Solution Of Cauchy Problem Research Articles

Stable cylindrical Lévy processes and the stochastic Cauchy problem

In this work, we consider the stochastic Cauchy problem driven by the canonical $\alpha $-stable cylindrical Lévy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.

On the regularization of the Cauchy problem for a system of linear difference equations

The article proposes unusual regularization conditions as well as a scheme for finding solutions of the linear Cauchy problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology. The problem posed in the article continues the study of the regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by S.G. Krein, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Cauchy problem has no inverse. In the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Cauchy problem for a system of difference equations in the critical case is found. The proposed regularization conditions, as well as the scheme for finding solutions to linear Cauchy problems for a system of difference equations in the critical case, are illustrated in details with examples. In contrast to the earlier articles of the authors, the regularization problem for a linear Cauchy problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a solution to the regularization problem.

Решение жестких задач Коши явными схемами с геометрически-адаптивным выбором шага

Решение жестких задач Коши явными схемами с геометрически-адаптивным выбором шага

Решение задачи Коши об охлаждении кусочно-однородного стержня

Решение задачи Коши об охлаждении кусочно-однородного стержня

On the Solutions of Fractional Cauchy Problem Featuring Conformable Derivative

In this study, we have obtained analytical solutions of fractional Cauchy problem by using q-Homotopy Analysis Method (q-HAM) featuring conformable derivative. We have considered different situations according to the homogeneity and linearity of the fractional Cauchy differential equation. A detailed analysis of the results obtained in the study has been reported. According to the results, we have found out that our obtained solutions approach very speedily to the exact solutions.

On Approximation of Multivalued Solution for Hamilton—Jacobi equation

Abstract The paper deals with the Cauchy problem for Hamilton—Jacobi equation with discontinuous w.r.t. state variable Hamiltonian. In this case we use the notion of M-solution proposed by Subbotin. We consider the sequence of auxiliary Cauchy problems for Hamilton— Jacobi equations with Lipschitz continuous w.r.t. phase variable Hamiltonians. We show that the sequence of distances between of graphs of solutions for auxiliary Cauchy problems and graph of M-solution tends to zero in metrics L1.

Solution of the singular Cauchy problem for a general inhomogeneous Euler–Poisson–Darboux equation

In this paper, we solve Cauchy problem for a general form of an inhomogeneous Euler–Poisson–Darboux equation, where Bessel operator acts instead of the each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, for a specially selected form of the initial conditions, the equation has a solution. The general form of the Euler–Poisson–Darboux equation with such conditions we will call the singular Cauchy problem.

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We deri ...

О приближенном решении задачи Коши для системы ОДУ посредством системы $1,\, x,\, \{\frac{\sqrt{2}}{\pi n}\sin(\pi nx)\}_{n=1}^\infty$

О приближенном решении задачи Коши для системы ОДУ посредством системы $1,\, x,\, \{\frac{\sqrt{2}}{\pi n}\sin(\pi nx)\}_{n=1}^\infty$

Решение начальной задачи для одной системы трёх дифференциальных уравнений с производной дробного порядка

Решение начальной задачи для одной системы трёх дифференциальных уравнений с производной дробного порядка

Exponential convergence to time-periodic viscosity solutions in time-periodic Hamilton-Jacobi equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.

The conditions of existence with probability one of generalized solutions of Cauchy problem for the heat equation with a random right part

The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random conditions is a classical problem of mathematical physics. In this paper we consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space Subφ(Ω). The conditions of existence with probability one generalized solution of the problem are investigated.

Trefftz energy method for solving the Cauchy problem of the Laplace equation

The inverse Cauchy problem of Laplace equation is hard to solve numerically, since it is highly ill-posed in the Hadamard sense. With this in mind, we propose a natural regularization technique to overcome the difficulty. In the linear space of the Trefftz bases for solving the Laplace equation, we introduce a novel concept to construct the Trefftz energy bases used in the numerical solution for the inverse Cauchy problem of the Laplace equation in arbitrary star plane domain. The Trefftz energy bases not only satisfy the Laplace equation but also preserve the energy, whose performance is better than the original Trefftz bases. We test the new method by two numerical examples.

