Solution of Two-Dimensional Helmholtz Equation By Multipole Theory Method

Application of the multipole theory method to the analysis of finlines

“…According to the applied laws of the MT method [9], the inhomogeneous field domain of finline is treated as a combination of three homogeneous regions ⍀ 1 , ⍀ 2 , and ⍀ 3 , as shown in Figure 1. Three inside poles of order 14 are placed at points O 1 , O 2 , and O 3 , respectively.…”

“…Recently, Zheng et al [9] have developed a new approach, the multipole theory (MT) method, for calculating two-dimensional (2D) Helmholtz equation boundary-value problem, and the MT method has been applied to the computation of the cutoff wavenumbers of waveguides with sharp metal edges [10] and the dielectric-loaded waveguides [11]. In this paper, the MT method is applied to the analysis of the cutoff frequencies of the finline, i.e., waveguides filled with dielectric substrate (thin dielectric) and sharp metal edges.…”

“…In this paper, the MT method is applied to the analysis of the cutoff frequencies of the finline, i.e., waveguides filled with dielectric substrate (thin dielectric) and sharp metal edges. The inhomogeneous field domain is treated as a combination of several simple homogeneous field domains satisfying the MT series expansion conditions [9]. To illustrate the application of the MT method, the cutoff frequencies of unilateral and bilateral finlines are given to validate the theory and to demonstrate the degree of its efficiency.…”

An efficient approach for solution of fractional-order Helmholtz equations

“…The equations occur in some phenomena, such as electromagnetic waves in fluids, vibrating lines, plates, walls, acoustics, magnetic fields, nuclear plants and the Lamb equation in geoscience. Consider a two-dimensional non-homogeneous isotropic medium whose velocity is c in euclidean space [11,12]. The wave result is μ(x, y) corresponding to the harmonic origin φ(x, y) vibrating at the specified fixed frequency ω > 0 satisfies the Helmholtz equation for the defined area R:…”

“…For instance, in linear acoustics, φ(x, y) can reflect a disturbance of the reference state pressure (Thompson and Pinsky, 1995 [12]). Conservation equations, which are also transformed into Helmholtz equations, often have various implementations in several physical problems, such as shear viscosity streams or fluids constrained inside thermophysical walls [11][12][13][14]. In recent decades, many of the numerical and analytical techniques have been implemented to solve fractional-order Helmholtz equations, such as He's homotopy perturbation technique [10], Laplace variational iteration technique [15], the reduced differential transform technique [16], the q-homotopy analysis transform technique [17], the spectral technique [17] and He's variational iteration technique [18].…”