Solution Of Helmholtz Equation Research Articles

A class of solutions of the n-dimensional generalized Helmholtz equation which describes generalized Weingarten hypersurfaces

In this paper, we introduce the \(n\)-dimensional generalized Helmholtz equation and present explicit solutions to this equation in terms of biharmonic functions, in particular, we get solutions that depend on holomorphic functions. Also, we present explicit radial solutions for this equation and we provide explicit solutions to the \(n\)-dimensional Helmholtz equation. In addition, as an application we introduced two classes of generalized Weingarten hypersurfaces, namely, the RSHGW-hypersurfaces and the RSGW-hypersurfaces, associated with solutions of the \(n\)-dimensional generalized Helmholtz equation and classify the RSHGW-hypersurfaces of rotation. For \(n=2\), we obtain a Weierstrass type representation for these surfaces which depend of three holomorphic functions and we classify the RSHGW-surfaces and the RSGW-surfaces of rotation.

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

The Helmholtz equation plays a crucial role in the study of wave propagation, underwater acoustics, and the behavior of waves in the ocean environment. The Helmholtz equation is also used to describe propagation through ocean waves, such as sound waves or electromagnetic waves. This paper presents the Elzaki transform residual power series method (E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document}T-RPSM) for the analytical treatment of fractional-order Helmholtz equation. To develop this scheme, we combine Elzaki transform (E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document}T) with residual power series method (RPSM). The fractional derivatives are described in Caputo sense. The E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document}T is capable of handling the fractional order and turning the problem into a recurrence form, which is the novelty of our paper. We implement RPSM in such a way that this recurrence relation generates the results in the form of an iterative series. Two numerical applications are considered to demonstrate the efficiency and authenticity of this scheme. The obtained series are determined very quickly and converge to the exact solution only after a few iterations. Graphical plots and absolute error are shown to observe the authenticity of this suggested approach.

Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side

Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side

A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations

In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method (ST-RPSM). This scheme uses a combination of the Shehu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare ST-RPSM results with exact solutions.

Asymptotics of the Helmholtz equation solutions in a two-layer medium with a localized right-hand side

Asymptotics of the Helmholtz equation solutions in a two-layer medium with a localized right-hand side

Helmholtz-equation solution in nonsmooth media by a physics-informed neural network incorporating quadratic terms and a perfectly matching layer condition

Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep-learning method, has achieved successful applications in solving a wide range of partial differential equations (PDEs), and there is still room for improvement on this front. For example, PINN can lead to inaccurate solutions when PDE coefficients are nonsmooth and describe structurally complex media. Thus, we solve the acoustic and visco-acoustic scattered-field (Lippmann-Schwinger) wave equation in the frequency domain with PINN instead of the wave equation to remove the source singularity. We first illustrate that nonsmooth velocity models lead to inaccurate wavefields when no boundary conditions are implemented in the loss function. Then, we add the perfectly matched layer (PML) conditions in the loss function to better couple the real and imaginary parts of the wavefield. Moreover, we design new neurons by replacing the classical affine function with a quadratic function in the argument of the activation function to better capture the nonsmooth features of the wavefields. We find that the PML condition and the quadratic functions improve the results including handling attenuation and discuss the reason for this improvement. We also illustrate that a network trained to predict a wavefield for a specific medium can be used as an initial model of the neural network for predicting other wavefields corresponding to PDE-coefficient alterations and improving the convergence speed accordingly. This pretraining strategy should find applications in iterative full-waveform inversion and time-lag target-oriented imaging when the model perturbation between two consecutive iterations or two consecutive experiments is small.

