Some Properties And The Related Boundary Value Problem Of The Teodorescu Operators Associated To The Helmholtz Equation And The Time-harmonic Maxwell Equations

In this paper,we study some properties of the Teodorescu operator associated with Helmholtz equation and N matrix operator in the quaternion analysis.Teodorescu oper-ator is a singular integral operator and it has been widely used in solving the boundary value problem of partial differential equations.In complex analysis and Rn space,the theory of the Teodorescu operator associated with the regular functions has been devel-oped very well.However,in the quaternion analysis,there are relatively few studies on the properties of the Teodorescu operator related to the Helmholtz equation and the N matrix operator.In this paper,we mainly study the boundedness,continuity of Teodor-escu operator associated with Helmholtz equation and N matrix operator.And we give the integral representation of the Riemann boundary value problem by using the Teodor-escu operator as stated above.Finally,using the relationship between the corresponding Cauchy type integral of the N matrix operator and the corresponding Cauchy type inte-gral of the Time-harmonic Maxwell equations,we give the integral representation of the Riemann boundary value problem related to the Time-harmonic Maxwell equations.In the first chapter,some preliminary knowledge are introduced,such as the quater-nion space and the algorithm of quaternion space.In the second chapter,firstly,some definitions and lemmas are given.Secondly,we prove the boundedness,continuity of the Teodorescu operator Tψ,α related to the Helmholtz equation and find its generalized derivative.Finally,we give the integral representation of the solution for the Riemann boundary value problem associated with the Helmholtz equation by using the Teodorescu operator as stated above.In the third chapter,we introduce the Teodorescu operator TN,α associated with the N matrix operator.By using the relevant conclusions of the second chapter,we prove the boundedness,continuity of the TN,α operator and so on,and give the integral representation of the solution for the Riemann boundary value problem related to N matrix operator.In the fourth chapter,we first introduce the Time-harmonic Maxwell equations,then we give the integral representation of the solution for the Riemann boundary value problem associated with the Time-harmonic Maxwell equations.