On source identification problem for a delay parabolic equation

In the present study, the inverse problem of a delay parabolic equation with nonlocal conditions is investigated. The stability estimates in Hölder norms for the solution of this problem are established

Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem

Bounded solutions of nonlinear hyperbolic equations with time delay

We consider the initial value problem \displaylines{ \frac{d^{2}u}{dt^{2}}+Au(t)=f(u(t),u(t-w)), \quad t>0, \cr u(t)=\varphi (t),\quad -w\leq t\leq 0 } for a nonlinear hyperbolic equation with time delay in a Hilbert space with the self adjoint positive definite operator A. We establish the existence and uniqueness of a bounded solution, and show application of the main theorem for four nonlinear partial differential equations with time delay. We present first and second order accuracy difference schemes for the solution of one dimensional nonlinear hyperbolic equation with time delay. Numerical results are also given

HE'S HOMOTOPY PERTURBATION METHOD FOR A GENERAL RICCATI EQUATION

WOS: 000272918400007The factorization method of the Hamiltonian in quantum mechanics is used to solve a particular type of Riccati equation. In this paper, He's homotopy perturbation method is applied to a general Riccati equation. The obtained solutions are in recursive sequence forms and, in many cases, these solutions lead to an exact solution. This property generally means that one can solve completely the eigenvalue problem for the Hamiltonian operator. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis

Stability of parabolic equations with unbounded operators acting on delay terms

In this article, we study the stability of the initial value problem for the delay differential equation \displaylines{ \frac{dv(t)}{dt}+Av(t)=B(t)v(t-\omega )+f(t),\quad t\geq 0,\cr v(t)=g(t)\quad (-\omega \leq t\leq 0) } in a Banach space E with the unbounded linear operators A and B(t) with dense domains $D(A)\subseteq D(B(t))$. We establish stability estimates for the solution of this problem in fractional spaces $E_{\alpha }$. Also we obtain stability estimates in Holder norms for the solutions of the mixed problems for delay parabolic equations with Neumann condition with respect to space variables