On the Notes of Quasi-Boundary Value Method for Solving Cauchy-Dirichlet Problem of the Helmholtz Equation
Benedict Barnes *
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
F. O. Boateng
Department of Interdisciplinary Studies, University of Education, Winneba, Kumasi, Ghana.
S. K. Amponsah
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
E. Osei-Frimpong
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
*Author to whom correspondence should be addressed.
Abstract
The Cauchy-Dirichlet problem of the Helmholtz equation yields unstable solution, which when solved with the Quasi-Boundary Value Method (Q-BVM) for a regularization parameter α = 0. At this point of regularization parameter, the solution of the Helmholtz equation with both Cauchy and Dirichlet boundary conditions is unstable when solved with the Q-BVM. Thus, the quasi-boundary value method is insufficient and inefficient for regularizing ill-posed Helmholtz equation with both Cauchy and Dirichlet boundary conditions. In this paper, we introduce an expression 1/(1+α2) ; α ∈ R, where α is the regularization parameter, which is multiplied by w(x; 1) and then added to the Cauchy and Dirichlet boundary conditions of the Helmholtz equation. This regularization parameter overcomes the shortcomings in the Q-BVM to account for the stability at α = 0 and extend it to the rest of values of R.
Keywords: Expresion 1/(1 α2), Q-BVM, ill-posed Helmholtz equation