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A timelike Cauchy problem and an inverse problem for general hyperbolic equations

Amirov A. | Yamamoto M.

Article | 2008 | Applied Mathematics Letters21 ( 9 ) , pp.885 - 891

We prove a Carleman estimate for hyperbolic equations with variable principal parts and present applications to the unique continuation and an inverse problem. Our Carleman estimate covers cases which the existing Carleman estimates do not treat. © 2007 Elsevier Ltd. All rights reserved.

An integral geometry problem along geodesics and a computational approach

Göolgeleyen I.

Article | 2010 | Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica18 ( 2 ) , pp.91 - 112

In this paper, we prove the existence, uniqueness and stability of the solution of an integral geometry problem (IGP) for a family of curves of given curvature. The functions in the statement of the curvature depend on two variables, which is occured especially in the case of IGP along geodesics. To prove the solvability of the problem, we reduce the IGP to an overdetermined inverse problem for the transport equation. We also develop a new symbolic algorithm to compute the approximate solution of the problem and present two computational experiments to show the accuracy of the algorithm. The results show that the proposed approach p . . .rovides highly accurate solutions and it is robust against data noises Daha fazlası Daha az

A Hölder stability estimate for inverse problems for the ultrahyperbolic Schrödinger equation

Gölgeleyen F. | Kaytmaz Ö.

Article | 2019 | Analysis and Mathematical Physics9 ( 4 ) , pp.2171 - 2199

In this article, we first establish a global Carleman estimate for an ultrahyperbolic Schrödinger equation. Next, we prove Hölder stability for the inverse problem of determining a coefficient or a source term in the equation by some lateral boundary data. © 2019, Springer Nature Switzerland AG.

NUMERICAL SOLUTION OF AN INVERSE PROBLEM FOR THE LIOUVILLE EQUATION

Golgeleyen, Fikret | Hasdemir, Muhammed

Article | 2019 | TWMS JOURNAL OF APPLIED AND ENGINEERING MATHEMATICS9 ( 4 ) , pp.909 - 920

We consider an inverse problem for the Liouville Equation. We present the solvability conditions and obtain numerical solution of the problem based on the finite difference approximation.

A generalization on the solvability of integral geometry problems along plane curves

Ustaoglu Z.

Article | 2013 | Boundary Value Problems2013 , pp.909 - 920

This paper is concerned with a general condition for the solvability of integral geometry problems along the plane curves of given curvatures. As two important results, the solvabilities of integral geometry problems along the family of circles with fixed radius and along the family of circles of varying radius centered on a fixed circle are given. By using some extension of the class of unknown functions, the proofs are based on the solvabilities of equivalent inverse problems for transport-like equation.©2013 Ustaoglu; licensee Springer.

Stability for some inverse problems for transport equations

Golgeleyen F. | Yamamoto M.

Article | 2016 | SIAM Journal on Mathematical Analysis48 ( 4 ) , pp.2319 - 2344

In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and a relaxed condition on the principal part. © 2016 Societ y for Industrial and Applied Mathematics.

An inverse problem for the general kinetic equation and a numerical method

Amirov A. | Gölgeleyen F. | Rahmanova A.

Article | 2009 | CMES - Computer Modeling in Engineering and Sciences43 ( 2 ) , pp.131 - 147

This paper has two purposes. The first is to prove existence and uniqueness theorems for the solution of an inverse problem for the general linear kinetic equation with a scattering term. The second one is to develop a numerical approximation method for the solution of this inverse problem for two dimensional case using finite difference method.© 2009 Tech Science Press.

Stability of inverse problems for ultrahyperbolic equations

Gölgeleyen F. | Yamamoto M.

Article | 2014 | Chinese Annals of Mathematics. Series B35 ( 4 ) , pp.527 - 556

In this paper, the authors consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation by some lateral boundary data. The authors prove Hölder estimates which are global and local and the key tool is Carleman estimate. © 2014 Fudan University and Springer-Verlag Berlin Heidelberg.

Solvability of an inverse problem for the kinetic equation and a symbolic algorithm

Amirov A. | Gölgeleyen F.

Article | 2010 | CMES - Computer Modeling in Engineering and Sciences65 ( 2 ) , pp.179 - 190

In this work, we derive the solvability conditions for an inverse problem for the kinetic equation and develop a new symbolic algorithm to obtain the approximate solution of the problem. The computational experiments show that proposed method provides highly accurate numerical solutions even subjecting to a large noise in the given data. Copyright © 2010 Tech Science Press.

On the solution of a coefficient inverse problem for the non-stationary kinetic equation

Yildiz M.

Article | 2009 | CMES - Computer Modeling in Engineering and Sciences45 ( 2 ) , pp.141 - 154

The solvability conditions of an inverse problem for the non-stationary kinetic equation is formulated and a new numerical method is developed to obtain the approximate solution of the problem. A comparison between the approximate solution and the exact solution of the problem is presented. Copyright © 2009 Tech Science Press.

On the solution of an inverse problem for an integro-differential transport equation

Gölgeleyen I.

Article | 2010 | CMES - Computer Modeling in Engineering and Sciences64 ( 1 ) , pp.71 - 89

In this paper, the solvability conditions for an inverse problem for an integro-differential transport equation are obtained and a numerical approximation method based on the finite difference method is developed. A comparison between the numerical solution and the exact solution of the problem is presented. Experimental results show that proposed method is robust to data noises. © 2010 Tech Science Press.

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