Theory of multiple scattering

Consideration is given to problems of obtaining exact and approximate solutions of kinetic equations in the multiple scattering problem. For cross sections which are rational functions of χ² (χ = 2sin(δ/2), δ is the scattering angle) exact solutions are obtained as a series in terms of Legendre polynomials. The limits of validity of the kinetic equation for the distribution function in terms of the variable q = 2sin(?/2) are refined [1] and the solutions of this equation are compared with the exact solutions of the Rutherford and Mott cross sections. The problem of convergence of approximate solutions in the form of a series in terms of Legendre polynomials and a series in powers of 1/B is solved. These approximations are obtained and their limits of validity are determined.     

The EPS method: A new method for constructing pseudospectral derivative operators

We develop a new type of derivative matrix for pseudospectral methods. The norm of these matrices grows at the optimal rate O(N²) for N-by-N matrices, in contrast to standard pseudospectral constructions that result in O(N⁴) growth of the norm. The smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation. The construction is based on representing the derivative operator as an integral kernel, and does not rely on the interpolating polynomials. In particular, we construct second derivative matrices that incorporate Dirichlet or Neumann boundary conditions on an interval and on the disk, but the method can be used to construct a wide variety of commonly used operators for solving PDEs and integral equations. The construction can be used with any quadrature, including traditional Gauss–Legendre quadratures, but we have found that by using quadratures based on prolate spheroidal wave functions, we can achieve a near optimal sampling rate close to two points per wavelength, even for non-periodic problems. We provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices, while resulting in a norm that is orders of magnitude smaller than the standard construction. To demonstrate the advantage of the new construction, we apply the method for solving the wave equation in constant and discontinuous media and for solving PDEs on the unit disk. We also present two compression algorithms for applying the derivative matrices in O(N log N) operations.     

Second-order time evolution of PN equations for radiation transport

Using polynomials to represent the angular variation of the radiation intensity is usually referred to as the P_N or spherical harmonics method. For infinite order, the representation is an exact solution of the radiation transport solution. For finite N, in some physical situations there are oscillations in the solution that can make the radiation energy density be negative. For small N, the oscillations may be large enough to force the material temperature to numerically have non-physical negative values. The second-order time evolution algorithm presented here allows for more accurate solutions with larger time steps; however, it also can resolve the negativities that first-order time solutions smear out. Therefore, artificial scattering is studied to see how it can be used to decrease the oscillations in low-order solutions and prevent negativities. Small amounts of arbitrary, non-physical scattering can significantly improve the accuracy of the solution to test problems. Flux-limited diffusion solutions can also be improved by including artificial scattering. One- and two-dimensional test results are presented.     

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Dynamic response of orthotropic plate using BEM with approximate fundamental solution

In order to overcome the difficulty in finding the fundamental solution in the boundary element method for dynamic problems of orthotropic plates, approximate fundamental solutions are traced. Two types of approximate fundamental solutions are considered: i.e., the series solution and the superposition solution. The former is made up of Hermitian orthogonal polynomials and the latter consists of an analytical fundamental solution for the static bending problem of a generalized isotropic plate and a series solution for correction. Some numerical examples amply show the feasibility and effectiveness of the method offered.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

Exact and approximate solutions to Schrödinger’s equation with decatic potentials

The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax ¹⁰ + bx ⁸ + cx ⁶ + dx ⁴ + ex ², a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.     

Surface electromagnetic waves supported by nano conducting layers with inhomogeneities in the conductivity profile

Conducting interfaces and nano conducting layers can support surface electromagnetic waves. Uniform charge layers of non-zero thickness and their asymptotic behavior toward conducting interfaces of infinitely small thicknesses, where the thin charge layer is modeled via a surface conductivity σ _s , are already studied. Here, the possible effects of inhomogeneity in the conductivity profile of the thin conducting layers are investigated for the first time and a new approximate yet accurate enough analytical formulation for mode extraction in such structures is given. In order to rigorously analyze the structure and justify the proposed approximate formulation, the Galerkin’s method with Legendre polynomial basis functions is applied, i.e. the transverse electric field for the TE polarized surface waves and the transverse magnetic field for the TM polarized surface waves are each expanded in terms of Legendre polynomials and then each eigenmode; subjected to appropriate boundary conditions, is sought in the complete space spanned by Legendre basis functions. The proposed approximate solution is then proved to be accurate. In particular, sinusoidal fluctuations are introduced into formerly uniform conductivity profiles and it is numerically demonstrated that surface electromagnetic waves supported by nano conducting layers are not much sensitive to the very shape of conductivity profiles.     