Isogeometric Solution of Helmholtz Equation with Dirichlet Boundary Condition in Regions with Irregular Boundary: Numerical Experiences

Isogeometric Solution of Helmholtz Equation with Dirichlet Boundary Condition in Regions with Irregular Boundary: Numerical Experiences

Electromagnetic Scattering by Bianisotropic Spheres

Electromagnetic fields in bulk bianisotropic media contain plane waves whose k-vectors can be found using the method of the index of refraction’s operator and belong to the Fresnel wave surfaces that fall into one of the five hyperbolic classes of the Durach et al. taxonomy of bianisotropic media. Linear combinations of vector spherical harmonics can be used as a set of solutions of vector Helmholtz equations in gyrotropic media to develop Mie’s theory of scattering by anisotropic spheres as accomplished by Lin et al. and Li et al. In this study, we introduced electromagnetic orbitals for bianisotropic media as linear combinations of vector spherical harmonics, which represent solutions of Maxwell’s equations in bianisotropic media. Using these bianisotropic orbitals, we developed a theory of the scattering of electromagnetic radiation by bianisotropic spheres with arbitrary effective material parameters and sizes. As a by-product, we obtained a simple expression for the expansion of a vector plane wave over vector spherical harmonics in a more compact form than the frequently used by Sarkar et al. We obtained the polarizability expressions in the Rayleigh limit in agreement with the results of the electrostatic approximation of Lakhtahia and Sihvola.

Parametrization-free locally-conformal perfectly matched layer method for finite element solution of Helmholtz equation

We present a novel Locally-Conformal Perfectly Matched Layer (LCPML) method for mesh truncation in the Finite Element Method (FEM), to solve wave equations derived from Maxwell's equations. The original LCPML method was proposed by Ozgun and Kuzuoglu in 2006 [16] by utilizing the concept of complex elements in which nodal coordinates of elements are replaced by complex coordinates obtained by using a complex coordinate-transformation. In the proposed LCPML method, named as LCPML-log method, a more efficient coordinate-transformation is used together with new FEM formulation and implementation. The proposed method offers the advantage that only a few (or even a single) PML layer is sufficient to get reliable results. In addition, it does not require any PML parameter tuning, and hence, the decaying wave behavior in PML is achieved self-adaptively. The numerical results of some detailed and systematic performance analyses are also presented.

Far-field analytical solutions of the non-homogeneous Helmholtz and wave equations for spatially non-localized sources.

Non-localized impulsive sources are ubiquitous in underwater acoustic applications. However, analytical expressions of their acoustic field are usually not available. In this work, far-field analytical solutions of the non-homogeneous scalar Helmholtz and wave equations are developed for a class of spatially extended impulsive sources. The derived expressions can serve as benchmarks to verify the accuracy of numerical solvers.

Full Waveform Inversion Based on an Asymptotic Solution of Helmholtz Equation

This study considers the full waveform inversion (FWI) method based on the asymptotic solution of the Helmholtz equation. We provide frequency-dependent ray tracing to obtain the wave field used to compute the FWI gradient and calculate the modeled data. With a comparable quality of the inverse problem solution as applied to the standard finite difference approach, the speed of the calculations in the asymptotic method is an order of magnitude higher. A series of numerical experiments demonstrate the approach’s effectiveness in reconstructing the macro velocity structure of complex media for low frequencies.

Numerical simulation of fractional-order two-dimensional Helmholtz equations

<abstract><p>In this paper, we investigate the exact solutions of several fractional-order Helmholtz equations using the homotopy perturbation transform method. We specify sufficient requirements for its convergence and provide error estimations. The homotopy perturbation transform method yields a quickly converging succession of solutions. Solutions for various fractional space derivatives are compared to present approaches and explained using figures. Appropriate parameter selection produces approximations identical to the exact answer. Test examples are provided to demonstrate the proposed approach's precision and competence. The results demonstrate that our system is appealing, user-friendly, dependable, and highly effective.</p></abstract>

On isolated singular solutions of semilinear Helmholtz equation

On isolated singular solutions of semilinear Helmholtz equation

Thin-layer solutions to the Helmholtz equation in three dimensions

This paper gives a brief overview of how boundary layer solutions of Helmholtz equation can reveal the structure of planar layers, caustics and foci in three dimensions. In particular, it describes solutions that are localised near straight lines, smooth and cusped caustics and hyperbolic umbilic points as well as solutions localised near smooth prescribed boundaries.

