The Eigenvalues of the Angular Spheroidal Wave Equation

Approximate relations are obtained between the eigenvalues λ and the ellipticity parameter c² of the angular spheroidal wave equation. Although based on WKBJ methods and the assumption that λ is large, the relations are useful throughout the complex c²-plane. They are exact at c² = 0, and reproduce the standard asymptotic formulas for λ when c² is large. At intermediate values of c², they provide approximations for the square-root branch points of the multivalued function λ(c²) in the complex c²-plane at which adjacent eigenvalues of the same class become equal in pairs. These branch points lie on an infinite sequence of distorted circular rings. Their exact locations have been computed for the first four rings for angular wavenumbers m = 0,…,4.     

Stability of peakons for the Degasperis‐Procesi equation

The Degasperis‐Procesi equation can be derived as a member of a one‐parameter family of asymptotic shallow‐water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa‐Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis‐Procesi equation on the line. By constructing a Lyapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations. © 2007 Wiley Periodicals, Inc.     

On the one‐dimensional relativistic induction equation

The induction equation of relativistic magnetohydrodynamics is considered as a singular perturbation problem for small magnetic diffusivity. When the quantities depend on a single space variable, the resulting hyperbolic equation may be studied with techniques of asymptotic analysis. Different approximations are found for initial, intermediate, and large times. The last case is the most difficult; the approximate magnetic flux function satisfies a certain parabolic equation. This equation is studied from the viewpoint of energy dissipation, providing clues on the behavior of the electric and magnetic fields. Copyright © 2013 John Wiley & Sons, Ltd.     

具有三个转向点方程渐近解的完全表达式

本文研究二阶线性常微分方程,λ为大参数,即具有三个转向点的方程。而。本文使用JL函数得到方程在转向点附近形式一致有效渐近解的完全表达式。     

Boundary function method to find the asymptotic solution of the plasma—sheath equation

We consider the asymptotic solution of the Tonks—Langmuir integro-different equation with an Emmert kernel, which describes the behavior of the potential both inside the main plasma volume and in a thin boundary layer. Equations of this type are singularly perturbed due to the small coefficient at the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the regular expansion series and in the boundary function expansion. The equation for the first coefficient of the regular series has only a trivial solution. Second-order differential equations are obtained for the first two boundary functions. The equation for the first boundary function is solved numerically on a discrete grid with locally uniform spacing. An approximate analytical expression for the first boundary function is obtained from the linearized equation. This solution adequately describes the behavior of the potential on small distances only. __________ Translated from Prikladnaya Matematika i Informatika, No. 19, pp. 21–40, 2004.     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

Asymptotic estimates for a solution of a singular perturbation optimal control problem on a closed interval under geometric constraints

An optimal control problem is considered for solutions of a boundary value problem for a second-order ordinary differential equation on a closed interval with a small parameter at the second derivative. The control is scalar and satisfies geometric constraints. General theorems on approximation are obtained. Two leading terms of an asymptotic expansion of the solution are constructed and an error estimate is obtained for these approximations.     

The exit problem for certain dynamical systems with small poisson disturbances

The mean first passage time from the inside to the outside of a finite interval for the solution of a differential equation perturbed by Poisson noise may have a discontinuity at one end of the interval. When the size ofthe perturbation is small, the magnitude of the discontinuity can be large and difficult to calculate. In this paper asymptotic approximations for the size of the discontinuity are derived in two cases of practical interest. The approximations show that the discontinuity grows exponentially.     

具有奇异系数的发展p-Laplace 方程

该文考虑Rⁿ内在球中和球面系数均有奇异的发展p-Laplac方程的初边值问题的可解性. 在一定条件下证明了轴对称整体解的存在性和不存在性.     

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.     

Uniform Asymptotic Approximations to the Solutions of the Orr-Sommerfeld Equation Part 2. The General Theory

This paper deals with the derivation of “first approximations” to the solutions of the Orr-Sommerfeld equation which are uniformly valid in a full neighborhood of the critical point. To this order the theory is remarkably simple. The essential elements in the theory are all well known from the older heuristic theories but its general structure is substantially different. The uniform approximations are also vastly simpler than the composite approximations obtained recently by the method of matched asymptotic expansions.     

