Two-dimensional Telegraph Equation Research Articles

Modifications of Schumann resonance spectra as an estimate of causative earthquake magnitude: The model treatment

Using a numerical model of the ELF radio wave scattering by seismogenic localized non-uniformity, we evaluate the earthquake (EQ) magnitude by using modifications in the power spectra of global electromagnetic (Schumann) resonance. The ionosphere non-uniformity is introduced as a disturbance in the vertical profile of atmosphere conductivity above the earthquake. Three types are considered of disturbance: the WHOLE, BOTTOM and TOP models, each depending on the EQ magnitude. The disturbance is axially symmetric; it varies along the radius according to the Gauss law with the scale corresponding to the EQ magnitude. The electromagnetic problem is solved by using the complex characteristic electric and magnetic heights relevant to conductivity profiles. These heights are found by solving the Riccati equation and are substituded into the two-dimensional telegraph equation (2DTE) for computing the vertical electric and two orthogonal magnetic fields in the Earth–ionosphere cavity. To avoid an impact of the field source coordinates, we apply the source model of independent lightning strokes uniformly distributed over the globe. The ionosphere non-uniformity is located above the North Pole, and the observer is positioned at the point of 63° N and 0° E, i.e., at 3 Mm distance from the disturbance. Comparison of power spectra in the regular and non-uniform cavities allowed us to single out their modifications at particular frequencies. Finally, equations are obtained connecting the EQ magnitude and the spectral modifications averaged over the frequency band covering four Schumann resonance modes. Computations indicate that reflections of radio waves from the localized seismogenic non-uniformity provide the noticeable impact when the EQ magnitude exceeds the M = 7 threshold. The BOTTOM model provides the highest impact on the Schumann resonance spectra, while the influence of the TOP model is the weakest one. The highest seismogenic effect exceeding the 20 dB level is observed in the meridional component HWE (the disturbance is found above the North Pole). Model data agree with available observations, and a further progress might be achieved in the case studies of particular EQs.

A reduced order extrapolating technique of solution coefficient vectors to collocation spectral method for telegraph equation

In this article, we mainly develop a reduced order extrapolating model for the solution coefficient vectors of the classical collocation spectral (CCS) scheme to the two-dimensional (2D) telegraph equation by means of a proper orthogonal decomposition (POD). Therefore, we first present the CCS scheme, offer the existence, stability, and error estimates to the SC solutions, and rewrite the CCS scheme into a matrix-form. We then build a reduced order extrapolating collocation spectral (ROECS) model and analyze the existence and stability as well as errors of the ROECS solutions by some matrix tools. We finally verify the reliability and validity of the ROECS model by means of two sets of numerical simulations for the magnetic field produced by two parallel wires with the same voltage.

Lagrange’s operational approach for the approximate solution of two-dimensional hyperbolic telegraph equation subject to Dirichlet boundary conditions

The key purpose of this study is to present two schemes based on Lagrange polynomials to deal with the numerical solution of second order two-dimensional telegraph equation (TDTE) with the Dirichlet boundary conditions. First, we convert the main equation into partial integro-differential equations (PIDEs) with the help of initial and boundary conditions. The operational matrices of differentiation and integration are then used to transform the PIDEs into algebraic generalized Sylvester equation. We compared the results obtained by the proposed schemes with Bernoulli matrix method and B-spline differential quadrature method which shows that the proposed schemes are accurate for small number of basis functions.

Application of the singular boundary method to the two-dimensional telegraph equation on arbitrary domains

The singular boundary method (SBM) is employed to solve the two-dimensional telegraph equation on arbitrary domains. The Houbolt finite difference method is used to discretize the time derivatives. The original equations are then split into a system of partial differential equations, which is solved using the method of particular solution, in combination with the singular boundary method to obtain the homogeneous solution. Finally, three numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method.

Analysis of a continuous Galerkin method with mesh modification for two-dimensional telegraph equation

In this paper, we investigate the convergence of a space–time continuous Galerkin (STCG) method for two-dimensional (2-D) telegraph equation. The variable time steps and spatial mesh structures are allowed, which are necessary for adaptive computations on unstructured mesh. We demonstrate existence and uniqueness of the numerical solution and give a priori estimate in L∞(L2) norm without space–time mesh restrictions. The analysis method presented here develops and improves the existing theories so that it can be used to study the more general time-dependent partial differential equations (PDEs). Some numerical experiments are provided, and compared with the standard finite element (FE) method to confirm the high efficiency and the high accuracy for the proposed scheme.

Numerical Solution of Space-time Fractional two-dimensional Telegraph Equation by Shifted Legendre Operational Matrices

Numerical Solution of Space-time Fractional two-dimensional Telegraph Equation by Shifted Legendre Operational Matrices

A Crank\u2013Nicolson collocation spectral method for the two-dimensional telegraph equations

In this paper, we mainly focus to study the Crank–Nicolson collocation spectral method for two-dimensional (2D) telegraph equations. For this purpose, we first establish a Crank–Nicolson collocation spectral model based on the Chebyshev polynomials for the 2D telegraph equations. We then discuss the existence, uniqueness, stability, and convergence of the Crank–Nicolson collocation spectral numerical solutions. Finally, we use two sets of numerical examples to verify the validity of theoretical analysis. This implies that the Crank–Nicolson collocation spectral model is very effective for solving the 2D telegraph equations.

