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British Journal of Mathematics & Computer Science, ISSN: 2231-0851,Vol.: 22, Issue.: 2

Original-research-article

On the Notes of Quasi-Boundary Value Method for Solving Cauchy-Dirichlet Problem of the Helmholtz Equation

Benedict Barnes1* , F. O. Boateng2, S. K. Amponsah3 and E. Osei-Frimpong4
1Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana.
2Department of Interdisciplinary Studies, University of Education, Winneba, Kumasi, Ghana.

Article Information
Editor(s):
(1) Jacek Dziok, Institute of Mathematics, University of Rzeszow, Poland.
(2) Andrej V. Plotnikov, Department of Applied and Calculus Mathematics and CAD, Odessa State Academy of Civil Engineering and Architecture, Ukraine.
Reviewers:
(1) Abdullah Sonmezoglu, Bozok University, Turkey.
(2) Seval Catal, Dokuz Eyll University, Turkey.
Complete Peer review History: http://www.sciencedomain.org/review-history/19213

Abstracts

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.

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