Showing papers on "Method of matched asymptotic expansions published in 2022"

Asymptotic expansions of solutions of singularly perturbed equations

Abstract: Рассматривается уравнение первого порядка в банаховом пространстве с малым параметром при производной и возмущением второго порядка малости в правой части. Строится решение задачи Коши в виде асимптотического разложения по степеням малого параметра методом Васильевой-Вишика-Люстерника. Оператор A в правой части вырожден: рассматривается случай обладания свойством иметь число 0 нормальным собственным числом и двумерным ядром; элементы ядра не имеют присоединенных. Получены формулы для вычисления компонент регулярной и погранслойной части разложения, а также условие регулярности вырождения. Доказывается асимптотичность разложения. Приводится иллюстрирующий пример. We consider a first-order equation in a Banach space with a small parameter at the derivativeand a second-order perturbation of smallness on the right-hand side. A solution to the Cauchy problem is constructedin the form of an asymptotic expansion in powers of a small parameter by the Vasilieva-Vishik-Lyusternik method.The operator A on the right-hand side is degenerate: we consider the case of possessing the property of having a number 0 by a normal eigenvalue and a two-dimensional kernel; core elements have no attached. Formulas for calculating the components of the regular and boundary layer parts of the expansion are determined. A condition for the regularity of degeneration is obtained. The expansion is shown to be asymptotic.An illustrative example is given.

Multisummability of Formal Solutions for a Family of Generalized Singularly Perturbed Moment Differential Equations

Abstract: The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.

Asymptotic Behavior of Solutions of Even-Order Differential Equations with Several Delays

Abstract: The higher-order delay differential equations are used in the describing of many natural phenomena. This work investigates the asymptotic properties of the class of even-order differential equations with several delays. Our main concern revolves around how to simplify and improve the oscillation parameters of the studied equation. For this, we use an improved approach to obtain new properties of the positive solutions of these equations.

Asymptotic proximity to higher order nonlinear differential equations

Abstract: Abstract The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.

Asymptotic Numerical Solutions for Second-Order Quasilinear Singularly Perturbed Problems

Abstract: For a second-order quasilinear singularly perturbed problem under the Dirichlet boundary conditions, we propose a new asymptotic numerical method, which involves two problems: a reduced problem with a one-side boundary condition and a novel boundary layer correction problem with a two-sided boundary condition. Through the introduction of two new variables, both problems are transformed to a set of three first-order initial value problems with zero initial conditions. The RungeeKutta method is then applied to integrate the differential equations and to determine two unknown terminal values of the new variables until they converge. The modified asymptotic numerical solution satisfies the Dirichlet boundary conditions. Some examples confirm that the newly proposed method can achieve a better asymptotic solution to the quasilinear singularly perturbed problem. For most values of the perturbing parameter, the present method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy within the entire domain.

Refined Asymptotic Expansions of Solutions to Fractional Diffusion Equations

Abstract: In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

Direct Scheme of Constructing Asymptotic Solution of Three-tempo Linear-Quadratic Control Problems with Weak Nonlinear Perturbations

Abstract: An asymptotic solution of a weakly nonlinearly perturbed linear-quadratic optimal control problem with threetempo state variables is constructed, using the direct scheme method, consisting of immediate substituting a postulated asymptotic solution into a problem condition and determining a series of control problems for finding asymptotics terms.

Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

Abstract: The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.

Asymptotic-numerical solvers for linear neutral delay differential equations with high- frequency extrinsic oscillations

Abstract: We present a method to compute efficiently and easily solutions of systems of linear neutral delay differential equations with highly oscillatory forcing terms. This method is based on asymptotic expansions in inverse powers of a perturbed oscillatory parameter. Each term of the asymptotic expansion is derived by recursion. The cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided and show that with few terms of the asymptotic expansion, the solutions are approximated with high accuracy.

Modeling of wave processes in porous media and asymptotic expansions

Abstract: Models of wave processes in porous periodic media are considered. It is taken into account that the corresponding wave equations depend on small parameters characterizing the microscale, density, and permeability of such media. The algorithm for determining asymptotic expansions for these equations is given. Estimates for the accuracy of such expansions are presented.

Numerical analysis of a singularly perturbed convection diffusion problem with shift in space

Abstract: We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.

Asymptotic Solutions of Resonant Nonlinear Singularly Perturbed Problems in the Case of Intersecting Eigenvalues of the Limit Operator

Abstract: Lomov’s regularization method is generalized to resonant, weakly nonlinear, singularly perturbed systems in the case of intersecting roots of the characteristic equation of the limit operator. For constructing asymptotic solutions, the regularization of the original problem by using normal forms developed by the authors is performed. In the absence of resonance, the regularizing normal form is linear, whereas in the presence of resonances, it is nonlinear. In this paper, the resonant case of a weakly nonlinear problem is considered. By using an algorithm of normal forms, we construct an asymptotic solution of any order (with respect to a parameter) and justify this algorithm.

Singularly Perturbed Solution of Non-Fourier Temperature Field with Mixed Boundary Value Dual-Phase Delay with Thermal Conductivity Jumps

Abstract: Based on the properties of laminates, a class of nonlinear singularly perturbed mixture equations with discontinuous coefficients over bounded domains is constructed by using a dual-phase delayed heat conduction model. First, the singular perturbation expansion method is used, combined with the corresponding boundary conditions, the partial differential equation method and the Laplace transform method are used to obtain the external solution, the boundary layer, and the corner layer. Secondly, the time-varying temperature field at the discontinuity is obtained, which leads to the asymptotic expansion of the solution. Finally, the consistent validity of the asymptotic solution is obtained through residual estimation.