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

ABSTRACTIn this paper, a combined compact finite difference method (CCD) together with alternating direction implicit (ADI) scheme is developed for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. The proposed CCD-ADI method is second-order accurate in time variable and sixth-order accurate in space variable. For the linear hyperbolic equation, the CCD-ADI method is shown to be unconditionally stable by using the Von Neumann stability analysis. Numerical results for both linear and nonlinear hyperbolic equations are presented to illustrate the high accuracy of the proposed method.

Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions

The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.

A Galerkin-like scheme to solve two-dimensional telegraph equation using collocation points in initial and boundary conditions

In this study, a Galerkin-like approach is presented in order to numerically solve two-dimensional hyperbolic telegraph equation. The method includes converting the equation to a finite number of algebraic equations by means of taking inner product of a set of three-variable monomials with a vector obtained from the equation in question. The initial and boundary conditions are also taken into account by a suitable utilization of collocation points. The resulting linear system is then solved, yielding a polynomial as the approximate solution. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution, is discussed briefly. The method and the residual correction technique are illustrated with two example problems. In order to evaluate the efficiency of the proposed scheme, the numerical results of both examples are compared with several methods.

Numerical solution of two-dimensional telegraph equations using Haar wavelets

We present here, three- dimensional Haar wavelet based method for solving well known two- dimensional telegraph equation, by approximating higher order mixed derivatives by a series of higher dimensional Haar wavelet functions, which are integrated subsequently to get wavelet approximation of the solution. Numerical examples have been solved to illustrate the accuracy and efficiency of the proposed Haar wavelet method. High accuracy of the results even in the case of a small number of collocation points have been observed.

A meshless technique based on the pseudospectral radial basis functions method for solving the two-dimensional hyperbolic telegraph equation

In this paper, the pseudospectral radial basis functions method is proposed for solving the second-order two-space-dimensional telegraph equation in regular and irregular domain. The proposed numerical method, which is truly meshless, is based on a time stepping procedure to deal with the temporal part of the solution combined with radial basis function differentiation matrices for discretizing the spatial derivatives. Here, we extended the pseudospectral radial basis functions method for two-dimensional hyperbolic telegraph equations in irregular domain. A cross-validation technique is used to optimize the shape parameter for the basis functions. Numerical results and comparisons are given to validate the presented method for solving the two-dimensional telegraph equation on both regular and irregular domains which show that the approximate solutions are in good agreement with the exact solution. The obtained results showed that the proposed method is easy to apply for multidimensional problems and equally applicable to both the regular and irregular domains.

An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(ź2 + h6) accurate, where ź, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.

Random Evolutions Are Driven by the Hyperparabolic Operators

We resolve the long-standing problem of describing the multidimensional random evolutions by means of the telegraph equations. This problem was posed by Mark Kac more than 50 years ago and has become the subject of intense discussion among researchers on whether the multidimensional random flights could be described by the telegraph equations similarly to the one-dimensional case. We give the exhaustive answer to this question and show that the multidimensional random evolutions are driven by the hyperparabolic operators composed of the telegraph operators and their integer powers. The only exception is the 2D random flight whose transition density is the fundamental solution to the two-dimensional telegraph equation. The reason of the exceptionality of the 2D-case is explained. We also show that, under the standard Kac’s condition, the governing hyperparabolic operator turns into the generator of the Brownian motion.

Investigation of the peculiarities of artificial electromagnetic signal propagation at high latitudes in the range 0.1–10 Hz

The results of measurements of the artificial electromagnetic signal in the range 0.1–10 Hz on the Spitsbergen archipelago are presented in this paper. The signal was emitted by a horizontal antenna located on the Kola Peninsula. Resonance spectra structures (RSSs) caused by the influence of the ionospheric Alfven resonator (IAR) were found in the artificial signal. The calculations of the field performed using the method of a two-dimensional telegraph equation for an anisotropic inhomogeneous model of the ionosphere allowed us to explain the main properties of the observed artificial RSS signal. Special attention was focused on the choice of the ionospheric model and calculation of the reflection coefficient for a plane wave normally incident on the ionosphere. The needed profiles of the ionospheric parameters were selected on the basis of the Upper Atmosphere Model (UAM) and IRI2007 model, while the atmospheric parameters were specified on the basis of MSIS-E-90. The advantages of the UAM model compared with the IRI for the description of the experimental results in a high-latitude region are shown in the paper.

