Problem with integral condition for a partial differential equation of the first order with respect to time

We investigate the problem with inhomogeneous integral condition for a homogeneous partial differential equation of the first order with respect to time and, in the general case, of infinite order with respect to the space variable with constant coefficients. We prove the existence and uniqueness of a solution of the problem in a class of quasipolynomials of the special form. We construct a solution of this problem with the use of the differential-symbol method. In the case of existence of a nonunique solution of the problem, we propose formulas for the construction of a particular solution of the problem.     

A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems

This paper presents a meshless method, which replaces the inhomogeneous biharmonic equation by two Poisson equations in terms of an intermediate function. The solution of the Poisson equation with the intermediate function as the right-hand term may be written as a sum of a particular solution and a homogeneous solution of a Laplace equation. The intermediate function is approximated by a series of radial basis functions. Then the particular solution is obtained via employing Kansa’s method, while the homogeneous solution is approximated by using the boundary radial point interpolation method by means of boundary integral equations. Besides, the proposed meshless method, in conjunction with the analog equation method, is further developed for solving generalized biharmonic-type problems. Some numerical tests illustrate the efficiency of the method proposed.     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

四元数分析中的T算子与两类边值问题

本文研究四元数分析中的非齐次 Ｄｉｒａｃ方程．引入了这类方程的分布解即 Ｔ算子,证明了Ｔ算子的一些性质并考察了非齐次Ｄｉｒａｃ方程的Ｄｉｒｉｃｈｌｅｔ边值问题,并将结果推广到高阶非齐次Ｄｉｒａｃ方程及这种方程的一类边值问题的情况．     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

n阶线性模糊微分方程的结构元解法

研究了n阶线性模糊微分方程的模糊初值问题,将n阶线性模糊微分方程转化成一阶线性模糊微分方程组,利用结构元方法将模糊线性微分方程组转化成两个分明的线性微分方程组,通过分明的线性微分方程组的解构造出原n阶线性模糊微分方程的解.最后,给出了具体的算例.     

Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative

A Hamilton–Jacobi equation with Caputo’s time fractional derivative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose, comparison principle as well as Perron’s method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time fractional derivatives with variable coe?cients.     

用逆微分算子法求n阶常系数线性一般非齐次微分方程特解公式

本文利用逆微分算子及其线性性质 ,给出了求 n阶常系数线性一般非齐次项微分方程特解公式 ,     

Problems for elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet problem for a class of second-order nonlinear elliptic partial differential equations with singular data. In particular, we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation.     

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

微分方程组(dX)/(dt)=AX+e~(αt)P_m(t)具有重特征根α时的特解结构定理及应用

对于常系数非齐线性微分方程组(dX)/(dt)=AX+eαt sum (Bktk) from k=0 to m,当n级方阵A具有s≥1重特征根α时,本文给出了其特解X(t)的结构定理和计算方法,将求特解X(t)的积分运算转化为简单的代数运算,解决了利用计算机求特解X(t)的计算问题.     

微分方程组（dX）/（dt）=AX＋e^αtPm（t）具有重特征根α时的特解结构定理及应用

对于常系数非齐线性微分方程组（dX）/（dt）=AX＋eαt sum （Bktk） from k=0 to m,当n级方阵A具有s≥1重特征根α时,本文给出了其特解X（t）的结构定理和计算方法,将求特解X（t）的积分运算转化为简单的代数运算,解决了利用计算机求特解X（t）的计算问题.     

Integrability and Self-Similarity in Transient Stimulated Raman Scattering

Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields. Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998     

Parameter Identification of a Finite Viscoplastic Material Model using different Deterministic Optimization Methods

The efficiency of two deterministic optimization strategies are discussed in order to solve the arising least squares minimization problem. A sequential quadratic programming method is compared with a trust region enhanced simplex method. In particular we focus on the effort to determine the necessary gradients for the sqp‐method and compare the computational costs of the identification process. In the later development the identification problem is extended to account for inhomogeneous deformations using the finite element method for simulation purpose. Here, displacement fields are determined via an optical measurement technique. The accompanying modification of the solution procedure results in an extended least squares functional. The sensitivity analysis, needed for the gradient based method, is embedded in a distinct format in the finite element context.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solutions.     

Boundary-value problem with mixed conditions for weakly nonlinear hyperbolic equations

In a cylindrical domain, we investigate the unique solvability of a problem with mixed boundary conditions for an inhomogeneous linear hyperbolic equation of higher order with coefficients variable with respect to space coordinates. To estimate from below the small denominators that appear in the construction of a solution of the problem, the metric approach is used. The obtained results are extended to the case where the equation is perturbed by a nonlinear summand.     

The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L 2

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.     

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.