Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation

Abstract

The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term 1 + α 2 m e m , where α > 0 is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness.

Solving the Helmholtz Equation Together with the Cauchy Boundary Conditions by a Modified Quasi‐Reversibility Regularization Method

“…Similar to this method, authors in [7][8][9] have used the Q − RRM to obtain regularized solution to the Helmholtz equation with various stability estimations in functional spaces. However, other techniques including the Divergence Regularization Method (DRM) and the Lavrentiev Regularization Method (LRM) developed by the authors in [10] and [11], respectively, have successfully produced regularized solutions of the Helmholtz equation with imposed Cauchy boundary conditions. The numerical schemes for restoring the stable solution of the Helmholtz equation have been observed by researchers across the globe.…”