Browsing by Author "Celebi, A. Okay"

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Citation - WoS: 10
Citation - Scopus: 10
Av Bitsadze's Observation on Bianalytic Functions and the Schwarz Problem
(Taylor & Francis Ltd, 2019) Aksoy, Umit; Begehr, Heinrich; Celebi, A. Okay; Mathematics
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.
Citation - WoS: 7
Citation - Scopus: 8
Av Bitsadze's Observation on Bianalytic Functions and the Schwarz Problem Revisited
(Taylor & Francis Ltd, 2021) Aksoy, Umit; Begehr, Heinrich; Celebi, A. Okay; Mathematics
The extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general. They do hold only for certain domains.
Dirichlet-Type Problems for n-poisson Equation in Clifford Analysis
(Taylor & Francis Ltd, 2022) Aksoy, Umit; Celebi, A. Okay; Mathematics
Iterated Dirichlet problem, also called as Riquier or Navier problem, and polyharmonic Dirichlet problem are studied for n-Poisson equation in Clifford analysis using iterated polyharmonic Green function and polyharmonic Green-Almansi type function appropriate for the boundary conditions of the problems.
Citation - WoS: 8
Citation - Scopus: 11
Schwarz Problem for Higher-Order Complex Partial Differential Equations in the Upper Half Plane
(Wiley-v C H verlag Gmbh, 2019) Aksoy, Umit; Begehr, Heinrich; Celebi, A. Okay; Mathematics
Linear and nonlinear elliptic complex partial differential equations of higher-order are considered under Schwarz conditions in the upper-half plane, Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schvvarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro-differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.
Citation - WoS: 6
Citation - Scopus: 8
Solution of Initial Value Problems of Cauchy-Kovalevsky Type in the Space of Generalized Monogenic Functions
(Birkhauser verlag Ag, 2010) Yueksel, Ugur; Celebi, A. Okay; Mathematics
This paper deals with the initial value problem of the type partial derivative(t)u(t, x) = Lu(t, x), u(0, x) = u(0)(x) where t is an element of R(0)(+) is the time, x is an element of R(n+1), u(0)(x) is a generalized monogenic function and the operator L, acting on a Clifford-algebra-valued function u(t, x) = Sigma(B) u(B)(t, x)e(B) with real-valued components u(B)(t, x), is defined by Lu(t, x) := Sigma(A,B,i) c(B,i)((A)) (t, x)partial derivative(xi) u(B)(t, x)e(A) + Sigma(A,B) d(B)((A)) (t, x)u(B)(t, x)e(A) + Sigma(A)gA(t,x)e(A). We formulate sufficient conditions on the coefficients of the operator L under which L transforms generalized monogenic functions again into generalized monogenic functions. For such an operator the initial value problem (0.1) is solvable for an arbitrary generalized monogenic initial function u(0) and the solution is also generalized monogenic for each t.