A robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation

Area of Science:

• Applied Mathematics
• Mathematical Physics
• Wave Propagation Modeling

Background:

• The Helmholtz equation is fundamental in modeling wave phenomena, including underwater acoustics and ocean wave behavior.
• Accurate analytical solutions are essential for understanding wave propagation in various environments.
• Fractional-order differential equations offer more precise models for complex wave dynamics.

Purpose of the Study:

• To introduce and validate the Elzaki transform residual power series method (ET-RPSM) for fractional-order Helmholtz equations.
• To demonstrate the method's capability in handling fractional derivatives using the Caputo sense.
• To showcase the efficiency and accuracy of the proposed analytical technique.

Main Methods:

• Combining the Elzaki transform (ET) with the residual power series method (RPSM).
• Utilizing the ET to transform the fractional Helmholtz equation into a recurrence relation.
• Applying RPSM to generate an iterative series solution from the recurrence relation.

Main Results:

• The ET-RPSM successfully provides analytical solutions for fractional-order Helmholtz equations.
• The method demonstrates rapid convergence to the exact solution within a few iterations.
• Numerical applications confirm the efficiency and authenticity of the proposed scheme.

Conclusions:

• The ET-RPSM is an effective and reliable technique for solving fractional-order Helmholtz equations.
• The integration of ET and RPSM offers a novel approach to analytical solutions in fractional calculus.
• The method's ability to handle fractional orders and produce quick, accurate results highlights its significance for wave propagation studies.