SECTION 12 - Green's Function

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

Artificial boundary conditions, which can be identified as Robin boundary conditions positioned at a complex space coordinate, are introduced in order to obtain pertinent approximations for the Green's functions in grounded layered media. These artificial boundary conditions include perfectly matched layers backed by perfectly electric or magnetic conductors. As a first result, we obtain analytical expressions for the differences of Green's functions subject to different boundary conditions. Since weighted sums of Green's functions are again Green's functions, the need arises to solve an optimization problem, in the sense of obtainingthe optimal weig hted mixture of Green's functions, as compared to the exact Green's function. Comprehensive eigenexpansions for the Green's functions are given in the general case, and a few examples illustrate the goodness of fit between the approximate Green's functions and the exact Green's function.

An addition formula for Green's functions

We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

Some results on boundary value problems for fuzzy differential equations with functional dependence

The Green’s function (GF) method, which makes use of GFs, is an important and elegant tool for solving a given boundary-value problem for the differential equation from a real engineering or physical field. Under a concentrated source, the solution of a differential equation is called a GF, which is singular at the source location, yet is very fundamental and powerful. When looking at the GFs from different physical and/or engineering fields, i.e. assigning the involved functions to real physical/engineering quantities, the GFs can be scaled and applied to large-scale problems such as those involved in Earth sciences as well as to nano-scale problems associated with quantum nanostructures. GFs are ubiquitous and everywhere: they can describe heat, water pressure, fluid flow potential, electromagnetic (EM) and gravitational potentials, and the surface tension of soap film. In the undergraduate courses Mechanics of Solids and Structural Analysis, a GF is the simple influence line or singular function. Dropping a pebble in the pond, it is the circular ripple traveling on and on. It is the wave generated by a moving ship in the opening ocean or the atom vibrating on a nanoscale sheet induced by the atomic force microscopy. In Earth science, while various GFs have been derived, a comprehensive review is missing. Thus, this article provides a relatively complete review on GFs for geophysics. In section , the George Green’s potential functions, GF definition, as well as related theorems and basic relations are briefly presented. In section , the boundary-value problems for elastic and viscoelastic materials are provided. Section is on the GFs in full- and half-spaces (planes). The GFs of concentrated forces and dislocations in horizontally layered half-spaces (planes) are derived in section in terms of both Cartesian and cylindrical systems of vector functions. The corresponding GFs in a self-gravitating and layered spherical Earth are presented in section in terms of the spherical system of vector functions. The singularity and infinity associated with GFs in layered systems are analyzed in section along with a brief review of various layer matrix methods. Various associated mathematical preliminaries are listed in appendix, along with the three sets of vector function systems. It should be further emphasized that, while this review is targeted at geophysics, most of the GFs and solution methods can be equally applied to other engineering and science fields. Actually, many GFs and solutions methods reviewed in this article are derived by engineers and scientists from allied fields besides geophysics. As such, the updated approaches of constructing and deriving the GFs reviewed here should be very beneficial to any reader.

In this article, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem where 1 < α < 2, 0 < β, η < 1, is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green's function, we establish some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear alternative, a fixed-point theorem on cones, and also we give uniqueness of a solution for a singular problem by a mixed monotone method.

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green's function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.

A survey is presented of recent related work (comprising numerical modeling and field experiments) on the determination of seafloor geoacoustic properties in shallow water from monochromatic pressure field measurements obtained using synthetic aperture and towed arrays. Two related techniques are discussed, their common foundation lying in the Hankel transform inversion of the pressure versus range data to obtain a depth-dependent Green's function versus horizontal wave number. Characteristic features of the Green's function, such as the peaks associated with normal mode propagation in the shallow water waveguide, are used to extract the geoacoustic properties (compressional and shear) as a function of depth in the bottom. INTRODUCTION Two related methods for determining geoacoustic properties in shallow water waveguides have recently been under development (Frisk and Lynch, 1984; Frisk et al., 1985; Kuperman et al., 1985; Werby and Tango, 1985). In both cases, the required input data are the magnitude and phase versus range of the pressure field due to a monochromatic (CW) point source. These data are numerically Hankel transformed to obtain the depth-dependent Green's function versus horizontal wave number. In the context of normal mode theory, the Green's function contains information about the nature of the discrete and continuous modal spectra as well as the plane-wave reflection coefficients of the waveguide boundaries. Specifically, the Green's function contains prominent peaks at horizontal wave numbers corresponding to the eigen-values for any trapped and virtual modes excited in the waveguide. The positions and magnitudes of these modal peaks are sensitive to the acoustic properties of the bottom, and this sensitivity is the basis of the two inverse methods for determining geoacoustic models in shallow water. The WHOI method utilizes an exact Hankel transform inversion of synthetic aperture array data obtained by towing a CW source away from two moored receivers. A geoacoustic model is obtained by computing the theoretical Green's function for various values of the bottom parameters and determining the parameter set which provides the best agreement with the experimental Green's function, particularly in the positions and relative magnitudes of the modal peaks. The NORDA technique involves an approximate Hankel transform inversion in which the beam formed output of a towed array due to the self-noise of the tow ship is computed. Layer sound velocity estimates are obtained from the identification of modal critical angles in the variable-offset beam former response. THEORY Consider a water column of thickness h, characterized by a constant density and a sound speed c(z), and bounded at the surface and bottom by horizontally stratified media as shown in Fig. 1. Then the spatial part of the acoustic pressure p, due to a point source at r =.0 and z = Zo with harmonic time satisfies the inhomogeneous Helmholtz equation(Mathematical equation available in full paper) The Green's function satisfies along with impedance boundary conditions at the surface and bottom. These boundary conditions can be ex pressed in terms of the plane-wave reflection coefficients RS(kr) and RB(kr) of the surface and bottom respectively (Frisk et al., 1980).

