Solving the 2D Maxwell equations by a Laguerre spectral method

A spectral method for solving the 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method is based on an expansion of the solution in terms of Laguerre functions in time. The operation of convolution of functions, which is part of the formulas describing the relaxation processes, is reduced to a sum of products of the harmonics. The Maxwell equations transform to a system of linear algebraic equations for the solution harmonics. In the algorithm, an inner parameter of the Laguerre transformis used. With large values of this parameter, the solution is shifted to high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution. Results of a comparison of the Laguerre method and a finite-difference method in accuracy both for a 2D medium structure and a layered medium are given. Results of a comparison of the spectral and finite-difference methods in efficiency for axial and plane geometries of the problem are presented.     

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.     

An analytical method for solving elastic system in inhomogeneous orthotropic media

In this paper, the three‐dimensional initial value problem for elastic system in inhomogeneous orthotropic media is considered and an analytical method is studied to solve this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type, whose solution is obtained by the method of successive approximations. Finally, using the real Paley‐Wiener theorem, it is shown that the solution of the initial value problem can be found by the inverse Fourier transform.     

A fully diagonalized spectral method using generalized Laguerre functions on the half line

A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method.     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Numerical modeling of seismic and acoustic-gravity waves propagation in an “Earth-Atmosphere” model in the presence of wind in the air

In this paper, an effective numerical algorithm for 2.5D seismic and acoustic-gravitational wave propagation is applied to a combined “Earth-Atmosphere” model in the presence of wind in the air. Seismic wave propagation in an elastic half-space is described by a system of first-order dynamic equations of elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere in the presence of wind is described by the linearized Navier-Stokes equations. The algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform with respect to a spatial coordinate combined with a finite difference method for the reduced problem.     

Stokes' first problem for a Newtonian fluid in a non‐Darcian porous half‐space using a Laguerre–Galerkin method

A Laguerre–Galerkin method is proposed and analysed for the Stokes' first problem of a Newtonian fluid in a non‐Darcian porous half‐space on a semi‐infinite interval. It is well known that Stokes' first problem has a jump discontinuity on boundary which is the main obstacle in numerical methods. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi‐infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations of the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A method of solving evolutionary problems based on the Laguerre step-by-step transform

In [1], Mikhailenko proposed a method of solving dynamic problems of elasticity theory. The method is based on the Laguerre transform with respect to time. In this paper, we propose a modification of this approach, applying the Laguerre transform to a sequence of finite time intervals. The solution obtained at the end of one time interval is used as initial data for solving the problem on the next time interval. To implement the approach, four parameters are chosen: a scale factor to approximate the solution by Laguerre functions, an exponential coefficient of a weight function that is used for finding a solution on a finite time interval, the duration of this interval, and the number of projections of the Laguerre transform. A way to find parameters that provide stability of calculations is proposed. The effect of the parameters on the accuracy of calculations when using second- and fourth-order difference schemes is studied. It is shown that the approach makes it possible to obtain a high-accuracy solution on large time intervals.     

A step-by-step method with Laguerre functions for solving time-dependent problems

A step-by-step modification of the well-known approach proposed by Mikhaylenko and Konyukh to solving dynamic problems is proposed. The approach is based on the Laguerre transform with respect to time. In this modification the Laguerre transform is applied to a sequence of finite time intervals. The solution obtained at the end of a time interval is used as the initial data for solving the problem on the next time interval. The method is illustrated by examples for the harmonic oscillator problem and the 1D wave equation. Accuracy and stability of the method are analyzed. This approach allows obtaining a solution of high accuracy on large time intervals.     

The numerical solution of an evolution problem of second order in time on a closed smooth boundary

We consider an initial value problem for the second-order differential equation with a Dirichlet-to-Neumann operator coefficient. For the numerical solution we carry out semi-discretization by the Laguerre transformation with respect to the time variable. Then an infinite system of the stationary operator equations is obtained. By potential theory, the operator equations are reduced to boundary integral equations of the second kind with logarithmic or hypersingular kernels. The full discretization is realized by Nyström's method which is based on the trigonometric quadrature rules. Numerical tests confirm the ability of the method to solve these types of nonstationary problems.     

Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross–Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre–Fourier spectral method with respect to the $(x,y)$ -variables as well as the Hermite spectral method in the $z$ -direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.     

Spectral methods using generalized Laguerre functions for second and fourth order problems

Spectral methods using generalized Laguerre functions are proposed for second-order equations under polar (resp. spherical) coordinates in ?² (resp. ?³) and fourth-order equations on the half line. Some Fourier-like Sobolev orthogonal basis functions are constructed for our Laguerre spectral methods for elliptic problems. Optimal error estimates of the Laguerre spectral methods are obtained for both second-order and fourth-order elliptic equations. Numerical experiments demonstrate the effectiveness and the spectral accuracy.     

