On the connection of the superposition and homogeneous-solutions methods in wave propagation problems in a semi-infinite rigidly-clamped elastic layer

The process of harmonic wave propagation is investigated in a semi-infinite rigidly-clamped elastic layer. An analytic solution of the problem is obtained by the superposition method. The wave field expansion in the form of a normal mode series for a corresponding infinite waveguide is established. According to residue theory, the explicit form of the expansion coefficients is established with physical requirements of the radiation conditions taken into account.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 3–10, 1988.     

Numerical solution of the Burgers’ equation by automatic differentiation

We compute the solution of the one-dimensional Burgers’ equation by marching the solution in time using a Taylor series expansion. Our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non-linear algebraic equations. Instead, we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the Taylor series. We illustrate the effectiveness of our method by solving four test problems with known exact solutions. The numerical solutions we obtain are in excellent agreement with the exact solutions, while being superior to other previously reported numerical solutions.     

Propagation of weakly guided waves in a Kerr nonlinear medium using a perturbation approach

The equations are represented in a simplified format with only a few leading terms needed in the expansion. The set of equations are then solved numerically using vector finite element method. To validate our algorithm, we analyzed a two-dimensional rectangular waveguide consisting of a linear core and nonlinear identical cladding. The exact nonlinear solutions for three different modes of propagations, TE0, TE1, and TE2 modes are generated and compared with the computed solutions. Next, we investigate the effect of a more intense monochromatic field on the propagation of a “weak” optical field in a fully three-dimensional cylindrical waveguide.     

Analytical solution for the fully coupled static response of variable stiffness composite beams

An analytical solution is presented for the 3D static response of variable stiffness non-uniform composite beams. Based on Euler-Bernoulli theory, a set of governing differential equations are obtained, in which four degrees of freedom are fully coupled. For the variable stiffness beam, the governing field equations have variable coefficients reflecting the stiffness variation along the beam. Using the direct integration technique, the general analytical solution is derived in the integral form and the closed-form expressions of the obtained solutions are presented employing a series expansion approximation. The series expansion representation enables the proposed approach to be applicable for variable stiffness composite beams with arbitrary span-wise variation of properties. As an alternative solution, the Chebyshev collocation method is applied to the proposed formulation to verify the results obtained from the analytical solution. A number of variable stiffness composite beams made by fibre steering with various boundary conditions and stacking sequences are considered as the test cases. The static response are presented based on the analytical solution and Chebyshev collocation method and excellent agreement is observed for all test cases. The proposed model presents a reliable and efficient approach for capturing the complicated behaviour of variable stiffness non-uniform composite beams.     

An exact solution for variable coefficients fourth-order wave equation using the Adomian method

This paper presents a novel approach for the analysis of a fourth-order parabolic equation dealing with vibration of beams by using the decomposition method. In this regard, a general approach based on the generalized Fourier series expansion is applied. The obtained analytic solution is simplified in terms of a given set of orthogonal basis functions. The result is compared with the classical modal analysis technique which is widely used in the field of structural dynamics. It is shown that the result of the decomposition method leads to an exact closed-form solution which is equivalent to the result obtained by the modal analysis method. Some examples are given to demonstrate the validity of the present study.     

Expansion of random field gradients using hierarchical matrices

We present two expansions for the gradient of a random field. In the first approach, we differentiate its truncated Karhunen-Loève expansion. In the second approach, the Karhunen-Loève expansion of the random field gradient is computed directly. Both strategies require the solution of dense, symmetric matrix eigenvalue problems which can be handled efficiently by combining hierachical matrix techniques with a thick-restart Lanczos method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation modes in a nonlinear optical system with distributed field rotation

We consider a model of a nonlinear optical system with distributed field rotation described by a functional-differential diffusion equation. An existence theorem is proved for periodical spatially nonhomogeneous traveling-wave solutions, which are generated from a spatially homogeneous stationary solution by an Andronov-Hopf (cycle-generating) bifurcation. A series expansion of the solution in powers of a small parameter is obtained and a stability condition is given. Simulation results are used to discuss the properties of the model. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 89–99.     

The Ground State Energy of the Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.     