Applying the dual operator formalism to derive the zeroth-order boundary function of the plasma-sheath equation

The article constructs the asymptotic solution of the Tonks-Langmuir integro-differential equation with an Emmert kernel, which describes the potential both in the bulk plasma and in a narrow boundary layer. Equations of this type are singularly perturbed, because the highest order (second) derivative is multiplied by a small coefficient. The asymptotic solution is obtained by the boundary function method. The second-order differential equation describing the behavior of the zeroth-order boundary function is investigated using the dual operator formalism — an analog of the conjugate operator in the linear theory. The application of this formalism has produced an asymptotic solution and has also made it possible to propose a number of homogeneous discrete three-point schemes for solving the equation. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 76–90, 2005.     

Some recent results on a free boundary problem for microelectromechanical systems

The mathematical modeling of idealized microelectromechanical systems leads to a nonlinear singular elliptic-parabolic free boundary problem involving non-smooth domains and depending on a parameter λ reflecting the applied voltage across the device. Recent results derived in [1,4] are presented which confirm physically important and expected features: For small λ there is (at least) one stable stationary solution and solutions to the evolution equation exist globally in time, while for large λ no stationary solution exists and the evolution equation has no global solution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation and Stability Theory of Periodic Solutions For Integrodifferential Equations

This paper is concerned with a nonlinear integrodifferential equation (the delay logistic equation) governing the growth dynamics of a single species N(t) for time t > 0. This equation contains a positive parameter λ. Suppose that there exists a positive equilibrium solution N = c which is stable for all small values of λ. Assume also that this solution loses stability as λ is increased past a critical value λ*. This will correspond to a simple pure imaginary conjugate pair of roots of a characteristic equation associated with the linearized stability of N = c at λ = λ*. Then we will construct a unique bifurcating time periodic solution of the equation as a Taylor series in a parameter ε. Furthermore this solution exists either for supercritical values of the parameter (λ > λ*) or for subcritical values (λ < λ*). The stability behavior of this small periodic solution can be characterized according to whether the bifurcation is supercritical or subcritical-supercritical solutions are stable, but subcritical solutions are unstable. Therefore these results are analogous to Hoprs bifurcation theorem for autonomous systems of differential equations.     

Estimates and exact expressions for lyapunov exponents of stochastic linear differential equations

We consider the asymptotic behavior of the solutions of a stochastic linear differential equation driven by a finite states Markov process. We consider the sample path Lyapunov exponent λ and the p-moment Lyapunov exponents g(p) for positive p. We derive relations between X and g{p\ which are extensions to our situation of results of Arnold [1] in a different context. Using a Lyapunov function approach, an exact expression forg(2) and estimates for g(p) are obtained, thus leading to upper and lower bounds for λ     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

Taylor expansions of the value function associated with a bilinear optimal control problem

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided.     

Testing the equality of several linear regression models

The linear regression models are widely used in different research fields, and often there is the need to analyze if there are similarities between two or more different linear models or to verify if a given relation between two variables remains the same in different intervals of time, in particular in cases where small differences might make a big difference. Motivated by these problems the authors consider a test of equality of k linear regression models which is a simultaneous test of equality of slopes, intercepts and variances. In order to overcome the extreme difficulties that exist in the use of the exact distribution of the likelihood ratio test (LRT) statistic and to make this test reliable and easy to use, we propose the use of near-exact distributions to approximate the distribution of the LRT statistic, under \(H_0\), in the balanced case, and of new asymptotic approximations for the unbalanced case. The near-exact approximations are built by approximating one factor of an adequate factorization of the characteristic function of the logarithm of the LRT statistic and may be easily implemented. The asymptotic approximations are developed using an expansion for the ratio of gamma functions. The quality of these approximations is analyzed and confirmed. Power studies are conducted in order to better assess the performance of the test. Finally to illustrate the applicability of the test we consider a real data set of gross domestic product at market prices and final consumption expenditure in European countries and one tests the existence of similarities between countries.     

Asymptotic Behavior of Solutions to the Cahn-Hilliard Equation in Spherically Symmetric Domains

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. The asymptotic behavior of solutions to the Cahn-Hilliard equation with Dirichlet boundary conditions and the associated stationary problem have been studied. In particular, it is proved that the only possible stable equilibrium solutions in spherically symmetric domains are spherically symmetric and monotone in the radial direction.     

A Note on the p-Laplacian Equation with Singular Coefficients

We consider the solvability of the Dirichlet problem in a unit n-ball for the p-Laplacian equation with singular coefficients which are singular not only as |x| tends to 1 but also as |x| tends to 0. The existence and regularity of positive radial solutions are proved under some conditions related to parameters p, r, λ and q.