Comparison of global storm activity rate calculated from Schumann resonance background components to electric field intensity E0 Z

This work presents the results of a comparison between the global storm activity rate I RS and electric field intensity E 0 Z . The permanent analysis of the I RS may become an important tool for testing Global Electric Circuit models. I RS is determined by a new method that uses the background component of the first 7 Schumann resonances (SR). The rate calculations are based on ELF observations carried out in 2005 and 2006 in the observatory station “Hylaty” of the Jagiellonian University in the Eastern Carpathians (Kułak, A., Zięba, S., Micek, S., Nieckarz, Z., 2003. Solar variations in extremely low frequency propagation parameters: I. A two-dimensional telegraph equation (TDTE) model of ELF propagation and fundamental parameters of Schumann resonances, J. Geophys. Res., 108, 1270, doi:10.1029/2002JA009304). Diurnal runs of the I RS rate were compared with diurnal runs of E 0 Z amplitudes registered at the Earth's surface in the Geophysical Observatory of the Polish Academy of Sciences in Świder (Kubicki, M., 2005. Results of Atmospheric Electricity and Meteorological Observations, S. Kalinowski Geophysical Observatory at Świder 2004, Pub. Inst. Geophysics Polish Academy of Sciences, D-68 (383), Warszawa.). The days with the highest values of the correlation coefficient ( R) between amplitudes of both observed parameters characterizing atmosphere electric activity are shown. The seasonal changes of R, I RS and E 0 Z are also presented.

Solar variations in extremely low frequency propagation parameters: 1. A two‐dimensional telegraph equation (TDTE) model of ELF propagation and fundamental parameters of Schumann resonances

The primary purpose of the research presented is to study solar variations in extremely low frequency (ELF) propagation parameters using Schumann resonance (SR) data from our irregular observations carried out in the East Carpathian mountains. The natural electromagnetic ELF fields in the Earth‐ionosphere cavity (the global resonator) are strongly dependent on its resonance and propagation properties as well as on the global thunderstorm activity. Thus a determination of the global resonator parameters from Schumann resonances needs information about such signal properties, which depend only on physical conditions of the Earth‐ionosphere cavity. To seek such signal properties, we consider the two‐dimensional telegraph equation (TDTE) approach for describing propagation of ELF electromagnetic waves in a two‐dimensional spherical transmission line model of the Earth‐ionosphere cavity. We have set up a model consisting of a network of finite, homogeneous, one‐dimensional transmission lines covering the sphere; with the square root dependence of the attenuation rate from frequency as a sufficient model for the study of the solar influence on global properties of the Earth‐ionosphere cavity. Using this model, we have constructed useful formulae and algorithms connecting the observed parameters of Schumann resonances with the attenuation rate. This enables us to study the solar influence on the attenuation rate from diurnal observations of the NS‐magnetic component of the ELF field measured at one station. As the measurements were carried out during both the minimum and the maximum of the solar cycle 23, we present in the work of Kuak et al. [2003] evidence for a change in the ELF attenuation rate in the Earth‐ionosphere cavity with solar activity.

Solar variations in extremely low frequency propagation parameters: 2. Observations of Schumann resonances and computation of the ELF attenuation parameter

Observations of resonant electromagnetic fields caused by global lightning activity are employed in determining the averaged parameters of the lower ionosphere. Using the two‐dimensional telegraph equation (TDTE) transmission line model described by Kuak et al. [2003], we have computed the attenuation rate of the Earth‐ionosphere waveguide from diurnal observations of the N‐S magnetic component of the ELF field performed irregularly for 6 years in the East Carpathian mountains. As the measurements were carried out during both the minimum and the maximum of the solar cycle 23 we present how solar activity influence the first Schumann resonance frequency and the attenuation rate. The analysis of all the data indicates that the first Schumann resonance frequency increases from 7.75 Hz at solar minimum to about 7.95 Hz at solar maximum while the global mean attenuation rate α at 8 Hz varies from 0.31 dB/Mm at minimum to about 0.26 dB/Mm at maximum.

Formation of a Resonance Structure of the Local Inductance of the Ionosphere at Frequencies 0.1–10 Hz

We solve the problem of determination of the local inductance of the ionosphere, which is a parameter entering a two-dimensional telegraph equation. Day- and nighttime models of the ionosphere at altitudes from 0 to 2000 km for different solar activity are presented. It is shown that the resonance structure of the local inductance is formed by interference of reflections from the upper and lower ionosphere at altitudes above 1000 km and below 300 km, respectively. We demonstrate that reflections from the upper layers of the ionosphere disappear at frequencies higher than 8 Hz.

Solving a Two-Dimensional Telegraph Equation with Anisotropic Parameters

We solve a two-dimensional telegraph equation with anisotropic parameters, which models the propagation of electromagnetic waves in the Earth–ionosphere waveguide, in the frequency range 0.1-30 Hz. The results are generalized to allow for the Earth's sphericity and the horizontal inhomogeneity of the waveguide. It is shown that the resonance character of reflection from the ionosphere at frequencies below 10 Hz becomes pronounced for the horizontal magnetic-field components and for the vertical electric-field component of a horizontal dipole. In the case of low solar activity under nighttime conditions, the oscillations in the frequency dependences of the field components are much more pronounced compared with the case of high solar activity.