The error in solving a wave equation based on a scheme with weights

The error in approximating a Cauchy problem for a two-dimensional wave equation based on a scheme with weights is studied. The dependence of the approximation error on the time step and weight parameter is considered. With this aim in view, the difference of second-order spatial derivatives in the wave equation is approximated, while the time derivative is preserved continuous; and an analytical solution of a Cauchy problem for a system of ordinary differential equations is obtained as the decomposition in the orthonormal basis consisting of the eigenvectors of the operator of the second difference derivative with respect to the spatial variables. Based on this solution, the errors of the approximation of the wave problem by three-layer difference schemes are studied and the conditions for the stability of a three-layer difference scheme are obtained. It is established that, when simulating the propagation of oscillation processes using difference methods, the oscillation frequency values differ from the real ones and depend on the weight parameter and time step. The optimal values of the weight parameter with which the deviation of the oscillation frequency for the difference scheme is minimal are obtained. The dependences of the approximation error on weight and spatial step are derived. The optimal values of the weight parameter with which the schemes are of the second and fourth order of accuracy with respect to the time step are found.

An inverse problem for the equation of membrane's vibration

A mathematical model for membrane's vibration process is used in this paper. The model is based on seeking a solution of the second-order hyperbolic differential equation. A new inverse problem is set and investigated in two versions. In the first version the known data are as follows: the coefficient defining the phase velocity, the starting data of the Cauchy problem, the Cauchy problem solution on two given planes, derivatives of the solution along the vector being normal to these planes. The challenge has been in localizing the support of the right-hand side of the equation for vibrations. The algorithm permitting to find the bounded domain containing the unknown support was designed. In the second version the algorithm refers to the case where the coefficient defining the phase velocity is unknown but an interval of its possible values is known. A series of runs was performed to illustrate the proposed model.

Half space problem for Euler equations with damping in 3-D

In this paper, we consider the half space problem for Euler equations with damping in 3-D. We restudy the fundamental solution for the Cauchy problem to obtain an exponentially sharp pointwise structure and a clear decomposition of the singular-regular components. Later, both Green's function for initial boundary value problem and fundamental solutions for Cauchy problems are investigated in the transformed domain after Laplace transform. The symbols are obtained and a connection between Green's function and fundamental solutions are established for the pointwise space–time structure of Green's function. Finally, the sharp estimates for Green's function together with a priori estimates from the energy method for high order derivatives result in the nonlinear stability of the solution and also the decaying rates.

Spaces of Smooth and Generalized Vectors of the Generator of an Analytic Semigroup and Their Applications

For a strongly continuous analytic semigroup {e tA }t ≥ 0 of linear operators in a Banach space B, we study some locally convex spaces of smooth and generalized vectors of its generator A, as well as the extensions and restrictions of this semigroup to these spaces. We extend the Lagrange result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on the description of the set of all vectors x∈𝔅 for which the limit $$ {\lim}_{n\to \infty }{\left(I+\frac{tA}{n}\right)}^nx $$ exists and coincides with e tA x. Moreover, we present a brief survey of particular problems whose solutions are required for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator, the representation of solutions of operator-differential equations on an open interval and the investigation of their boundary values, and the existence of solutions of an abstract Cauchy problem in various classes of analytic vector-valued functions.

Feynman and quasi-Feynman formulas for evolution equations

New methods for obtaining representations of solutions of the Cauchy problem for linear evolution equations, i.e., equations of the form u t '(t, x) = Lu(t, x), where the operator L is linear and depends only on the spatial variable x and does not depend on time t, are proposed. A solution of the Cauchy problem, that is, the exponential of the operator tL, is found on the basis of constructions proposed by the author combined with Chernoff’s theorem on strongly continuous operator semigroups.

The Solution Of Nonhomogen Abstract Cauchy Problem by Semigroup Theory of Linear Operator

In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.

A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations

The generalized finite difference method (GFDM) is a relatively new domain-type meshless method for the numerical solution of certain boundary value problems. The method involves a coupling between the Taylor series expansions and weighted moving least-squares method. The main idea here is to fully inherit the high-accuracy advantage of the former and the stability and meshless attributes of the latter. This paper makes the first attempt to apply the method for the numerical solution of inverse Cauchy problems associated with three-dimensional (3D) Helmholtz-type equations. Numerical results for three benchmark examples involving Helmholtz and modified Helmholtz equations in both smooth and piecewise smooth 3D geometries have been analyzed. The convergence, accuracy and stability of the method with respect to increasing the number of scatted nodes inside the whole domain and decreasing the amount of noise added into the input data, respectively, have been well-studied.