Numerical solution of the differential equation describing the behavior of the zeroth-order boundary function

The asymptotic solution of the integro-differential plasma-sheath equation is considered. This equation is singularly perturbed because of the small coefficient multiplying 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 form of both a regular series expansion and an expansion in boundary functions. The equation for the first coefficient of the regular series has only a trivial solution. A numerical algorithm is considered for the solution of the second-order differential equation describing the behavior of the zeroth-order boundary function. The proposed algorithm efficiently solves the boundary-value problem and produces a well-behaved solution of the Cauchy problem. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 24–35, 2006.     

Asymptotic partial decomposition for diffusion with sorption in thin structures

The method of partial asymptotic decomposition of a domain is applied to a model non-linear equation set in a rod structure. The estimates of error are provided. An asymptotic expansion of solution is constructed and justified.     

A perturbation solution for forced convection in a porous-saturated duct

Fully developed forced convection through a porous medium bounded by two isoflux parallel plates is investigated analytically on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied for small values of the Darcy number. For the case of large Darcy number the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. With the velocity distribution determined, the energy equation is solved using the same asymptotic technique. The results for limiting cases are found to be in good agreement with those available in the literature and the numerical results obtained here.     

Asymptotic decay of solutions of the liouville equation under perturbations

We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale. Translated frommatematicheskie Zametki, Vol. 68, No. 2, pp. 195–209, August, 2000.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

一类小周期椭圆方程的三重尺度渐近分析

利用三重尺度方法对一类小周期椭圆方程进行了三重尺度渐近展开分析,构造了对应的三重尺度形式渐近展开式,得到了均匀化常数和均匀化方程.在形式渐近展开的基础上,构造了对应边值问题解的三重尺度渐近近似解,并分析了对应三重尺度形式渐近误差估计.     

Homogenization of a boundary condition for the heat equation

An asymptotic analysis is given for the heat equation with mixed boundary conditions rapidly oscillating between Dirichlet and Neumann type. We try to present a general framework where deterministic homogenization methods can be applied to calculate the second term in the asymptotic expansion with respect to the small parameter characterizing the oscillations. Received August 20, 1999 / final version received March 1, 2000?Published online June 21, 2000     

Front motion in the parabolic reaction-diffusion problem

A singularly perturbed initial-boundary value problem is considered for a parabolic equation known in applications as the reaction-diffusion equation. An asymptotic expansion of solutions with a moving front is constructed, and an existence theorem for such solutions is proved. The asymptotic expansion is substantiated using the asymptotic method of differential inequalities, which is extended to the class of problems under study. The method is based on well-known comparison theorems and is a development of the idea of using formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

A parametric transformation at numerical integration of ODE

The initial value problem for ordinary differential equation (ODE) is investigated when the linear parametric transformation (rotation of coordinate axes) is applied. It is shown that for transformed equation the principal term of asymptotic error expansion of numerical method can be minimized by an angle of rotation. The dependence of the optimal angle φ_opt(λ)on λis plotted for the model equation solved by linear multistep methods and Runge-Kutta methods.     

半线性奇摄动椭圆型方程边值问题的渐近解

本文运用了边界层函数构造了一类半线性奇摄动椭圆型方程边值问题解的渐近展开式，并证明了该展开式达到任一精度的一致有效性．     

Behavior at infinity of a solution to a matrix differential-difference equation

We obtain an asymptotic expansion for a solution to a nonhomogeneous retarded- or neutraltype differential-difference equation. The case of unbounded delays is considered. The influence is accounted for the roots of the characteristic equation. We establish the exact asymptotics for the remainder depending on the asymptotic properties of the free matrix term of the equation.     

On a singularly perturbed initial value problem in the case of a double root of the degenerate equation

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.     

On Complicated Expansions of Solutions to ODES

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.     

Energy change to insertion of inclusions associated with a diffusive/convective steady‐state heat conduction problem

The topological derivative concept has been successfully applied in many relevant physics and engineering problems. In particular, the topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations. In this paper, the topological asymptotic analysis of the energy shape functional associated with a diffusive/convective steady‐state heat equation is developed. The topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form with help of a non‐standard adjoint state. Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas. The obtained result is new and can be applied in the context of topology design of heat sinks, for instance. Copyright © 2015 John Wiley & Sons, Ltd.     

Behavior at infinity of a solution to a differential-difference equation

We obtain an asymptotic expansion for a solution to an mth order nonhomogeneous differential-difference equation of retarded or neutral type. Account is taken of the influence of the roots of the characteristic equation. The exact asymptotics of the remainder is established depending on the asymptotic properties of the free term of the equation.     

半线性奇摄动反应扩散方程初边值问题解的渐近性(英文)

本文运用边界层以及角点层函数法构造了一类半线性奇摄动反应扩散方程初边值问题解的渐近展开式 ,并用微分不等式方法证明了该展式达到任一精度的一致有效性     

Local theory of extrapolation methods

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.     

Analytic and Numerical Study of N-Waves Governed by the Nonplanar Burgers Equation

In this article, evolution of N-waves under the nonplanar Burgers equation, which takes into account geometrical expansion or contraction, is treated analytically. An exact asymptotic solution, generalizing that for the planar Burgers equation, is given for the case of expansion. An approximate treatement, using a balancing argument, gives asymptotic analytic results for both expansion and contraction. The analysis is fortified by an accurate numerical solution of the problem. This study is brought in close conjunction with the earlier work of Crighton and Scott [13] and Sachdev, Joseph and Nair [3].     

Exact and chapman-enskog solutions for the carleman model

The Chapman-Enskog procedure is applied to the Carleman model of the Boltzmann equation. It has been proved that the Carleman equations possess a solution on the time interval on which a smooth solution of the fluid-like equation exists. The calculations have been performed up to the first order i.e., to the Navier-Stokes-like equation. It has been shown that in this case a difference between an exact solution and the Chapman-Enskog solution is of order ?². Extension of the results to higher orders is also possible. This gives a justification of the Chapman-Enskog procedure as an asymptotic expansion method.     

Hopf Bifurcation and Stability of Periodic Solutions of Differential-difference and Integro-differential Equations

Periodic solutions bifurcating from a steady state of differential-differenceand integrodifferential equations are studied. An algorithmfor determining the asymptotic orbital stability, directionof bifurcation, period, and asymptotic form of these solutionsis presented. This algorithm is applied to the Hutchinson-Wrightequation and to an equation with two delays. Finally some integro-differentialequations, modelling two and three trophic level competition,introduced by May, are discussed. Some numerical work supportingthe theory is described.