Homogeneous Partial Differential Equations Research Articles

Factoring systems of linear PDEs with finite-dimensional solution spaces

A D-finite system is a finite set of linear homogeneous partial differential equations in several independent and dependent variables, whose solution space is of finite dimension. Let L be a D-finite system with rational function coefficients. We present an algorithm for computing all hyperexponential solutions of L, and an algorithm for computing all D-finite systems whose coefficients are also rational functions, and whose solutions are contained in the solution space of L.

The discontinuous enrichment method for multiscale analysis

Computation naturally separates scales of a problem according to the mesh size. A variety of improved numerical methods are described by multiscale considerations, differing in the treatment of the unresolved, fine scales. The discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework. The method thus combines relative ease of implementation with improved numerical performance. These properties are demonstrated for both multiscale wave and transport problems, pointing to the potential of considerable savings in computational resources.

A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime

We present a discontinuous Galerkin method (DGM) for the solution of the Helmholtz equation in the mid-frequency regime. Our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non-conforming field that contains free-space solutions of the homogeneous partial differential equation to be solved. Hence, for the Helmholtz equation, the enrichment field is chosen here as the superposition of plane waves. We enforce a weak continuity of these plane waves across the element interfaces by suitable Lagrange multipliers. Preliminary results obtained for two-dimensional model problems discretized by uniform meshes reveal that the proposed DGM enables the development of elements that are far more competitive than both the standard linear and the standard quadratic Galerkin elements for the discretization of Helmholtz problems.

Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications

AbstractGiven a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain Ω in Rn and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by “analytic” and “meromorphic” solutions of the equation Lu = 0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

Experimental investigation of friction-induced noise in disc brake systems

This paper presents experimental and analytical investigations examining the influence of the interfacial forces between a rotating disc and a friction element on the generation of chatter and squeal. Due to inevitable misalignment between the element and disc surface, a kinematic constraint instability known as sprag-slip is created. The experimental measurements include time history records of normal and friction forces, time variation of the friction coefficient, and acceleration of the friction element. The time history records of interfacial forces revealed short periods of high frequency component. It is found that the friction force is non-Gaussian and that its power spectral density covers a wide frequency band. The dependence of the root mean square of the friction coefficient on the relative velocity is found to have a negative slope at lower disc speeds. Depending on the direction of disc rotation, it is found that the friction velocity curve for clockwise disc speed is completely different from counter-clockwise rotation. The associated noise is also different in pitch and frequency content. The analytical modelling emulates the dynamics of the friction element. The transverse motion of the friction element is described by a homogeneous partial differential equation with non-homogeneous boundary conditions. The analysis shows that the normal force appears as a coefficient of the stiffness term, while the friction force appears as a non-homogeneous term. Since the normal force varies randomly as observed experimentally, it acts as a parametric noise to the friction element, and results in parametric instability in the form of squeal or vibration.

Generalized Partial Differential Equation and Fermat's Last Theorem

The equivalence of Fermat's Last Theorem and the non-existence of solutions of a generalized n th order homogeneous hyperbolic partial differential equation in three dimensions and periodic boundary conditions defined in a cubic lattice is demonstrated for all positive integer, n > 2. For the case n = 2, choosing one variable as time, solutions are identified as either propagating or standing waves. Solutions are found to exist in the corresponding problem in two dimensions.

Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients

We consider the Cauchy problem with respect to for a homogeneous linear partial differential equation with constant coefficients in two independent variables . We show that the relative smoothness with respect to and of analytic and ultradifferentiable solutions of the Cauchy problem depends essentially on the value of and, as a rule, is completely determined by it. We also obtain rather general uniqueness theorems and find conditions which guarantee that the particular solution constructed depends both continuously and linearly on the initial functions.

On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.

