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In this article, with the help of Legendre wavelets a hybrid approach is presented for the numerical solution of the Helmholtz equation which has a complex solution. The present hybrid approach is based on the Legendre Wavelet Collocation Method (LWCM). Initially, the Helmholtz equation is converted to the coupled equation with suitable transformation, later all the derivatives in the equations including boundary conditions are approximated with the help of Legendre Wavelets. Later, with the help of the Legendre wavelet operational matrix of integration, a system of linear algebraic equations are formed for obtaining the numerical solution of Helmholtz equation. The aforementioned proposed algorithm is very simple and can easily be executed in any computer-oriented language efficiently. To demonstrate the efficiency and performance of the newly established technique, the method is tested on various well-known examples from previous literature including other Legendre wavelet-based methods. The obtained results produce better accuracy and widespread use of the newly developed technique for a range of benchmark problems which have complex solutions.

A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution