Asymptotic Expansion Of Solution Research Articles

Resonance phenomena in curved quantum waveguides coupled via windows

A system of two curved waveguides mutually coupled via a small window is considered by solving a boundary value problem with the Dirichlet conditions at the external boundaries and the Neumann conditions at an interface with the window. Asymptotic expansions (with respect to the window size) for a resonance related to the window and for a bound state related to the waveguide curvature are obtained using the method of matching of the asymptotic expansions of solutions of the boundary-value problem.

ASYMPTOTIC EXPANSIONS OF THE SOLUTIONS TO THE HEAT EQUATIONS WITH HYPERFUNCTIONS INITIAL VALUE

We will derive the asymptotic expansions of solutions of the heat equation with hyperfunctions initial data.

The solvability for a class of singularly perturbed quasi-linear differential system

The solvability for a class of singularly perturbed Robin problem of quasi-linear differential system is considered. Using the boundary layer corrective method the formal asymptotic solution is constructed. And using the theory of differential inequality the uniform validity of the asymptotic expansions for solution is proved.

Singularly perturbed solution of coupled model in atmosphere-ocean for global climate

A box model of the interhemispheric thermohaline circulation (THC) in atmosphere-ocean for global elimate is considered. By using the multi-scales method, the asymptotic solution of a simplified weakly nonlinear model is discussed. Firstly, by introducing first scale, the zeroth order approximate solution of the model is obtained. Secondly, by using the multi-scales, the first order approximate equation of the model is found. Finally, second order approximate equation is formed to eliminate the secular terms, and a uniformly valid asymptotic expansion of solution is decided. The multi-scales solving method is an analytic method which can be used to analyze operation sequentially. And then we can also study the diversified qualitative and quantitative behaviors for corresponding physical quantities. This paper aims at providing a valid method for solving a box model of the nonlinear equation.

Topological derivatives for optimization of plane elasticity contact problems

Topological derivatives for plane elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions to elasticity boundary value problems in domains with small holes and determine the explicit asymptotic expansions of solutions with respect to small parameter, which describes the radius of internal hole. The elastic potentials of Muskhelishvili give us an explicit solution in the ring C ( ρ , R ) = { ρ < | x | < R } in the form of complex valued series. The series depends on the small parameter, the radius ρ of the ring, and we are interested in the behavior of the series for the passage ρ → 0 . Such analysis leads to the expansion of the elastic energy in the form E ( ρ , R ) = E ( 0 , R ) + ρ 2 E 1 ( R ) + ρ 4 E 2 ( R ) + ⋯ , where E 1 ( R ) is used to determine the first-order topological derivatives of shape functionals, and E 2 ( R ) can be used to determine the second-order topological derivatives of shape functionals. In this paper the first-order term E 1 ( R ) is given; however, the method is general and can be used to determine the subsequent terms of the energy expansion and the topological derivatives of higher order.

Numerical Verification and Comparison of Error of Asymptotic Expansion Solution of the Duffing Equation

A numerical order verification technique is applied to demonstrate that the asymptotic expansions of solutions of the Duffing equation obtained respectively by the Lindstedt-Poincaré(LP) method and the modified Lindstedt-Poincaré(MLP) method are uniformly valid for small parameter values. A numerical comparison of error shows that the MLP method is valid whereas the LP method is invalid for large parameter values.

Asymptotic expansion of solutions to an inverse problem of parabolic type

We consider an abstract linear parabolic problem, with a scalar part $f$ of the source term which is unknown. This lack of information is compensated by the knowledge, for any time, of the value of a certain functional $\Phi$ when applied to the solution. Under suitable assumptions, we prove a result of global existence and uniqueness of the solution $(u,f)$. Moreover, if the coefficients of the system admit an asymptotic expansion, the same holds for $(u,f)$. The abstract results are applied to general parabolic mixed Cauchy-boundary value problems.