Direct calculation of two-dimensional collection probability in pn junction solar cells, and study of grain-boundary recombination in polycrystalline silicon cells

A new method for the direct calculation of the two-dimensional collection probability in pn junction solar cells is presented. Based on its reciprocity properties, the inhomogeneous continuity equation for excess carriers is transformed to a homogeneous partial differential equation (PDE) for the probability of carriers being collected in the external circuit. The new PDE is easier to solve and directly gives the short-circuit current. The method is applied to study the impact of grain-boundary recombination on the performance of polycrystalline silicon solar cells. A critical grain width of four times the carrier diffusion length in the base is found to be the limiting boundary between polycrystalline behavior and monocrystalline behavior of the cell. The sensitivity of short-circuit AM1.5 collection efficiency to the grain width Wg, the grain-boundary recombination velocity Sg, minority-carrier diffusion lengths, and surface recombination velocities, is quantified for a variety of cell types and recombination parameters. The sensitivity analysis indicates that AM1.5 collection efficiency is most sensitive to grain width in narrow grain cells, and to base diffusion length in wide grain cells.

A ‘‘compact Green function’’ approach to the time-domain direct and inverse problems for a stratified dissipative slab

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.

Designing localized waves

In this paper we re-interpret a recently introduced method for obtaining non-separable, localized solutions of homogeneous partial differential equations. This reinterpretation is in the form of a geometrical consideration of the algebraic constraint that the Fourier transforms of such solutions must satisfy in the transform domain (phase space). With this approach we link two classes of localized, non-separable solutions of the homogeneous wave equation, and examine the transform domain characteristic that determines the space-time localization properties of these classes. This characterization allows us to design classes of solutions with better localization properties. In particular, we design and discuss the properties of several novel subluminal and superluminal solutions of the homogeneous wave equation. We also design families of non-separable, localized, subluminal and superluminal solutions of the Klein-Gordon equation by using the same technique.

A method for constructing solutions of homogeneous partial differential equations: localized waves

We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

An effective boundary element method for inhomogeneous partial differential equations

A method for removing the domain or volume integral arising in boundary integral formulations for linear inhomogeneous partial differential equations is presented. The technique removes the integral by considering a particular solution to the homogeneous partial differential equation which approximates the inhomogeneity in terms of radial basis functions. The remainder of the solution will then satisfy a homogeneous partial differential equation and hence lead to an integral equation with only boundary contributions. Some results for the inhomogeneous Poisson equation and for linear elastostatics with known body forces are presented.

Hyperholomorphic functions and second order partial differential equations in 𝑅ⁿ

Hyperholomorphic functions in R n {R^n} with n ≥ 2 n \geq 2 are introduced, extending the hitherto considered hyperholomorphic functions in R 2 {R^2} . A Taylor formula is derived, and with it a unique representation theorem is proved for hyperholomorphic functions that are real analytic at the origin. Hyperanalyticity is seen to be generally a consequence of hyperholomorphy and real analyticity combined. Results for hyperholomorphic functions are applied to gradients of solutions of second order homogeneous partial differential equations with constant coefficients. Polynomial solutions of such a second order equation are obtained by a matrix algorithm. These polynomials are modified and combined to form polynomial bases for real analytic solutions. It is calculated that in such a basis there are ( m + n − 3 ) ! ( 2 m + n − 2 ) / m ! ( n − 2 ) ! (m + n - 3)!(2m + n - 2)/m!(n - 2)! homogeneous polynomials of degree m m .

Transient evolution of steady‐state shock profiles for sonic booms in a relaxing atmosphere

When a shock wave propagates through a uniform medium over a long distance, there is an asymptotic tendency for the portion of the waveform that includes a sudden pressure rise (shock) to approach an invariant form F(x−Vl), where Vis the shock speed and where the function F, in addition to the indicated argument, depends only on the net pressure jump Psh and the local properties of the atmosphere (including relaxation times). The present paper is concerned with the broad question of how long it takes (or how far the shock must propagate) before this asymptotic limit is realized, given some other nonasymptotic profile at some initial time. Although this question has in principle been answered by Benton and Platzman [Q. Appl. Math. (1972)] for the one‐dimensional Burgers' equation, new techniques are required when internal relaxation is to be taken into account. A perturbation theory is used here, with the shock profile taken as the steady‐state profile plus a perturbation, which must asymptotically decay to zero. This perturbation, to lowest order, is found to satisfy a set of linear homogeneous partial differential equations with nonconstant coefficients. With the choice of l and ξ = x − Vl as the independent variables, the coefficients depend only on ξ. Consequently, the equations can be solved for any given initial value problem with the use of Laplace or Fourier transforms. The asymptotic decay of the inverse Fourier transforms to zero is developed by well‐known techniques such as described in the text by Bleistein and Handelsman [Dover, New York (1986)]. [Work supported by NASA‐LRC and by the William E. Leonhard endowment to Penn State Univ. The author acknowledges the advice of A. D. Pierce.]When a shock wave propagates through a uniform medium over a long distance, there is an asymptotic tendency for the portion of the waveform that includes a sudden pressure rise (shock) to approach an invariant form F(x−Vl), where Vis the shock speed and where the function F, in addition to the indicated argument, depends only on the net pressure jump Psh and the local properties of the atmosphere (including relaxation times). The present paper is concerned with the broad question of how long it takes (or how far the shock must propagate) before this asymptotic limit is realized, given some other nonasymptotic profile at some initial time. Although this question has in principle been answered by Benton and Platzman [Q. Appl. Math. (1972)] for the one‐dimensional Burgers' equation, new techniques are required when internal relaxation is to be taken into account. A perturbation theory is used here, with the shock profile taken as the steady‐state profile plus a perturbation, which must asymptotically decay t...