Advanced efficient iterative methods to the Helmholtz equation

Parallel computing has recently gained widespread acceptance as a means of handling very large computational data. Since iterative methods are appealing for large systems of equations, and they are the prime candidates for implementations on parallel architectures, We presented based on exploration, through virtual technology having 30 cores, in literature solutions of Helmholtz equation is available up to 12 cores by Jacobi method, here we increased the number of cores and virtual machine having 30 cores first time used to find the solution of Helmholtz equation, our findings are encouraging and found that parallel computing by OpenMP implementations is effective on current supercomputing as well as virtual machine platforms and that is an auspicious programming model to use for applications to be run on emerging and future platforms with accelerated nodes.

Numerical Solution of a One-Dimensional Nonlocal Helmholtz Equation by Perfectly Matched Layers

We consider the computation of a nonlocal Helmholtz equation by using Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Particularly for a typical kernel function $\gamma_1(s)=\frac12 e^{-| s|}$, we obtain the Green's function of the nonlocal Helmholtz equation, and use the Green's function to further prove that (i) the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the nonlocal Helmholtz solution itself decays exponentially outside some domain. Based on our theoretical analysis, the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems. Finally, numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.

Application of asymptotic solution of Helmholtz equation in Full Waveform Inversion

This study considers the Full Waveform Inversion (FWI) method based on the asymptotic solution of the Helmholtz equation. We provide frequency-dependent ray tracing to get the wavefield used to compute the FWI gradient and calculate the modelled data. With a comparable quality of the inverse problem solution as applied to the standard finite-difference approach, the speed of calculations in the asymptotic method is an order of magnitude higher. A series of numerical experiments demonstrate the approach's effectiveness in reconstructing the macro velocity structure of complex media for low frequencies.

Exact Solutions of the 3D Fractional Helmholtz Equation by Fractional Differential Transform Method

In this work, we applied the fractional reduced differential transform method (FRDTM) to find the exact solutions of the three-dimensional fractional Helmholtz equation (FHE) and compared our outcomes with the tenth-order approximate solutions for diverse fractional orders. Different values of fractional derivatives are signified explicitly in three dimensions. Outcomes are denoted in the tables for nonfractional exact and approximate solutions and fractional approximate solutions for different fractional orders. Two examples are given to prove the proficiency of the suggested method, to resolve diverse categories of fractional partial differential equations.

Nonparaxial pulse propagation in a planar waveguide with Kerr–like and quintic nonlinearities; computational simulations

This article investigates some novel solitary wave solutions of the cubic-quintic nonlinear Helmholtz equation by implementing two recent analytical (Khater II and generalized exponential methods) techniques. As a result of seeing the nonparaxial effect, this model was developed to show how a pulse propagates in a planar waveguide with Kerr-like and quintic nonlinearities as well as spatial dispersion. When the traditional approximation of a slowly changing envelope fails, this effect takes over. Some novel soliton wave solutions are obtained and formulated in distinct forms such as rational, hyperbolic, and trigonometric forms. The obtained solutions are represented through various figures’ types to show the physical and dynamical behavior of the nonparaxial pulse propagation. The model’s features are demonstrated, such as its intriguing bright, anti-bright, periodic, singular, gray, chirped anti-dark, and dark solitary waves depending upon the nature of nonlinearities. The solutions’ originality is discussed by comparing it with those constructed in recently published articles. All built results are checked their accuracy by putting them back into the original model by employing Mathematica 12.

A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution

In this article, with the help of Legendre wavelets a hybrid approach is presented for the numerical solution of the Helmholtz equation which has a complex solution. The present hybrid approach is based on the Legendre Wavelet Collocation Method (LWCM). Initially, the Helmholtz equation is converted to the coupled equation with suitable transformation, later all the derivatives in the equations including boundary conditions are approximated with the help of Legendre Wavelets. Later, with the help of the Legendre wavelet operational matrix of integration, a system of linear algebraic equations are formed for obtaining the numerical solution of Helmholtz equation. The aforementioned proposed algorithm is very simple and can easily be executed in any computer-oriented language efficiently. To demonstrate the efficiency and performance of the newly established technique, the method is tested on various well-known examples from previous literature including other Legendre wavelet-based methods. The obtained results produce better accuracy and widespread use of the newly developed technique for a range of benchmark problems which have complex solutions.