Browsing by Author "Barnes,Benedict"

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    Divergence regularization method for solving ill-posed Helmholtz equation
    (KNUST, 2016-06) Barnes,Benedict
    n this work, we introduce Divergence Regularization Method (DRM) for regularizing the Cauchy problem of the Helmholtz equation where the boundary deflection is not equal to zero in Hilbert space H. The DRM incorporates a positive integer scaler which homogenizes inhomogeneous boundary deflection in Cauchy problem of the Helmholtz equation to ensure the existence and uniqueness of solution for the equation. The DRM employs its regualarization term (1 + α2m)em to restore the stability of the regularized Helmholtz equation, and guarantees the uniqueness of solution of Helmholtz equation when it is imposed by Neumann boundary conditions in the upper half-plane. The DRM gives better stability approximation when compared with other methods of regularization for solving Cauchy problem of the Helmholtz equation where the boundary deflection is zero. In the process, we introduce Adaptive Wavelet Spectral Finite Difference (AWSFD) method to obtain the approximated solutions of the regularized Helmholtz equation with regularized Cauchy boundary conditions, regularized Neumann boundary conditions in the upper half-plane, and finally with regularized both Dirichlet and Cauchy boundary conditions where the boundary deflection is equal to zero. The AWSFD method captures the boundary points to obtain approximated solution of Helmholtz equation. This method reduces the Helmholtz equation in two dimensions to one dimension which is then solve spectrally using a suitable wavelet basis. The solutions by AWSFD method confirms the analytic solutions of regularized Helmholtz equation by DRM. The norm of relative error between the analytic solution by DRM and the approximated solution by AWSFD method is minimal. Moreover, we introduce interpolation scheme in the AWSFD method to obtain the approximated solutions of the regularized Helmholtz equation with above boundary conditions.