A. Okay Çelebi

Complex methods for partial differential equations

Preface. 1. A reflection principle and its applications J. Witte. 2. On some problems for first order elliptic systems in the plane D.Q. Dai. 3. Differential-operator solutions for complex partial differential equations O. Celebi, S. Sengul. 4. On a generalized Riemann-Hilbert Boundary value problem for second order elliptic systems in the plane M. Akal. 5. Boundary value problems of the theory of generalized analytic functions G. Manjavidze, G. Akhalaia. 6. On well-posedness of problems for nonclassical systems of equations D.Kh. Safarov. 7. An application of the periodic Riemann boundary value problem to a periodic crack problem X. Li. 8. Initial and boundary value problems for singular differential equations and applications to the theory of cusped bars and plates G. Jaiani. 9. Multidimensional logarithmic residues and their applications L.A. Aizenberg. 10. The Neumann problem for the inhomogeneous pluriharmonic system in polydiscs A. Mohammed. 11. Second order Cauchy-Pompeiu representations H. Begehr. 12. On a class of second order elliptic overdetermined systems A. Dzhuraev. 13. Boundary spinors and values of holomorphic functions J. Cnops. 14. Two approaches to non-commutative geometry V.V. Kisil. 15. Some partial differential equations in Clifford analysis E. Obolashvili. 16. Generalized monogenic functions satisfying differential equations with anti-monogenic right-hand sides W. Tutschke, U. Yuksel. 17. Complex analytic method forhyperbolic equations of second order G.-C. Wen. 18. Remarks on the solvability of Dirichlet problems in different function spaces F. Rihawi. 19. Complex methods in the theory of initial value problems W. Tutschke. 20. Optimal balls for solving fixed-point problems in Banach spaces T. Tutschke. 21. Wavelet transform of operators and functional calculus V.V. Kisil.

Mixed boundary value problems for higher-order complex partial differential equations

Abstract In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

Nonoscillation and asymptotic behaviour for third order nonlinear differential equations

In this paper we consider the equation y‴+q(t)y′α + p(t)h(y)=0, where p, q are real valued continuous functions on [0, ∞) such that q(t) ≥ 0, p(t) ≥ 0 and h(y) is continuous in (−∞, ∞) such that h(y)y > 0 for y ≠ 0. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.

The cauchy–kowalewski theorem in the space of pseudoholomorphic functions

In this article we prove the Cauchy–Kowalewski theorem for the initial-value problem in the space PD (E) of pseudoholomorphic functions in the sense of Bers.

A Boundary Value Problem for Generalized Analytic Functions in Wiener-type Domains

In this article we investigate the conditions under which the solution of the boundary value problem exists in the class of generalized analytic functions defined on a Wiener-type domain D ⊂ ℂ

On the nonoscillatory behavior of solutions of third order differential equations

We are interested in nonoscillatory behavior of solutions of differential equations of the form where p(t) ≥ 0, q(t) ≤ 0 are real valued continuous functions on [0,∞), ∝ > 0 is the ratio of odd integers and h(y) is continuous on (-∞,∞) such that h(y)y>0 for y ≠ 0. We obtain sufficient contitions so that all solutions of the considered equation are nonoscillatory.

Some relations among the classes of pseudoholomorphic functions

In this article, first we define an operation on the space of generating pairs and investigate the properties of this operation. Secondly we have derived the class of functions which involves the derivatives of , where and is the class of – pseudoholomorphic functions in D.

An optimal control problem with nonlinear elliptic state equations

Abstract In this article some of the results for optimal control of linear systems have been generalized to a nonlinear case. This is achieved by employing standard techniques of the nonlinear theory. After demonstrating the existence of optimal controls, finite element method is used to discretize the problem. The resulting finite dimensional problem is solved by a special algorithm. The theoretical discussions are completed by proving that approximate solutions are reduced to exact solutions as the element size tends to zero. This study is closed by a presentation and a discussion of several related numerical results.

A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem

ABSTRACT According to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.

Dirichlet Problems with Nonsmooth Boundary

In this paper, we give a survey on the solubility of the Dirichlet problems without imposing restrictions on the boundaries of the domains.