Abstract
In this article, we first propose a kind of mixed boundary value problem for the inhomogeneous Cimmino system, which consists of first order linear partial differential equations in $\mathbb{R}^{4}$ . Then, by using the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, we transform the problem as stated above into a problem related to the ψ-hyperholomorphic functions in quaternionic analysis. Moreover, we show the boundedness, Holder continuity, and generalized derivatives of a kind of singular integral operator ${}^{\psi } T_{\mathbb{C}^{2}}[g]$ related to ψ-hyperholomorphic functions in quaternionic analysis. Lastly, the solution of the mixed boundary value problem for the inhomogeneous Cimmino system is explicitly described.
Highlights
The skew field of quaternions H gives an example of a noncommutative Clifford algebra with minimal dimension
Quaternionic analysis is regarded as a broadly accepted branch of classical analysis offering a successful generalization of complex analysis
It studies functions defined on domains in R or R with values in the skew field of real quaternions H
Summary
The skew field of quaternions H gives an example of a noncommutative Clifford algebra with minimal dimension. Quaternionic analysis is regarded as a broadly accepted branch of classical analysis offering a successful generalization of complex analysis It studies functions defined on domains in R or R with values in the skew field of real quaternions H. We will study a kind of mixed boundary value problem for the inhomogeneous Cimmino system ); we obtain an integral representation of the solution of the mixed boundary value problem by using the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of a Cimmino system’s solutions. Suppose the imaginary unit of C is identified with the basis element i in quaternion algebra space H, for arbitrary z ∈ C, we have z = x + ix and its complex conjugate z = x – ix.
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