Linear Combination Of Solutions Research Articles

Analytical study of the pantograph equation using Jacobi theta functions

The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y′(x)=ay(qx)+by(x), where a and b are two non-zero real or complex numbers. When 0<q<1 and y(0)=1, the associated Cauchy problem admits a unique power series solution, ∑n≥0(−a/b;q)nn!(bx)n, that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.

A localized collocation solver based on fundamental solutions for 3D time harmonic elastic wave propagation analysis

This paper presents the localized collocation solver based on fundamental solutions to 3D elastic wave propagation analysis. In the proposed collocation solver, the approximated solution at the considered node is represented by a linear combination of the solutions at a few nearest nodes in the stencil support of the considered node instead of the whole discretization nodes. The related weighting functions are determined by solving a system of linear equations, which is constructed as a linear combination of fundamental solutions in the related stencil support. Therefore, the proposed collocation solver produces sparse resultant matrix, which makes it possible to perform large-scale elastic wave propagation simulations on a desktop computer. Furthermore, it avoids highly ill-conditioned dense matrix encountered in most of collocation methods. The efficiency and accuracy of the proposed method have been verified under several benchmark examples.

Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations

The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS.

Dirac, Majorana and Weyl particles in the oscillating de Sitter universe, reflection from the cosmological barrier

It is known that the geometry of the Lobachevsky space acts on the fields of particles with spins 0, 1/2, 1 as an ideal mirror distributed in space. The depth of penetration of the field in such a medium increases with increasing field energy. Since the Lobachevsky model is a constituent element in some cosmological models, this property means that in such models it is necessary to take into account the effect of the presence of a “cosmological mirror”; it must lead to a redistribution of the particle density in space. The earlier analysis assumed the static nature of the space-time geometry. In this article, we generalize the research of the spin 1/2 field in the case of the oscillating model of the de Sitter universe. The Dirac equation is solved in the non-static quasi-Cartesian coordinates. At this, we substantially use the diagonalization of a generalized helicity operator. The wave functions of the particle are nontrivially time-dependent; however the effect of a complete reflection of the particles from an effective potential barrier is preserved. For the real Majorana 4-spinor field, the similar results are valid. For the solutions describing the reflection effect to be constructed, we must use linear combinations of solutions with opposite helicities. Such combinations are forbidden for 2-component Weyl particles, for this reason such particles cannot be reflected by the cosmological barrier.

Exact solutions of the Grad–Shafranov equation via similarity reduction and applications to magnetically confined plasmas

We derive exact solutions of a linear form of the Grad–Shafranov (GS) equation, including incompressible equilibrium flow, using ansatz-based similarity reduction methods. The linearity of the equilibrium equation allows linear combinations of solutions in order to obtain axisymmetric MHD equilibria with closed and nested magnetic surfaces which are favorable for the effective confinement of laboratory plasmas. In addition, employing the same reduction methods we obtain analytical solutions for several non-linear forms of the GS equation. In this context analytic force-free solutions in both linear and nonlinear regimes are also derived.

On the First Time that a 3-D Bessel Bridge Hits a Boundary

In this work we derive the density of the first time that a 3-D Bessel bridge started at y > 0, reaches a linear level a + b·t for t ≥ 0 from above, y > a. As well as from below y < a. The reason as to why a distinction has to be made will be clarified. Next, a sort of equivalence relationship between the 3-D Bessel bridge, the 3-D Bessel process, and the Brownian bridge is derived. In order to do so, we first introduce a new classification of SDEs in terms of solutions of Burgers’ equation. For instance, these processes fall within the same class. This follows since one can show that the local drift of each is a solution to Burgers’ equation. However, the local drift of the 3-D Bessel bridge, is also the sum of two solutions of Burgers’ equation. In general, linear combinations of solutions of Burgers’ equation are not solutions themselves.

Remarks on the overlapping-sphere method for molecular orbitals

In preference to the overlapping-sphere multiple-scattering method for solving the self-consistent-field problem for a molecule or atomic cluster, the authors recommend a cellular method. Each nucleus is surrounded by a cell, and the cells taken together fill all space and do not overlap. The cells in such a case as a diatomic molecule extend out to infinity. For an approximation one can assume a spherically symmetrical potential in each cell. The general solution of Schroedinger's equation within a cell is then set up as a linear combination of solutions of the spherical problem for a single energy, but for different l and m values. In order to have the functions regular at the nucleus and at infinity, one must join two independent solutions for each l value, one regular at the origin and the other at infinity, over the surface of a sphere of radius r/sub 1/, on which the functions are made continuous, but the normal derivatives are discontinuous. The coefficients of the functions are so chosen that the sum is continuous with continuous derivative over the spheres of radius r/sub 1/, and over the planes separating neighboring atoms. In practice, one uses a finite sum of functions, perhapsmore » for i = 0, 1, ..., 10, and applies the conditions of continuity at enough points to determine the coefficients. By expressing the potential and the radial functions as power series in r, or in r - r/sub 0/, one can get a convenient analytic solution of Schroedinger's equation. The various steps involved in this method have been studied, but they have not yet been combined into a computer program, so that this paper is regarded as a progress report.« less

Computing the Continuous-spectrum Linearised Bounded Standing Wave on a Plane Bed of Arbitrary Slope

