Solution of Dirichlet problem for a rectangular region in terms of elliptic functions

In this paper, the free vibration of an annular membrane consisting of three con- centric segments is considered. The frequency equation and mode shapes are obtained by the use of the Green's function method. A numerical example to vibration problem of non- homogeneous annular membrane is presented. solution of the problem and vibration analysis of membranes with discontinuously varying thickness is presented. In reference (2) the finite element method in the analysis was also used. The free vibration problem of annular membrane with many discontinuous variation of the density is the subject of paper (3). Although the formulation of the problem deals with the vibration of membrane which consist of m segments (each of constant density), the numerical examples concern the antisymmetric modes of composite membrane consisting of two segments. In this case the frequency equation is obtained by setting the determinant of a 4x4 matrix of coefficients to a derived system of equation, which equals zero. In many papers various methods are applied to solution of the eigenvalue problems. The authors of paper (4) in solving the eigenproblem for annular membrane propose the method of fundamental solution. In this method the free space Green's function are ap- plied. In this paper the free vibration problem of a composite annular membrane con- sisting of three segments of constant densities is presented. The solution of the problem (frequency equation, mode shapes) is derived by using the properties of Green's functions corresponding to the Helmholtz operator in an annular domain. An example of numerical frequency analysis is given.

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.

By analogy with the Green function for the Dirichlet problem for a domain , one considers the Green type function for the Neumann problem for . We obtain the Green type function for the positive half-space of R n and use it to solve the Neumann problem. The cases where is the positive half space of R 2 and of R 3 are analyzed in details under new hypotheses.

In this paper, the torque-free rotational motion of a general rigid body is developed analytically and is applied to the flipping motion of a T-handle spinning in zero gravity that can be seen in videos on the internet. This flipping motion is known both as the Dzhanibekov effect (after the cosmonaut who reported it) and more recently the tennis racket effect. The presentation is self-contained, accessible to students, and is complementary to the treatment found in most texts in that it involves a time-dependent analytical solution in terms of elliptic functions as opposed to a development based on conservation laws. These two complementary approaches are interesting and useful in different ways. In the present approach, the Euler rigid-body equations are derived and then solved as differential equations that are satisfied by Jacobi elliptic functions. This is analogous to solving the spring–mass harmonic oscillator problem by turning Newton's laws into differential equations that are satisfied by sine and cosine functions. The Jacobi functions are closely related to these trigonometric functions and are only slightly more complicated. They are defined as geometrical ratios on a reference ellipse and developed geometrically without reference to power series or complex variables. However, because these functions are less familiar, they are introduced in a short Appendix where their main properties are derived. Also, a link is provided to a Mathematica script for animating the analytical solution to the present problem.

The surface Helmholtz integral equation (SHIE) for the acoustic pressure on an object whose surface S vibrates harmonically with a given normal velocity, is known to fail by nonuniqueness at certain “characteristic” frequencies. These frequencies are identical to the eigenfrequencies of the homogeneous Dirichlet problem for a volume of fluid enclosed by the surface S. This paper deals with nondegenerate characteristic frequencies, that is, frequencies at which exactly one independent solution of the interior homogeneous Dirichlet problem exists but no nontrivial solutions of the interior homogeneous Neuman problem exist. The problem is reformulated in terms of operators on the Hilbert space of functions which are square integrable over S. The eigenvectors of the (skew-Hermitian) Green's function operator for the interior Neumann problem provide an orthonormal basis for the space and, hence, the unknown Green's function operator for the exterior Dirichlet problem can be completely specified by its matrix elements with respect to these eigenvectors. The nonuniqueness failure of the SHIE is found to result from the indeterminacy of a single element of this infinite matrix. An expression for the indeterminate matrix element is obtained directly from the SHIE by a perturbation technique yielding a formal solution of the problem.

A new coarse-mesh computational method for the numerical solution of heat conduction and fluid flow problems is formally developed and applied to sample problems. The method is based upon formal use of Green's functions, which are defined locally over subdomains of the original system under consideration. The formal development of the local Green's function method for the solution of heat conduction problems is presented and discussed. Numerical solutions of sample problems for one-dimensional heat conduction with constant thermal conductivity, one-dimensional heat conduction with temperature-dependent thermal conductivity, and two-dimensional heat conduction with constant thermal conductivity are given, and these results are compared with results obtained using the finite difference and finite element methods. The formal development of the local Green's function method for the solution of fluid flow problems is then also presented and discussed; the numerical solution of a sample problem for simple one-dimensional incompressible fluid flow with viscous heating is also given.

A projective method is used for the numerical solution of the plane interior Dirichlet problem for a simply connected region bounded by a curve of bounded rotation. The convergence of the method is proved and its application to a specific problem is shown.

In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

The extension of the heaviside expansion theorem to the equations of engineering and physics in curvilinear orthogonal coordinates

The existence of infinitely many solutions for an autonomous elliptic Dirichlet problem involving the p-Laplacian is investigated. The approach is based on variational methods.

The strongly elliptic system with constant m × m matrix-valued coefficients for a vector-valued functions u = (u 1, … , u m ) in the half-space as well as in a domain with smooth boundary ∂ Ω and compact closure is considered. A representation for the sharp constant in the inequality is obtained, where |·| is the length of a vector in the m-dimensional Euclidean space, , and ‖·‖ p is the L p -norm of the modulus of an m-component vector-valued function, 1 ≤p ≤∞. It is shown that where is a point at ∂ Ω nearest to x ∈ Ω, u is the solution of Dirichlet problem in Ω for the strongly elliptic system with boundary data from , and is the sharp constant in the aforementioned inequality for u in the tangent space to ∂ Ω at . As examples, Lamé and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula is derived, where 1 < p < ∞.

An addition formula for Green's functions

As already mentioned in Chapter 1, we refer to mathematical models that can be expressed as systems of ordinary differential equations and deal with the solution of problems in mechanics and, in general, in applied sciences. Therefore, it is important to deal with mathematical aspects related to the analysis of differential equations and with the solution of initial and boundary-value problems. This chapter refers to the above topics and is organized in seven sections: Section 2.2 deals with a formal classification of ordinary differential equations. Linear and nonlinear properties are identified together with the analysis of some additional properties of the system. Physical interpretations are often related to this type of classification. Section 2.3 deals with the mathematical statement of problems, which is obtained by linking the evolution equation to the conditions that are necessary to its solution. If these conditions are given at the initial value of the independent variable, say t = 0, one has an initial-value problem, also called a Cauchy problem. Otherwise, if these quantities are given at both ends of the range of the independent variable, say t = 0 and t = T, one has a boundary-value problem. The solution of problems has to satisfy both the differential equation and the initial and/or the boundary conditions. This section is mainly devoted to the analytic treatment of the Cauchy problem and will report the basic theorems which provide the conditions necessary to ensure existence, uniqueness, and regularity of the solution. Section 2.4 deals with solution methods of the initial-value problem. Analytic methods can be developed in order to solve linear problems, while computational schemes are needed for nonlinear problems. Actually, this matter is proposed as a concise guide to mathematical methods and to the related literature. The aim of this guide is to provide the necessary elements for the interaction with the scientific computation based on Mathematica ® Section 2.5 deals with the representation of solutions and applications both in the linear and in the nonlinear case. Here, a systematic use of scientific computation is developed. As we shall see, the computation is useful not only in the nonlinear case, but also for the fast treatment of linear problems. Section 2.6 deals with boundary-value problems for ordinary differential equations and the problems related to their solution, both analytical and numerical. The last section suggests some problems which are proposed to the reader’s attention for practicing with mathematical methods.