A Boundary Value Problem of the Helmholtz Equation
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چکیده
This chapter is devoted to the investigation of the Dirichlet problem for the Helmholtz equation in a bounded domain. In Section 1, we briefly explain why and how boundary valued problems for the Helmholtz equation are solved. Using the properties ofDα, Tα, Fα as well as the projections defined by the singular Cauchy operator the Dirichlet problem for classical α−hyperholomophic functions are solved. This work is presented in Section 2. An orthogonal decomposition of L2(Ω,H(C)) for a generalization of the Laplacian is considered in Section 3. Analogous works are done in Section 4 for classical α−metaharmonic functions instead of α− hyperholomophic functions . Finally, by induction in Section 5, the existence and the unique solution to the boundary value problem for the n− th order Helmholtz equation are given.
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تاریخ انتشار 2005