Non-axisymmetric consolidation of poroelastic multilayered materials with anisotropic permeability and compressible constituents

This paper presents an analytical layer-element solution to non-axisymmetric consolidation of multilayered poroelastic materials with anisotropic permeability and compressible constituents. By applying Fourier expansions, Hankel transforms and Laplace transforms to the state variables involved in the governing equations of poroelasticity with respect to the circumferential, radial and time coordinates, respectively, the analytical layer-element (i.e. a symmetric stiffness matrix) is derived, which describes the relationship between the transformed generalized stresses and displacements at the surface (z = 0) and those at an arbitrary depth z, considering the corresponding boundary and continuity conditions at the layer interfaces, the global stiffness matrix of a multilayered system is assembled in the transformed domain. The actual solutions in the physical domain are acquired by applying numerical quadrature schemes for the inversion of the Laplace–Hankel transform. Finally, numerical calculation is presented to investigate the influence of layering and poroelastic material parameters on consolidation process.     

A hybrid method for pricing European options based on multiple assets with transaction costs

The problem of pricing European options based on multiple assets with transaction costs is considered. These options include, for example, quality options and options on the minimum of two or more risky assets. The value of these options is the solution of a nonlinear parabolic partial differential equation subject to a final condition given by the payoff function associated with the option. A computationally efficient method to solve this final-value problem is proposed. This method is based on an asymptotic expansion of the required solution with respect to the parameters related to the transaction costs followed by the numerical solution of the linear partial differential equations obtained at each order in perturbation theory. The numerical solution of these linear problems involves an implicit finite-difference scheme for the parabolic equation and the use of the fast Fourier sine transform to solve the resulting elliptic problems. Numerical results obtained on test problems with the method proposed here are shown and discussed.     

变厚度圆薄板非对称非线性弯曲问题

首先将直角坐标系中的横向变厚度薄板的大挠度方程,转化到极坐标系中的变厚度圆薄板的非对称大挠度方程·　此方程和极坐标系中径向、切向两个平衡方程联立求解·　将物理方程和中面应变非线性变形方程,代入3个平衡方程,可得用3个变形位移表示的3个非对称非线性方程·　用Fourier级数表示的解代入基本方程,获得相应的基本方程·　在周边夹紧边界条件下,用修正迭代法求解·　作为算例,研究了余弦形式载荷作用下的问题,还给出了载荷与挠度的特征曲线,曲线依据变厚度参数变化而变化,其结果和物理概念完全吻合·     

A study of the deflection of an isotropic shell under the action of an abruptly applied concentrated normal force

We consider the perturbed deformed state of a thin shallow isotropic shell of nonnegative Gaussian curvature under the action of an abruptly applied concentrated normal force. Using the integral Fourier transform method with respect to the geometric coordinates and the Laplace transform with respect to time we obtain a solution of this problem in the form of double series in generalized Fresnel sine and cosine integrals. We give numerical results for the initial reaction of the shell at the point of application of the load. We give a comparison with known results. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 58–63, 1991.     

Legendre-Laguerre coupled spectral element methods for second- and fourth-order equations on the half line

Some Legendre spectral element/Laguerre spectral coupled methods are proposed to numerically solve second- and fourth-order equations on the half line. The proposed methods are based on splitting the infinite domain into two parts, then using the Legendre spectral element method in the finite subdomain and Laguerre method in the infinite subdomain. C⁰ or C¹-continuity, according to the problem under consideration, is imposed to couple the two methods. Rigorous error analysis is carried out to establish the convergence of the method. More importantly, an efficient computational process is introduced to solve the discrete system. Several numerical examples are provided to confirm the theoretical results and the efficiency of the method.     

A spectral method of solving two-dimensional integral equations of convolution type

Two-dimensional integral equations of convolution type can be solved by the spectral method in the Chebyshev-Laguerre polynomial basis. We construct an exact parametric representation of the generalized Laguerre spectrum of the required solution with respect to the known Laguerre spectrum of the right-hand side and the kernel of the equation.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 84–88.     

The solution of the two-dimensional sine-Gordon equation using the method of lines

The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor–corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.     

On a method for solving the inverse scattering problem on the line

A method for solving the inverse scattering problem on the line is proposed. It is based on a Fourier‐Laguerre series representation of the integral transmutation kernel. Substitution of the representation into the Gel'fand‐Levitan‐Marchenko equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential.