Pseudoparabolic equations with convection

The existence of solutions of pseudoparabolic equations withconvection by using discretization along characteristics isshown. The uniqueness of the solution of a pseudoparabolic equationis proved for a linear elliptic part and for a space dimensionN 4.     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

一类一维非线性伪抛物型方程的反问题

将Riemann函数方法与不动点理论有效地结合起来,研究了一类一维非线性伪抛物型方程的后向热流问题,得出了反问题解的存在唯一性结论.     

一类具奇异右端项的伪抛物方程的摄动问题

本文讨论了一类具奇异右端项的伪抛物方程的初边值问题的摄动,证明了摄动问题广义解的存在性及极限性态,并得到了当ε趋于零时,摄动问题的解在一定意义下收敛于原问题的解.     

Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels

We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.     

Third order homogeneous weakly hyperbolic equations with nonanalytic coefficients

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.     

Delayed quasilinear evolution equations with application to heat flow

In this paper we show local and global existence for a class of (hyperbolic) quasilinear equations perturbed by bounded delay operators. In the last section, the abstract results are applied to a heat conduction model (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter , and a discontinuous drift. The proof of this result is based on the Girsanov theorem for the fractional Brownian motion.     

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Dedicated to S. P. Novikov on his 65 birthdayOn leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.     

Weak exponential schemes for stochastic differential equations with additive noise

^** Email: cmora{at}ing-mat.udec.cl This paper develops weak exponential schemes for the numericalsolution of stochastic differential equations (SDEs) with additivenoise. In particular, this work provides first and second-ordermethods which use at each iteration the product of the exponentialof the Jacobian of the drift term with a vector. The articlealso addresses the rate of convergence of the new schemes. Moreover,numerical experiments illustrate that the numerical methodsintroduced here are a good alternative to the standard integratorsfor the long time integration of SDEs whose solutions by thecommon explicit schemes exhibit instabilities.     

Nonelliptic Schrödinger equations

Summary Nonelliptic Schr?dinger equations are defined as multidimensional nonlinear dispersive wave equations whose linear part in the space variables is not an elliptic equation. These equations arise in a natural fashion in several contexts in physics and fluid mechanics. The aim of this paper is twofold. First, a brief survey is made of the main nonelliptic Schr?dinger equations known by the authors, with emphasis on water waves. Second, a theory is developed for the Cauchy problem for selected examples. The method is based on linear estimates which are strongly related to the dispersion relation of the problem.     

On uniqueness criteria for systems of ordinary differential equations

We present some new uniqueness criteria for the Cauchy problem

     

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain. In many practice applications, we cannot know the initial temperature during heat process, therefore we consider a non-characteristic Cauchy problem for the heat equation. The method of fundamental solutions is applied to solve this problem. Due to ill-posedness of this problem, we first discretize the problem and then regularize it in the form of discrete equation. Numerical tests are conducted for showing the effectiveness of the proposed method.     

Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces . Though the case of the derivative index s=0 has been treated in our previous work, the case s≠0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of for the existence of local and global solutions for initial data .     

Predictor-corrector pseudospectral methods for stochastic partial differential equations with additive white noise

Commonly used finite-difference numerical schemes show some deficiencies in the integration of certain types of stochastic partial differential equations with additive white noise. In this paper efficient predictor-corrector spectral schemes to integrate these equations are discussed. They are all based on the discretization of the system in Fourier space. The nonlinear terms are treated using a pseudospectral approach so as to speed up the computations without a significant loss of accuracy. The proposed schemes are applied to solve, both in one and two spatial dimensions, two paradigmatic continuum models arising in the context of nonequilibrium dynamics of growing interfaces: the Kardar-Parisi-Zhang and Lai-Das Sarma-Villain equations. Numerical results about the Lai-Das Sarma-Villain equation in two spatial dimensions have not been previously reported in the literature.     

Tricomi problem for strongly degenerate equations of parabolic-hyperbolic type

A new integral representation of solutions of a Tricomi problem for a strongly degenerate system of equations of parabolic-hyperbolic type is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp.385–392, September, 1999.     

Nonlinear equations with unbounded heat conduction and integrable data

We consider a class of quasi-linear diffusion problems involving a matrix A(t,x,u) which blows up for a finite value m of the unknown u. Stationary and evolution equations are studied for L ¹ data. We focus on the case where the solution u can reach the value m. For such problems we introduce a notion of renormalized solutions and we prove the existence of such solutions.