Analytical results for 2‐D non‐rectilinear waveguides based on a Green's function

We consider the problem of wave propagation for a 2‐D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of the matrix riemann problem to the study of excitation of natural waves in an acoustic waveguide by a piston radiator

Stationary excitation of natural waves in a flat acoustic waveguide with absolutely soft lateral walls by a piston radiator is investigated. In the case where the length of the radiator is equal to the halved width of the waveguide, the exact analytic solution is obtained. The representation of the acoustic field in the waveguide in the form of an expansion over the natural waves of the waveguide is provided.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 129–132.     

Solution of a stochastic Darcy equation by polynomial chaos expansion

This paper deals with solving a boundary value problem for the Darcy equation with a random hydraulic conductivity field.We use an approach based on polynomial chaos expansion in a probability space of input data.We use a probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. The computational complexity of this algorithm is determined by the order of the polynomial chaos expansion and the number of terms in the Karhunen–Loève expansion. We calculate various Eulerian and Lagrangian statistical characteristics of the flow by the conventional Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method over the directMonte Carlo algorithm.     

A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions

In this paper we propose a new finite element realization of the Perfectly Matched Layer method (PML-method). Our approach allows to deal with a wide class of polygonal domains and with certain types of inhomogeneous exterior domains. Among the covered inhomogeneities are open waveguide structures playing an essential role in integrated optics. We give a detailed insight into implementation aspects. Numerical examples show exponential convergence behavior to the exact solution with the thickness of the PML sponge layer.     

Unsteady Couette flow of viscous fluid under a non-uniform magnetic field

The aim of this letter is to construct the analytic solution for unsteady Couette flow in the presence of an arbitrary non-uniform applied magnetic field. The flow is induced by a generalized velocity given to the lower plate. The perturbed eigenfunction expansion method is employed to develop a series solution for small magnetic field.     

Spectral parameter power series analysis of isotropic planarly layered waveguides

This work addresses the analysis of an isotropic planarly layered waveguide consisting of an inhomogeneous core that is enclosed between two homogeneous layers forming the cladding. The analysis relies on an auxiliary one-dimensional spectral problem that is intimately linked with the scalar wave equation for planarly layered media. We construct the Green function of the waveguide as an expansion involving the eigenfunctions of the continuous and the discrete spectrum of the auxiliary problem. From the eigenvalues of the discrete spectrum, we calculate the allowed propagation constants of the guided modes. The Spectral Parameter Power Series (SPPS) method [Math. Method Appl. Sci. 2010;33: 459–468] leads us to analytic expressions for the eigenfunctions of the auxiliary problem in the form of power series of the spectral parameter. In addition, we obtain an SPPS representation for the dispersion relation without making any kind of approximation or discretisation to the core of the waveguide. The SPPS analysis here presented is well suited for its numerical implementation, since all these series can be truncated due to their uniform convergence.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Laguerre polynomial approach for solving linear delay difference equations

This paper presents a numerical method for the approximate solution of mth-order linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre series expansion approach. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple.     

Symmetry,validation, and scattering matrices for time–domain scattering in waveguides

We outline a method to compute the solution in the frequency–domain for scattering in a waveguide by exploiting symmetry. The method is illustrated by considering a simple scattering example, where soft hard boundary conditions are alternated. We show how the straightforward mode matching or eigenfunction matching solution can be easily converted to scattering and transmission matrices when symmetry is exploited. We then show how the solution for two scatterers can be found explicitly, using symmetry which allows validation of our subsequent solution by scattering matrices. We also give a series of identities which the scattering matrix must satisfy for further numerical validation. Using these frequency–domain solutions we compute the time-domain scattering by incident Gaussian wave–packets.     

Diffraction of normal modes in composite and stepped elastic waveguides

A method of calculating the diffraction of Lamb waves at the vertical junctions of elastic waveguides is proposed, the effectiveness of which is ensured by taking into account the nature of the singularity of the solution at corner points. The property of the generalized orthogonality of normal modes plays a key role. It enables the coefficients of the expansion of the wave field in the modes to be expressed in terms of the displacements and stresses along the junction line. Numerical results are presented which show how the transmission coefficients and the energy distribution depend on the height of the step and frequency for a stepped waveguide, attached to an undeformed substrate.