On Riemann Boundary Value Problem for Polyanalytic Functions on the Real Axis

Wang, Yufeng; Du, Jinyuan

doi:10.1016/s0252-9602(17)30251-5

Cited by 22 publications

(23 citation statements)

References 3 publications

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“…A great deal of interest has arisen in various kinds of boundary value problems for polyanalytic functions (e.g. [8,14,18,19]). A generalization of polyanalytic functions is the so-called metaanalytic functions [6], which satisfy a polynomial of the Cauchy-Riemann operator with roots.…”

Section: Introductionmentioning

confidence: 99%

“…A generalization of polyanalytic functions is the so-called metaanalytic functions [6], which satisfy a polynomial of the Cauchy-Riemann operator with roots. On the other hand, different types of boundary conditions need to be treated differently as boundary value problems, for example, Riemann BVPs [8,18], Hilbert BVPs [19], Haseman BVPs [10,15,20], Carleman BVPs [12,16], Dirichlet BVPs [1,4], Schwarz BVPs [3], compound BVPs [12], mixed BVPs [7,9,13], etc. In 2001, Fatulaev studied a Haseman BVP on the circle for a special class of metaanalytic functions [10].…”

Section: Introductionmentioning

confidence: 99%

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Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

2010

Journal of Mathematical Analysis and Applications

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In this article, we reconsider the mixed boundary value problem on the unit circle for a pair of metaanalytic and analytic functions as in Du and Wang (2008) [9]. By adopting appropriate transformations, we convert the problem into two independent boundary value problems for analytic functions. We then obtain expressions of solution and condition of solvability for the mixed boundary value problem. The forms of the solutions and the condition of solvability here are rather dissimilar to those in Du and Wang (2008) [9]. But the equivalence is established at the end of this article.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

2010

Journal of Mathematical Analysis and Applications

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“…In recent time, various kinds of BVPs for polyanalytic functions and generalized polyanalytic functions have widely been investigated, see for example [4,[7][8][9]13,14]. In general, two methods are used to deal with BVPs of polyanalytic functions.…”

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confidence: 99%

“…One approach is to make use of the so-called poly-Cauchy operator [4,7]. The other is to transform the BVPs for polyanalytic functions into equivalent BVPs of analytic functions or systems of analytic functions by the decomposition theorem for polyanalytic functions [13].…”

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confidence: 99%

A Dirichlet problem for polyharmonic functions

Begehr

Wang

2007

Annali di Matematica

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In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given.

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“…In recent time, a large number of investigations on various boundary value problems (simply, BVPs) for polyanalytic functions, metaanalytic functions have widely been published, refer to papers [6,9,10,12,21,22] and references there. However, the investigations on Dirichlet problems for polyharmonic functions (simply, PHD problems) just appeared in recent two years [3,4,7].…”

Section: Introductionmentioning

confidence: 99%

Decompositions of functions and Dirichlet problems in the unit disc

Guo

Wang

2010

Journal of Mathematical Analysis and Applications

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In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.

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On Riemann Boundary Value Problem for Polyanalytic Functions on the Real Axis

Cited by 22 publications

References 3 publications

Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

A Dirichlet problem for polyharmonic functions

Decompositions of functions and Dirichlet problems in the unit disc

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