Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach

The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations.