## Divergence regularization method for solving ill-posed Helmholtz equation

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##### Date

June, 2016

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##### Abstract

In 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 + 2m)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 AdaptiveWavelet 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.

##### Description

A thesis submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.