Solving the Axisymmetric Dirichlet Potential using the One-Variable Hankel Transform of the I-Function

Abstract

The study presents a novel and efficient approach to solving the axisymmetric Dirichletpotential problem by employing the one-variable Hankel transform of the I-function. The Dirichletpotential problem, a fundamental concept in mathematical physics, arises in various fields includingelectromagnetism, fluid dynamics, and heat conduction. Traditional solutions often involve complexmathematical manipulations and extensive computational efforts.In this research, we leverage the powerful tool of the one-variable Hankel transform applied to the Ifunction,a special function in mathematical analysis. By transforming the governing equations into theHankel domain, we simplify the problem significantly, reducing it to a manageable form. The transformedequations are solved analytically, leading to explicit solutions for the axisymmetric Dirichlet potential.The efficiency and accuracy of the proposed method are demonstrated through comprehensivenumerical simulations and comparisons with existing solutions. Using the Hankel transform of an Ifunctionin one variable, we have solved the famous Axisymmetric Dirichlet problem for a half-space inthis study. When dealing with cylindrical coordinates and boundary value issues, the Hankel transform isa powerful tool. We have solved the Axisymmetric Dirichlet problem in a half space defined by thefollowing equations