Asymptotic expansions of high-frequency multiple scattering iterations for sound hard scattering problems

Abstract: We consider the two-dimensional high-frequency plane wave scattering problem in the exterior of a finite collection of disjoint, compact, smooth, strictly convex obstacles with Neumann boundary conditions. Using integral equation formulations, we determine the H\"{o}rmander classes and derive high-frequency asymptotic expansions of the total fields corresponding to multiple scattering iterations on the boundaries of the scattering obstacles. These asymptotic expansions are used to obtain sharp wavenumber dependent estimates on the derivatives of multiple scattering total fields which, in turn, allow for the optimal design and numerical analysis of Galerkin boundary element methods for the efficient (frequency independent) approximation of sound hard multiple scattering returns. Numerical experiments supporting the validity of these expansions are presented.

Singularly Perturbed Integral Equations with Rapidly Oscillating Inhomogeneities

Abstract: Singularly perturbed integro-differential equations with a rapidly oscillating right-hand side are considered. The main goal of this work is to generalize the Lomov regularization method and to reveal the influence of the rapidly oscillating right-hand side on the asymptotics of the solution to the original problem in the presence of an integral operator. Various applied problems related to the properties of media with a periodic structure lead to the need to study differential equations with rapidly oscillating inhomogeneities. Equations of this type are often found in applications such as electrical systems under the influence of high frequency external forces. The presence of such forces creates serious problems for the numerical integration of the corresponding differential equations. Therefore, asymptotic methods are usually applied to such equations, the most well-known of which are the Lomov regularization method and the Feshchenko–Shkil–Nikolenko splitting method. The splitting method is especially effective when applied to equations with a rapidly oscillating inhomogeneity, and in the case of an inhomogeneity containing both fast and slow components, the Lomov regularization method turned out to be the most effective. However, both of these methods were developed mainly for singularly perturbed equations that do not contain an integral operator. The transition from differential equations to integro-differential equations requires a significant restructuring of the regularization method’s algorithm. The integral term generates new types of singularities in solutions that differ from the previously known ones, which complicates the development of an algorithm for the regularization method.

Asymptotic solutions of boundary value problem for singularly perturbed system of differential-algebraic equations

Abstract: This paper deals with the boundary value problem for a singularly perturbed system of differential algebraic equations of the second order. The case of simple roots of the characteristic equation is studied. The sufficient conditions for existence and uniqueness of a solution of the boundary value problem for system of differential algebraic equations are found. Technique of constructing the asymptotic solutions is developed.

Asymptotic analysis of Sturm-Liouville problem with Robin and two-point boundary conditions

Abstract: We analyze the initial value problem and get asymptotic expansions for solution. We investigate the characteristic equation for Sturm-Liouville problem with one classical Robin type boundary condition and another two-point nonlocal boundary condition. Finally, we obtain asymptotic expansions for eigenvalues and eigenfunctions.

Asymptotic estimations of the solution for a singularly perturbed equation with unlimited boundary conditions

Abstract: The paper studies a two-point boundary value problem with unlimited boundary conditions for a linear singularly perturbed differential equation. Asymptotic estimates are given for a linearly independent system of solutions of a homogeneous perturbed equation. Auxiliary, so-called boundary functions, the Cauchy function are defined. For sufficiently small values of the parameter, estimates for the Cauchy function and boundary functions are found. An algorithm for constructing the desired solution of the boundary value problem has been developed. A theorem on the solvability of a solution to a boundary value problem is proved. For sufficiently small values of the parameter, an asymptotic estimate for the solution of the inhomogeneous boundary value problem is established. The initial conditions for the degenerate equation are determined. The formula is determined; the phenomena of the initial jump are studied.

To substantiate the asymptotics of solving the Cauchy problem for a singularly perturbed weakly nonlinear transport equation

Abstract: A theorem is proved on the uniform estimation of the residual term of the asymptotic expansion with respect to a small parameter of the solution of the initial problem for a singularly perturbed differential operator weakly nonlinear transport equation in the critical case.

Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. I -- Method and Justification

Abstract: In this paper, we provide a systematic methodology for calculating multi-order asymptotic expansion of blow-up solutions near blow-up for autonomous ordinary differential equations (ODEs). Under the specific form of the principal term of blow-up solutions for a class of vector fields, we extract algebraic objects determining all possible orders in the asymptotic expansions. Examples for calculating concrete multi-order asymptotic expansions of blow-up solutions are finally collected.

Asymptotics of a multizonal internal layer solution to a piecewise-smooth singularly perturbed equation with a triple root of the degenerate equation

Abstract: A singularly perturbed boundary value problem for a stationary equation of reaction-diffusion type in the case when reactive term undergoes discontinuity along some curve that is independent of the small parameter is studied. This is a new class of problems with triple roots of the degenerate equation, which leads to the formation of complex multizonal internal layers in the neighborhood of the discontinuity curve. By the method of asymptotic differential inequalities and matching asymptotic expansion, the existence of a contrast structure solution is proved. Using a different modified boundary layer function method, the asymptotic representation of point itself and this solution are constructed.

Study of asymptotic solutions of systems of singularly perturbed differential equations with turning points

Abstract: We study a system with a small parameter at the highest derivatives. Using model operator Airy–Langer for defined regular function. Received the conditions of construction an uniform asymptotic solution for a given system.

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

Abstract: The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.