Channel modeling is a challenging vital step towards the development of diffusion-based molecular communication networks (DMCNs). Analytical approaches for diffusion channel modeling are limited to simple and specific geometries and boundary conditions. Also, simulation- and experiment-driven methods are very time-consuming and computationally complex. In this paper, the channel model for DMCN employing the fundamental concentration Green's function (CGF) is characterized. A general homogeneous boundary condition framework is considered that includes any linear reaction systems at the boundaries in the environment. To obtain the CGF for a general DMCN including multiple transmitters, receivers, and other objects with arbitrary geometries and boundary conditions, a semi-analytical method (SAM) is proposed. The CGF linear integral equation (CLIE) is analytically derived. By employing the numerical method of moments, the problem of CGF derivation from CLIE is transformed into an inverse matrix problem. Moreover, a sequential SAM is proposed that converts the inversion problem of a large matrix into multiple smaller matrices reducing the computational complexity. Particle-based simulator confirms the results obtained from the proposed SAM. The convergence and run time for the proposed method are examined. Further, the error probability of a simple diffusion-based molecular communication system is analyzed and examined using the proposed method.

from a unified point of view, namely, the extensive use of Green's function. Our main interest is concerned with the first mixed boundary problem (for definition, see ?4) for f linear or nonlinear in u, Harnack theorems, etc. Most of our methods are known. Green's function for the heat equation in rectangular domains was constructed already by E. E. Levi [21] (in particular ?7) in 1907 and was used by him to derive some existence theorems. An extensive use of Green's function for more general parabolic equations was made by Gevrey in his fundamental paper [16] (in particular ??4, 4*, 24, 28, 39, 40, 41). The analogue of the Harnack convergence theorem was first proved for the heat equation in Levi's paper [21, pp. 386-387]. Nonlinear existence problems were considered by Gevrey [16, ?28] by reducing them with the aid of Green's function to integral equations and then applying successive approximations. A more detailed survey on the older literature concerned with Green's function is to be found in the book of Ascoli-BurgattiGiraud [I]. In this paper we use all the above mentioned methods and a few new ones to treat more general problems than those considered in earlier papers. Essential use of Dressel's fundamental solutions for general linear second order parabolic equations [10; 11] enables us to perform this extension. We give below a brief description of our results and their connection to previous papers. In ?1 we construct Green's function for linear second order parabolic equations with smooth coefficients in an (n+1)-dimensional rectangle. In this construction we employ the fundamental solutions constructed by Dressel [11]. Green's function for the heat equation in one dimension was constructed by Levi [21] by reflecting the fundamental solution t-12 exp(_-.2/4t) with respect to the x-variable; our method is an extension of Levi's method.

With an explicit expression for the Green's function, Eq. (19), we have a common method for obtaining the radial parts of the Green's functions for a broad range of physical problems which lead to the Whittaker equation (listed in part in the introduction). This analysis emphasizes the special role played by the Whittaker equation in physical applications. Since the symmetry of the Green's function — an important property — is a consequence of the self-adjoint nature of the corresponding differential operator, the Whittaker equation is apparently the most general second-order equation with one regular singularity and one irregular singularity which retains its value for inhomogeneous physical problems, i.e., for problems leading to inhomogeneous differential equations.

Analytical modeling of squeeze film damping for rectangular elastic plates using Green's functions

A new type of Green function approach is introduced for time-domain scattering for a stratified dissipative slab. The new Green functions are defined as mappings from the transmitted field to the split fields (i.e., the right-moving and left-moving fields) at any point within the stratified slab. Unlike the usual Green functions, the present Green functions have compact support in the time variable. The new technique is illustrated for the case of normally incident electromagnetic waves on a stratified slab where the permittivity, permeability, and conductivity vary with the depth. The linear and homogeneous partial differential equations (PDEs) for these ‘‘compact Green functions’’ are derived. The PDEs together with the initial and boundary conditions are well suited for a numerical treatment. Numerical results for the compact Green functions are presented in the direct problem, while in the inverse problem the permittivity and conductivity (or the permeability and conductivity) are reconstructed simultaneously using the reflection and transmission data for the first round trip. The present method is useful and attractive for both the direct and inverse problems due to its simplicity, high speed, and high accuracy in numerical computations.

This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.