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Harmonic Functions, Conformal Mapping, and Some Applications
Published in A. David Wunsch, ® Companion to Complex Variables, 2018
Harmonic functions and conformal mapping are two of the richest and most interesting topics in complex variable theory. It is our good fortune that harmonic functions, which are always the real and imaginary parts of analytic functions, can be used to describe two-dimensional configurations of fluid flow, heat transfer, and electrical fields. Historically, conformal mapping achieved importance as a means of solving two-dimensional problems in these branches of engineering and physics. Although numerical methods on modern computers have rendered this application of less importance, conformal mapping still provides us with solutions to certain canonical problems that can be used to verify the correctness of computer-generated solutions. This is analogous to the practice of learning to perform integrations in elementary calculus courses even though, for example, the MATLAB® Symbolic Mathematics Toolbox will perform them for you. One must know what kinds of answers to expect from a computer and to have a means of checking them.
Preliminaries
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
Maximum principleThe Hopf maximum principle for elliptic partial differential equations and differential inequalities generalizes the classical maximum-minimum principle for harmonic functions. That principle, in turn, can be thought of as generalizing the following simple fact from calculus: if a continuous function y defined on a closed interval [a,b] satisfies y′′=0 on the open interval (a,b), then y cannot achieve a local maximum or local minimum value at a point in (a,b) unless y is identically constant. This follows because y is a linear function on [a,b]. Corresponding results with significant consequences hold for solutions y to certain ordinary differential equations and differential inequalities. Two of those results are presented here, in adequate generality for our purposes. The pioneering work of Eberhard Hopf is far deeper than what is suggested here.
Differentiation of Functions of a Complex Variable
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
at each point in D. Laplace's equation and harmonic functions play an important role in physics and technology. For example, the static temperature distribution on a domain D, and electric potential fields on charge-free domains, are harmonic functions. The connection between analytic and harmonic functions, which we will study in this section, is exploited in various applications of analytic functions.
Probabilistic global well-posedness for a viscous nonlinear wave equation modeling fluid–structure interaction
Published in Applicable Analysis, 2022
Jeffrey Kuan, Tadahiro Oh, Sunčica Čanić
By taking the Fourier transform of (10) in the x and y variables but not the z variable, we can establish that where ξ denotes the frequency variable corresponding to the x and y variables. For more details, see the explicit calculation in Ref. [1]. Then, by taking the Fourier transform of (9) in the x and y variables and using (11) and (12), we obtain By taking the inverse Fourier transform, this gives the Neumann boundary condition for the harmonic function π in terms of (derivatives) of u. Recall that the Dirichlet-Neumann operator for the lower half plane with the vanishing boundary condition at infinity is given by ; see Ref. [2]. By inverting this operator, we see that the Neumann-Dirichlet operator with the same boundary condition at infinity is given by the Riesz potential with a Fourier multiplier . Therefore, by applying the Neumann-Dirichlet operator to (the inverse Fourier transform of) (13), we obtain the desired result in (8).
On non-homogeneous Robin reflection for harmonic functions
Published in Applicable Analysis, 2022
Murdhy Aldawsari, Tatiana Savina
Let normal derivative of a harmonic function equals on an arc of the unit circle, where α and β are constants. Then formula (14) implies Function is an example of such harmonic function. Here is a harmonic function defined in the neighborhood of the unit circle, whose normal derivative vanishes on the circle.
Solutions to a two-dimensional, Neumann free boundary problem
Published in Applicable Analysis, 2020
J. A. Gemmer, G. Moon, S. G. Raynor
We say that a harmonic function v on a Lipschitz domain D satisfies Neumann boundary conditions weakly along an open set if for every , possibly with a boundary condition along .