Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

n阶常系数非齐次线性微分方程的降阶解法

给出一阶线性非齐次微分方程的积分因子解法,避免了常数变易法带来的不便和不自然;给出,n阶常系数非齐次线性微分方程的降阶解法,可以看出,高阶常系数线性非齐次微分方程最终都可以归结为求解一阶线性微分方程,从而避免了待定系数法求非齐次方程特解的繁琐,并最终说明了一般微积分教材中只给出两种类型常系数非齐次线性微分方程的待定系数解法的原因．     

Method for constructing solutions of linear ordinary differential equations with constant coefficients

For Cauchy problems involving linear differential equations with constant coefficients, a new method for constructing solutions without determining the roots of the characteristic equation is proposed. Formulas for the differentiation of the solution with respect to the equation coefficients are derived, and an approximate analytical solution is found.     

一类高阶混合型方程周期解存在唯一的充要条件

该文利用Fourier级数理论讨论了具有两个偏差变元的常系数线性高阶混合型泛函微分方程的周期解,得到了用系数与时滞表示的周期解存在、唯一的充要条件.     

Integro-differential Burgers equation. Solvability and homogenization

We consider the integro-differential Burgers equation which appears in nonlinear acoustics. The integral term in the right-hand side describes the relaxation (memory) effects. The global existence and uniqueness of solution is proved. The smoothness of the solution is studied.In the case when the coefficients of the equation rapidly oscillate, we replace it by the homogenized equation with constant coefficients and prove the error estimates.     

On <Emphasis Type="Italic">n</Emphasis>-arithmetical triangles constructed for polynomial coefficients

A fundamental solution system to a finite order linear difference equation with constant coefficients is considerably used during the investigation of the n-arithmetical triangles properties. The constructed fundamental solution system is explicitly expressed via the coefficients of the difference equation. We prove the symmetric and unimodal properties of the polynomial coefficients sequence and obtain the formulas for these coefficient calculations.     

Backlund Transformation and Exact Solution to Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the homogeneous balance principle, we derive a Backlund transformation (BT) to (3+1)-dimensionaI Kadomtsev-Petviashvili (K-P) equation with variable coefficients if the variable coefficients are linearly dependent. Based on the BT, the exact solution of the (3+1)-dimensional K-P equation is given. By the same method, we derive a BT and the solution to (2+1)-dimensional K-P equation. The variable coefficients can change the amplitude of solitary wave, but cannot change the form of solitary wave.     

Baecklund Transformation and Exact Solution to Kadomtsev-Petviashvili Equation with Variable Coefficients

By using homogeneous balance principle,we derive a Baecklund trans-formation(BT) to (3 1)-dimensional Kadomtsev-Petviashvili( K-P) equation with variable coefficients if the variable coefficients are linearly dependent.Based on the BT,the exact solution of the (3 1)-dimensional K-P equation is given.By the same method,we derive a BT and the solution to (2 1)-dimensional K-P equation,The variable coefficients can change the amplitude of solitary wave,but cannot change the form of solitary wave.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

A Stability Estimate for a Solution in the Inverse Problem of Finding the Coefficients of Lower Derivatives

We consider the problem of finding the coefficients of the first derivatives in a second-order hyperbolic equation. The additional information is the trace of a solution and its normal derivative on the lateral surface of the cylindrical domain of some direct problem for the original equation. The impulse point source lies outside the domain in which the sought coefficients are determined and is a parameter of the problem. We suppose that the number of sources for which the trace of a solution is given coincides with the number of the coefficients to be determined. The main result of this article is a stability estimate for a solution to the inverse problem under consideration.     

Existence theorem for a stochastic wave equation with discontinuous unbounded coefficients

We prove an existence theorem for stochastic hyperbolic equations with measurable locally bounded coefficients. A solution of a stochastic hyperbolic equation is understood as a martingale solution of the stochastic inclusion corresponding to the equation.     

Nontrivial solvability of elliptic equations in divergence form with complex coefficients

The stationary Fokker-Planck-Kolmogorov equation with complex diffusion coefficients and a complex vector-field is examined on a torus. Under suitable conditions for the diffusion coefficients, it is proven that a nontrivial solution exists and the solution space is multidimensional in some cases.     

On long-time asymptotics of solutions of parabolic equations with increasing leading coefficients

Sharp sufficient conditions on the coefficients of a second-order parabolic equation are examined under which the solution of the corresponding Cauchy problem with a power-law growing initial function stabilizes to zero. An example is presented showing that the found sufficient conditions are sharp. Conditions on the coefficients of a parabolic equation are obtained under which the solution of the Cauchy problem with a bounded initial function stabilizes to zero at a power law rate.     

用无限阶Toeplitz矩阵求常系数微分方程的级数解

无限阶Toeplitz矩阵的属于0的特征向量可递推地求得,可表示常系数齐次微分方程的解.用它的逆可求得常系数非齐次微分方程的特解.     

Exact solutions of KdV equation with time-dependent coefficients

In this paper, we study the Korteweg-de Vries (KdV) equation having time dependent coefficients from the Lie symmetry point of view. We obtain Lie point symmetries admitted by the equation for various forms for the time-dependent coefficients. We use the symmetries to construct the group-invariant solutions for each of the cases of the arbitrary coefficients. Subsequently, the 1-soliton solution is obtained by the aid of solitary wave ansatz method. It is observed that the soliton solution will exist provided that these time-dependent coefficients are all Riemann integrable.     

On an Optimal Filtration Problem for One-Dimensional Diffusion Processes

We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Itô equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula.     

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.     

Error bounds for numerical solutions of ordinary differential equations

Summary An a posteriori error bound, for an approximate solution of a system of ordinary differential equations, is derived as the solution of a Riccati equation. The coefficients of the Riccati equation depend on an eigenvalue of a matrix related to a Jacobian matrix, on a Lipschitz constant for the Jacobian matrix, and on the approximation defect. An upper bound is computable as the formal solution of a sequence of Riccati equations with constant coefficients. This upper bound may sometimes be used to control step length in a numerical method.     

A necessary condition for a power series to be a formal solution of a singular linear differential equation of order k

We obtain a necessary condition on the coefficients of a formal power series, which is a formal solution of a nontrivial singular linear differential equation of order k, with analytic coefficients and prove a “uniqueness” theorem for the differential equation.