Boundary element approach for identification of point forces of distributed parameter systems

Abstract Recently the identification of an external force applied to distributed parameter systems has become increasingly important in relation to the analysis and control of geophysical and environmental phenomena. A boundary element approach is developed for identifying the external force applied to a number of points on the domain of a distributed parameter system, which is described by a non- homogeneous partial differential equation of parabolic (diffusion) type. Firstly, an integral equation corresponding to the given partial differential equation is derived. The solution of the integral equation at several reference points is computed by using the boundary element method, and then the locations and magnitudes of the external force are identified with the number of application points by minimizing the sum of the squares of relative errors. Numerical examples are presented to illustrate the usefulness of the proposed method.

Mode-Dependent Finite-Difference Discretization of Linear Homogeneous Differential Equations

A new methodology utilizing the spectral analysis of local differential operators is proposed to design and analyze mode-dependent finite-difference schemes for linear homogeneous ordinary and partial differential equations. We interpret the finite-difference method as a procedure for approximating exactly a local differential operator over a finite-dimensional space of test functions called the coincident space, and show that the coincident space is basically determined by the nullspace of the local differential operator. Since local operators are linear and approximately with constant coefficients, we introduce a transform domain approach to perform the spectral analysis. For the case of boundary-value ordinary differential equations (ODEs), a mode-dependent finite-difference scheme can be systematically obtained. For boundary-value partial differential equations (PDEs), mode-dependent 5-point, rotated 5-point, and 9-point stencil discretizations for the Laplace, Helmholtz, and convection-diffusion equations are developed. The effectiveness of the resulting schemes is shown analytically, as well as by considering several numerical examples.

Operator products in 2-dimensional critical theories

A noteworthy feature of certain conformally invariant 2-dimensional theories, such as the Ising and 3-state Potts models at the critical point, is the existence of “degenerate primary fields” associated with nullvectors of the Virasoro algebra. Such fields are endowed with a remarkably simple multiplication table under the operator product expansion, known as the fusion rules. In addition, correlation functions made up of these fields satisfy a system of linear homogeneous partial differential equations. We show here that these two properties are intimately related: for any n-point function, the number of conformally invariant solutions to the system of equations equals the number of times that the identity operator appears in the fusion of all n fields in the correlator. This theorem permits the calculation of some apparently intractable correlation functions. Finally, we generalize these ideas to the Neveu-Schwarz sector of superconformal theories.

Recursive construction of a sequence of solutions to homogeneous linear partial differential equations with constant coefficients

For second order homogeneous partial difference equations with constant coefficients in n variables, it is always possible to construct a generating function of exponential form containing n − 1 arbitrary parameters, from which a sequence of solutions ( P i 1,…, i n − 1 ) can be derived recursively. The determination of a solution u = Σ⋯ Σai 1,…, i n − 1 Pi 1,…, i n − 1 that approximately satisfies given boundary conditions, for a given problem, is then possible by solving a linear least squares problem. Once the a i 1,…, i n − 1 are known, the evaluation of u and its partial derivatives can be done very easily, using a second recursion, that can be derived from the recursion used for the determination of the P i 1,…, i n − 1 . Some classical examples will be treated, for which the relevant formulae are given completely.