Perturbation theory and the Rayleigh quotient

The characteristic frequencies ω of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ε, it is shown that when an approximate eigenfunction is known with an error O(ε^N), the Rayleigh quotient gives the frequency with an error O(ε^2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theory.     

Legendre multiwavelet collocation method for solving the linear fractional time delay systems

In this article the Legendre multiwavelet basis with aid of collocation method has been applied to give approximate solution for fractional delay systems. The properties of Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of algebraic system. Numerical results and comparison with exact solutions in the cases when we have exact solution are given in test examples in order to demonstrate the applicability and efficiency of the method.     

大平板瞬态热传导问题的一种新的近似解法

１引言大平板瞬态热传导问题有着广泛的工程应用背景。对于复杂的初边值条件或含内热源问题,以及工程上常见的多层复合平壁对象,分析求解难度很大甚至无法求解。在此类情况下往往采用数值方法。但是单纯的数值解不便于理解影响该问题的各种参数的物理意义。因此,各种近似分析方法得到了发展［１,２］。但在近似精度上,往往难以对整个时间坐标范围都达到较高的精度,这就使得近似解更多地局限于定性分析。此外,对于不同的初边值条件或含内热源问题,近似解的形式相异,降低了解的通用性。增加了求解的工作量。本文提出一种基于矩阵理论…     

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x?0 and x,y?0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann-Hilbert problem in the one-dimensional case, and a nonlocal Riemann-Hilbert problem in the two-dimensional case.     

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.     

An approximate method for calculating scattering and absorption of light by fractal aggregates

A simplified version of the coupled dipole method (CDM) is proposed which allows one to reduce the initial system of 3N×3N equations to a simpler system of N×N equations. The method neglects depolarization effects in the interaction of dipoles but, unlike the mean field approximation, it takes into account local fluctuations of the scalar amplitudes of the excited dipole moments. Simple analytic solutions are obtained for integrated cross sections averaged over aggregate orientations. It is shown by the example of ballistic fractal aggregates that this method provides the accuracy close to that of a standard CDM, being substantially less time-consuming. In the case of biospheres, the approximate method is compared with the exact results of the multipole expansion.     

Exact solution of a model for πd induced reactions and its application to pion-nucleus reactions

Reduction techniques are applied to πd elastic scattering and π-absorption in a theory without anti-nucleons. In a one-pion approximation we derive two sets of exact coupled-channel equations for respectively the amplitudes T_πd,NN, T_{Nδ, NN} and T_{πd, πd}, T_{d, NΔ}- Alternatively we express in terms of the absorption amplitude T_{πd, NN} available solutions for a three-body problem restricted to the πd and NΔ channels. We explicitly demonstrate that our model (which comes close to the one of Thomas, Mizutani and Koltun) strictly respects the Pauli principle and avoids double-counting. Using the same technique we determine amplitudes for the (π, 2N) reaction and for π (in)elastic scattering on general nuclei in terms of amplitudes amongst the NN, NNπ channels. Both the elastic amplitude and the πA optical potential are shown to decompose into a multiple scattering part based on an input πN amplitude without the P₁₁ partial wave and calculable absorption corrections.     

Non-adiabatic effects in the time dependent level crossing problem

We discuss the exact solution of the time-dependent Schrödinger equation for a system of two crossing levels with a residual interaction. In contrast to the familiar Landau-Zener (LZ) solution used in most applications, we allow for more general boundary conditions; in particular we treat explicitly the case of afinite interval around the crossing point. The exact jumping probability is shown to be extremely sensitive to these boundary conditions; in many realistic cases it is found to be smaller than the LZ value by several orders of magnitude. We also compare the exact excitation energy to the one obtained in the usual cranking approach.