All asymptotic expansions of solutions to the sixth Painlevé equation

All asymptotic expansions of solutions to the sixth Painlevé equation

Approximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory

Approximation of one-dimensional stochastic differential equations and their addi- tive functionals by dynamical systems with piecewise-constant random coefficients is obtained. We calculate asymptotic expansion of solution in terms of the step of discretization Δ.

Expansion of solutions to the fifth Painlevé equation in a neighborhood of a nonsingular point

For the fifth Painlevé equation, the asymptotic expansions of solutions are obtained in a neighborhood of a nonsingular point of the independent variable by using methods of power geometry. Four families of expansions are found. All of these expansions are convergent series in integral powers with complex coefficients.

Expansion of solutions of the sixth Painlevé equation near a regular point

For the sixth Painlevé equation, by using power-geometry methods, all asymptotic expansions of solutions are obtained in a neighborhood of a regular point of the independent variable. All of these expansions are convergent series in integer powers with constant complex coefficients. Five families of expansions are found.

Asymptotic Solutions of a Fourth Order Differential Equation

In this paper, we derive uniform asymptotic expansions of solutions to the fourth order differential equation where x is a real variable and λ is a large positive parameter. The solutions of this differential equation can be expressed in the form of contour integrals, and uniform asymptotic expansions are derived by using the cubic transformation introduced by Chester, Friedman, and Ursell in 1957 and the integration-by-part technique suggested by Bleistein in 1966. There are two advantages to this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we extend the validity of the uniform asymptotic expansions to include all real values of x.

Asymptotics for critical Rosenau type equations with nonconvective nonlinearites on a half-line

We study nonlinear pseudodifferential equations on a half-line with nonanalytic symbol where where Qm and Pm are polynomials of order m ∈ N such that ReK(p)>0 for all Rep = 0 and The aim of this article is to prove the global existence of solutions to the initial boundary value problem and to find the main term of the asymptotic representation of solutions in critical case, when the nonlinear term of equation has the time decay rate less than that of the linear terms. Also in this study for the first time we give some general approaches to obtain the global existence of solution of initial boundary value problem in critical nonconvective case and elaborate on the general sufficient conditions to obtain asymptotic expansion of solution.

The singular function boundary integral method for biharmonic problems with crack singularities

We use the singular function boundary integral method (SFBIM) to solve two model fracture problems on the plane. In the SFBIM, the solution is approximated by the leading terms of the local asymptotic solution expansion, which are also used to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of the divergence theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The main advantage of the method is that the leading stress intensity factors (SIFs) are calculated directly together with the Lagrange multipliers, i.e. no post-processing of the numerical solution is necessary. The numerical results for the two model problems show the fast convergence of the method and compare well with those of the collocation Trefftz method.

Stability Estimates in Inverse Problems for Hyperbolic Equations

We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse problems are given.

A nonlinear wave equation associated with nonlinear boundary conditions: Existence and asymptotic expansion of solutions