The problem is revisited of continuous-spectrum small-amplitude bounded standing waves over a plane beach of arbitrary incline. Three documented approaches are compared and discussed vis-??-vis computational difficulties and this reveals, in particular, that the computation in established models, for particular beach angles, is optimised by seeking solutions to a functional integro-difference equation in a L2 space. This simplification is found to be impossible for general beach angles and developed instead is an alternative method initially based on Cauchy Principal value convergence of the inversion integral but improved (by linear combination of solutions) and developed into a comprehensive computational package which is validated by three fundamental tests. A single f77 routine is provided for the user through a web-link. Comparisons with established but hitherto un-computed methods are found numerically to favour the present development. Examples presented include both very steep and very shallow beaches and both potentials and velocities are calculated and displayed throughout the water column.

Weld analysis using an inverse-problem approach based on quasi-one-dimensional interpolation

In this article, interpolation properties are examined that are inherent in the physical character of heat-deposition processes such as welding and that are relevant to using an inverse-problem approach for their analysis. This study considers interpolation properties of both parametric and non-parametric representations of the temperature field for heat-deposition processes. In the case of parametric representations of the temperature field, linear combinations of solutions to the heat-conduction equation are adopted as interpolation functions rather than strictly explicit physical-model representations of heat-deposition processes. In the case of non-parametric representations of the temperature field, interpolation procedures that are applicable to discrete representations of the temperature field are adopted. Issues related to parameter optimization and the practical application of methods based on the inverse-problem approach related to interpolation properties are also discussed.

A multi‐domain weighted residual method for the one‐electron Schrödinger equation: Application to H+2

The Schrodinger equation is solved for a single electron moving in the coulombic field of some arbitrary configuration of nuclei. Space is partitioned by centering a sphere on each of the individual nuclei without any overlap or touching of the spheres, i.e., muffin‐tin spheres. All regions are treated by a weighted residual technique, which is a more general approach than the variational method. Outside the spheres, both the wavefunction and its product with the potential energy function are expanded as a linear combination of solutions taken from the modified Helmholtz equation (M.H.E.). A basis set is prepared by solving the M.H.E. repeatedly for a select set of eigenvalues and boundary conditions, using a boundary integral technique. Inside any sphere, the wavefunction is written as a linear combination of terms, each a product of a radial function and a spherical harmonic. The radial factor is written as product of an exponential and a power series. For either region, an alternate basis set is chosen...

Application of the edge function method to rock mechanics problems

The edge function method is considered as an alternative to conventional numerical schemes for the solution of plane problems in rock mechanics. The essence of the approach is the approximation of the solution by a linear combination of solutions of the field equations. The unknowns in the linear combination are obtained from a system of equations which follows from the approximation of the boundary conditions by a boundary Galerkin energy method. No mesh generation is required over the domain or boundary of the problem. Previous edge function work in anisotropic elasticity is enhanced by the incorporation of a special solution for the effect of gravity. Examples are presented to illustrate the applicability of the method in determining stresses in various rock mechanics problems. A high level of accuracy is achieved with a relatively small number of degrees of freedom. Convergence is rapid because of the inclusion of special analytic solutions to model stress concentrations. The inclusion of the gravity force does, however, lead to a small increase in the number of degrees of freedom needed to achieve acceptable results. The optimum use of the edge function method, at present, may be as a special element within more general finite element or discrete element codes.

Surface motion of topographical irregularities for incidentP, SV,and Rayleigh waves

AbstractTrefftz's method is applied to solve the diffraction of P, SV, and Rayleigh waves by surface irregularities in an elastic, isotropic and homogeneous half-space. The diffracted wave fields are constructed with linear combinations of solutions which form c-complete or T-complete (T after Trefftz) families for the wave equation. Boundary conditions are satisfied in the least-squares sense on the irregularity and on a finite interval of the half-space surface. Excellent agreement was found with the solution by Wong (1982).

Transport and Diffusion Calculations during Time- and Space-Varying Meteorological Conditions Using a Microcomputer

Small computers can now solve complex transport and diffusion problems. Making such calculations rapidly and on the spot would be very useful for decision makers during emergencies. This paper describes a microcomputer model that can simulate nonsteady state transport and diffusion and calculate mass-consistent flow fields in uneven terrain. Transport and diffusion calculations use a Puff model approach to simulate dispersion from a continuous point source. Three-dimensional wind fields are generated from linear combinations of solutions that are obtained ahead of time on a larger machine. A computer code has been written in BASIC and successfully run on an Apple II personal computer.

Sturm comparison and separation theorems for linear, second order, self-adjoint, ordinary, differential equations and for first order systems

Sturm comparison and separation theorems, for first order linear combinations of solutions of linear, second order, self-adjoint, ordinary, differential equations, are obtained as consequences of Sturm comparison and separation theorems for linear combinations of components of solutions of linear, first order systems of ordinary, differential equations.

Solution of Boundary Value Problems in Plane Strain

THIS method replaces direct solutions of the Airy relation by, in effect, linear combinations of solutions of this equation for the edge dislocation. So far its use has been investigated only in connexion with the stress fields of dislocations in finite bodies; it may well be that it is useful in other connexions as well.