Abdullah Said Erdogan

On the second order implicit difference schemes for a right hand side identification problem

In many physical phenomena, especially in temperature over-specification partial differential equation with an unknown source function appears. The present paper is devoted to the study of the well-posedness of the approximate solution of a right-hand side identification problem for a parabolic equation. The second order of accuracy implicit difference scheme is presented. The coercive stability estimates for the solution of this difference scheme are obtained. The theoretical statements for the solution of this difference schemes are supported by results of numerical experiments.

A Note on the Inverse Problem for a Fractional Parabolic Equation

For a fractional inverse problem with an unknown time-dependent source term, stability estimates are obtained by using operator theory approach. For the approximate solutions of the problem, the stable difference schemes which have first and second orders of accuracy are presented. The algorithm is tested in a one-dimensional fractional inverse problem.

A Note on the Right-Hand Side Identification Problem Arising in Biofluid Mechanics

The inverse problem of reconstructing the right-hand side (RHS) of a mixed problem for one-dimensional diffusion equation with variable space operator is considered. The well-posedness of this problem in Holder spaces is established.

On a first order partial differential equation with the nonlocal boundary condition

In this paper, the initial value problem for the first order partial differential equation with the nonlocal boundary condition is studied. In applications, the stability estimates for the first order partial differential equation with the nonlocal boundary condition are obtained. The finite difference method for the initial value problem for hyperbolic equations with nonlocal boundary conditions is applied. In practice, the stability estimates for the solution of the difference scheme of the problem for hyperbolic equations with nonlocal boundary conditions are obtained.

On the numerical solution of the diffusion equation with variable space operator

In present paper numerical schemes are developed for obtaining approximate solutions to the mixed problem for one-dimensional diffusion equation with variable space operator.The method of lines semidiscretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations. The partial derivative with respect to the space variable is approximated by the first and second-order finite-difference approximation. For the solution of the resulting system of first-order ordinary differential equations we apply the first and second order of accuracy difference schemes. Stability estimates for the solution of these difference schemes are established. Numerical techniques are developed by applying a procedure of the solution of first order linear difference equation with matrix coefficients. The algorithms are tested on a model problem in biofluid mechanics. Two regions are considered close to the endothelial cells (EC) which can be modeled taking the mixed problem for one-dimensional diffusion equation with variable space operator.

Stability of Implicit Difference Scheme for Solving the Identification Problem of a Parabolic Equation

We consider the inverse problem of reconstructing the right side of a parabolic equation with an unknown time dependent source function. Numerical solution and well-posedness of this type problem with local boundary conditions considered previously by A.A. Samarskii, P.N. Vabishchevich and V.T. Borukhov. In this paper, we focus on studying the stability of the problem with nonlocal conditions. A stable algorithm for the approximate solution of the problem is presented.

Well-Posed and Ill-Posed Boundary Value Problems for PDE 2013

equations. The issue covers a wide variety of problems for different classes of ordinary and partial differential equations, as well as dynamic equations on time scales. The topics discussed in the contributed papers are traditional for qualitative theory of differential equations. The issue contains papers on the global well-posedness of the viscous twocomponent Camassa-Holm system, local and global existence of solutions for a generalized Camassa-Holm equation, global solutions for the Cauchy problem of a Boussinesq-type equation, exact asymptotic expansion of singular solutions for the � 2 � 1� -D Protter problem, on the regularity for variational evolution integrodifferential in equalities, right-hand side identification problem arising in biofluid mechanics, regularized solutions of optimal control problem in a hyperbolic system, generalized localization of Fourier inversion associated with an elliptic operator for distributions and Kamenev-type oscillation criteria for the secondorder nonlinear dynamic equations with damping on-time scales. Furthermore, classification

On the unique solvability of an inverse problem for a semilinear parabolic equation

In the present paper, unique solvability of an inverse problem of source identification governed by a semilinear parabolic equation is investigated. Moreover, for the numerical solution of this problem finite difference method is applied. The convergence estimate for the solution of the difference scheme is presented.

On the numerical solution of a diffusion equation arising in two-phase fluid flow

In present paper, difference schemes are constructed for obtaining approximate solutions of the inverse problem for one-dimensional diffusion equation arising in two-phase fluid flow.

On the unique solvability of an inverse problem for a semilinear equation with final overdetermination

This paper deals with existence and uniqueness of the solution of an inverse problem for a semilinear equation subject to a final overdetermination in a Banach space. Moreover, the first order of accuracy Rothe difference scheme is presented for the numerical solution of this problem. The existence and uniqueness result for this difference scheme is given. This difference scheme is applied on a particular example and some numerical results are given.