The Analysis of Fractional-Order Helmholtz Equations via a Novel Approach

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.


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In general, fractional calculus has been a significant field of applied mathematics in the last several centuries. Good results than classical derivatives are obtained by modeling real phenomena with fractional derivative and fractional integral. In the modeling of certain physical phenomena, some important applications can be traced, visco-elasticity damping, especially signal processing, electronics, genetic algorithms, communication, transport systems, robotics, biology, chemistry, physics and finance. In the field of fractional calculus, several researchers are focusing on several significant discoveries and contributions [13,14]. For most observers and scholars, fractional calculus is an important research field due to its interesting applications, and the investigation of fractional-order partial differential equations (PDEs) has attracted specific attention from various researchers. In light of this, a number of approaches have been used to solve different linear and nonlinear fractional PDEs. For instance, for solving the time-fractional and anomalous mobile-immobile solution transport process, the local meshless approach is used [5][6][7][8]15,16]. The Helmholtz equation or reduced wave equation is an elliptical partial differential equation (PDE) derived directly from the wave model. Helmholtz equation is a PDE that signifies time-independent mechanical growth in the universe. The Helmholtz equation is one of the essential applied mathematics and physics equations. The Helmholtz equation results, which are usually created from the separation of variables, address important science phenomena. Equations occur in a number of phenomena, such as vibrating lines, electromagnetic Waves in fluids, walls, plates, Nuclear plants, acoustics, magnetic fields and the Lamb equation in geoscience. Consider a multi dimensional nonhomogenous isotropic medium whose velocity is c in euclidean space [9,10]. The wave outcome is µ(ξ, ζ) consistent to the harmonic origin φ(ξ, ζ) vibrating at the identified fixed frequency ω > 0 satisfies the Helmholtz equation for the defined field R: Where µ(ξ, ζ) is an appropriately differentiable function at the boundary of R, and φ(ξ, ζ) is a specified 13 function, λ > 0 is a constant value, and √ λ = ω c is a wave number with a wave length of 2π In recent years, fractional-order calculus has emerged as a potential tool in various science and where denotes the gamma function is Γ, ρ > 0, ζ > 0 and 0 < α < 1.
The Caputo generalized fractional-order ρ-Laplace transform derivative of a continuous function f is defined by [19].

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The Mittag-Leffler generalized function is given by Consider the functional equation where the function denote by f , linear and nonlinear function of ϕ show by L and N, respectively. Let us consider solution of (3) is of the form where m = 1, 2, 3 · · · , Now, linear and nonlinear form can be define as [28]: We achieve the result of (3) as Consider the initial value fractional-order equation with Caputo derivative as with initial condition ϕ(ξ, 0) = g(ξ), where N is a nonlinear function. Now on both sides using ρ-Laplace transformation of Eq. (8), we get applying (2), we achieve (N ϕ(ξ, ζ)).
The recurrence relation is given as, where m = 1, 2, 3 · · · , Hence, the result of (8) is of the form,

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Example 1. Consider the fractional Helmholtz equation is given as with ϕ(0, ζ) = ζ and ϕ ξ (0, ζ) = 0. Using on both side ρ-Laplace transformation of Eq.(34), we get from which, Implement on both sides inverse ρ-Laplace transformation of (36), we get Applying technique of new suggested method, we get Hence, the analytical result of Eq. (34) is given in terms of Mittag-Leffler function as: The exact solution of the problem 1 when α = 2, ϕ(ξ, ζ) = ζ cosh ξ. In the same producer the result of ζ-space can be determined through ITM as: with initial condition Thus the solution of the above (31) is obtained as in the case when α = 2, then the result through ITM is (24)  Example 2. Consider the fractional Helmholtz equation is given as with ϕ(0, ζ) = ζ and ϕ ξ (0, ζ) = 0. Using on both sides ρ-Laplace transformation of Eq.(34), we achieve Using on both sides inverse ρ-Laplace transformation of Eq. (36), we get Applying technique of new suggested method, we have . . ..

(29)
Hence, the approximate result of Eq. (34) is given in recursive of Mittag-Leffler function as: The exact solution of the problem 1 when α = 2, In the same producer the result of y-space can be calculated through ITM as: with the initial condition Thus the solution of the above (31) is obtain by in the case when α = 2, then the result through ITM is with ϕ(0, ζ) = ζ and ϕ ξ (0, ζ) = ζ + cosh(ζ). Using on both side ρ-Laplace transform of Eq.(34), we achieve from which, Implement on both side inverse ρ-Laplace transformation of (36), we get Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 March 2021 doi:10.20944/preprints202103.0256.v1 Applying technique of new suggested method, we have . . .

(38)
Hence, the approximate result of (34) is given in recursive of Mittag-Leffler function as: The exact solution of the problem when α = 2, ϕ(ξ, ζ) = ζe ξ + ξ cosh ζ.   then modified it to solve other fractional-order linear and nonlinear partial differential equations.