Consider the initial–boundary value problem for the nonlinear wave equation (1) { u t t − u x x + K | u | p − 2 u + λ | u t | q − 2 u t = F ( x , t ) , 0 < x < 1 , 0 < t < T , u x ( 0 , t ) = P ( t ) , − u x ( 1 , t ) = | u ( 1 , t ) | p 1 − 2 u ( 1 , t ) + | u t ( 1 , t ) | q 1 − 2 u t ( 1 , t ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , where p , p 1 , q 1 ≥ 2 , q > 1 , K , λ are given constants and u 0 , u 1 , F are given functions, and the unknown function u ( x , t ) and the unknown boundary value P ( t ) satisfy the following nonlinear integral equation (2) P ( t ) = g ( t ) + K 0 | u ( 0 , t ) | p 0 − 2 u ( 0 , t ) + | u t ( 0 , t ) | q 0 − 2 u t ( 0 , t ) − ∫ 0 t k ( t − s ) u ( 0 , s ) d s , where p 0 , q 0 ≥ 2 , K 0 are given constants and g , k are given functions. In this paper, we consider three main parts. In Part 1, under the conditions ( u 0 , u 1 ) ∈ H 1 × L 2 , F ∈ L 2 ( Q T ) , k ∈ W 1 , 1 ( 0 , T ) , g ∈ L q 0 ′ ( 0 , T ) , λ = 1 , K , K 0 ≥ 0 ; p , p 0 , q 0 , p 1 , q 1 ≥ 2 , q > 1 , q 0 ′ = q 0 q 0 − 1 , we prove a theorem of existence and uniqueness of a weak solution ( u , P ) of problem (1) and (2). The proof is based on the Faedo–Galerkin method and the weak compact method associated with a monotone operator. For the case of q 0 = q 1 = 2 , p , q , p 0 , p 1 ≥ 2 , in Part 2 we prove that the unique solution ( u , P ) belongs to ( L ∞ ( 0 , T ; H 2 ) ∩ C 0 ( 0 , T ; H 1 ) ∩ C 1 ( 0 , T ; L 2 ) ) × H 1 ( 0 , T ) , with u t ∈ L ∞ ( 0 , T ; H 1 ) , u t t ∈ L ∞ ( 0 , T ; L 2 ) , u ( 0 , ⋅ ) , u ( 1 , ⋅ ) ∈ H 2 ( 0 , T ) , if we make the assumption that ( u 0 , u 1 ) ∈ H 2 × H 1 and some others. Finally, in Part 3 we obtain an asymptotic expansion of the solution ( u , P ) of the problem (1) and (2) up to order N + 1 in three small parameters K , λ , K 0 .

Exact asymptotic expansions for solutions of multi-dimensional renewal equations

We derive expansions with exact asymptotic expressions for the remainders for solutions of multi-dimensional renewal equations. The effect of the roots of the characteristic equation on the asymptotic representation of solutions is taken into account. The resulting formulae are used to investigate the asymptotic behaviour of the average number of particles in age-dependent branching processes having several types of particles.

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.

Convective instability in wake intermediate asymptotics

The paper presents a multiscale analysis of the intermediate region of the two-dimensional convectively unstable wake past a bluff body. A recent asymptotic expansion solution was used as basic flow (Tordella &amp; Belan, Phys. Fluids, vol. 15, 2003, 1897). This solution was obtained by matching an inner to an outer flow, both of which are Navier–Stokes solutions. By introducing a spatio-temporal multiscaling into the instability problem, an inhomogeneous Orr–Sommerfeld equation and an associated modulation equation are obtained. The streamwise variation of the instability characteristics can then be deduced from the wave modulation, by considering the system to be perturbed by waves with a complex wavenumber that corresponds of the dominant saddle point of the local dispersion relation, taken at different positions downstream of the wake, and at different Reynolds numbers. The corrections of no parallelism are remarkable in the intermediate wake. When the disturbance is related to an early intermediate station, the corrections lead to absolute instability in the upstream portion of the intermediate wake, where, in addition, the spatial growth rate decreases. When the disturbance is related to a section in the far field, conditions of minimal temporal stability are reached about 20 body scales downstream. In the far field the temporal damping increases with the Reynolds number.

On the Eshelby-Kostrov property for the wave equation in the plane

This work deals with the linear wave equation considered in the whole plane R 2 \mathbb {R}^{2} except for a rectilinear moving slit, represented by a curve Γ ( t ) = { ( x 1 , 0 ) : − ∞ &gt; x 1 &gt; λ ( t ) } \Gamma \left ( t\right ) =\left \{ \left ( x_{1},0\right ) :-\infty &gt;x_{1}&gt;\lambda \left ( t\right ) \right \} with t ≥ 0. t\geq 0. Along Γ ( t ) , \Gamma \left ( t\right ) , either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. The latter have a simple geometrical interpretation, and in particular allow us to derive precise asymptotic expansions for solutions near the tip of the curve. In the Neumann case, we thus recover a classical result in fracture dynamics, namely the form of the stress intensity factor in crack propagation under antiplane shear conditions.