonzero normed linear space obtain orthogonal orthonormal PDE of problem piecewise continuous piecewise smooth pointwise proof reader real numbers Recall regular S-L system satisfies scalar second-order Show solution subinterval substitution Suppose Theorem variables vector space wave equation zero
Partial Differential Equations: Analytical and Numerical Methods, Volume 1
Mark S. Gockenbach
No preview available - 2002