Telegraph Equation Research Articles

Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem.

Dynamic water potential waves in unsaturated soils

Quantitative descriptions of the temporal and spatial variations of volumetric water content and total water potential in unsaturated soils have for more than nine decades focused on diffusive mechanisms as represented by the Richards equation (Richards, 1931). However, recent laboratory studies (Lo et al., 2017) have revealed short-time transient behavior of water in non-deforming unsaturated soils that is distinctly oscillatory, not diffusive. In this paper, we establish a theoretical framework to generalize the Richards equation in a systematic way to account for non-diffusive mechanisms giving rise to oscillatory behavior in the water content and total water potential. A partial differential equation is developed, with the total water potential as the dependent variable, based on the coupling of mass and linear momentum balance within the continuum theory of mixtures. This new differential equation becomes equivalent to the Richards equation when non-diffusive mechanisms can be neglected, but when this condition is not met, the governing equation describes damped propagating wave behavior of the total water potential. This dynamic wave differs from a poroelastic wave in respect to solid framework motions and wave speed, as well as time and length scales. For a rigid, homogeneous unsaturated soil, the governing equation reduces to the telegraph equation, which can be solved analytically. An inherent feature of our approach is the appearance of natural time and length scales characteristic of total water potential waves. Consideration of these natural scales for three representative soils of differing texture helps to prescribe more precisely the domain of applicability of the Richards equation.

General Solution of Telegraph Equation Using Aboodh Transform

This paper describes using the Aboodh integral transform in solving one-dimensional second-order hyperbolic telegraph equations with satisfying initial conditions by solving some telegraph equations with specific initial conditions and then generalizing the solution to the telegraph equation using the Aboodh integral transform.

Efficient numerical algorithms of time fractional telegraph‐type equations involving Hadamard derivatives

In this paper, we study the finite difference/iterative method for the fractional telegraph equation with Hadamard derivatives. The model is first transformed into an equivalent fractional integro‐differential equation, then the technic of exponential type meshes is adopted. The resulting discrete coefficients of Hadamard fractional integral own several graceful properties, which are crucial in numerical analysis. To conquer the difficulty caused by weak regularity, we also propose iterative schemes. The error estimate is equal to , where is the order of fractional derivative and is the number of iterations. All the theoretical results are confirmed by numerical experiments.

Real-Time Fault Location Using the Retardation Method

A new method for short-circuit fault location is proposed. The method is based on instantaneous signal measurement and its first and second derivatives, which are the novel elements of the current approach. The derivatives allow associating a precise time stamp to the occurrence of the fault. Due to retardation phenomena, the difference between the times in which a signal is registered in two detectors can be used to locate the fault. We offer several mathematical models to describe the fault. Although a description of faults in terms of a lumped circuit is useful for elucidating the methods for detecting the fault, this description will not suffice to describe the fault signal propagation; hence, a distributed models is needed, which is given in terms of the telegraph equations. Those equations were used to derive a transmission line transfer function, and an exact analytical description of the fault signal propagating in the transmission line was obtained. The analytical solution was verified both by numerical simulations and experimentally.

A note on “Analytical solution for the time-fractional telegraph equation by the method of separating variables”

Chen et al. (2008) investigated the analytical solution of the time-fractional telegraph equation with three kinds of nonhomogeneous boundary conditions by the method of separating variables. However, for the Sturm-Liouville eigenvalue problem derived under the Robin boundary conditions, the eigenvalues and eigenfunctions provided by the authors have some defects. In this note, we mainly analyze and modify the eigenvalues and eigenfunctions in detail and obtain the new analytical solution. Numerical experiments show that our new analytical solution is valid.

An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals

An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals

Direct computation of modal parameters for musical wind instruments

Parameters of modal expressions for the input impedance of wind instruments can be computed from the Telegrapher equations with radiating boundary conditions at the bell. One dimensional finite elements are used for spatial discretization. Provided that the models can be put under a specified form, the modal parameters can be computed directly by solving a generalized eigenvalue problem. Viscothermal effects can also be accounted for, as well as open or closed toneholes. Modal shapes can be visualized along the instrument. Parameters of modal expressions for the input admittance of flute-like instruments can also be computed.

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

Dynamic peculiarities of interphase boundaries and domain walls for non-spin domains

The analysis of magnetization oscillations (de Haas – van Alphen effect; dHvA effect) and features of the Fermi surface shape in three (3D)- and two-dimensional (2D) metals shows that the dynamic phenomena — the dynamics of domain walls and interphase boundaries during diamagnetic phase transitions — are described by a new nonlinear partial differential equation. The equation is a result of the inclusion of the case of multiple extremal cross sections of the Fermi surface in these metals. Our consideration indicates the possibility of the appearance of metastable non-spin domains (Condon domains) and first-order phase transitions to the ordered phase in the regime of dHvA oscillations for the two-frequency case. Sine–Gordon, Klein–Gordon, double sine–Gordon, time-dependent Ginzburg–Landau equations, and the telegraph equation are limiting cases of the derived equation. We show that particular moving kink-soliton solutions of the equation describe traveling wave fronts being moving domain walls and interphase boundaries in the Condon domain phase.

Two‐dimensional collocation method for generalized partial integro‐differential equations of fractional order with applications

In this paper, we present a new generalized model of partial integro‐differential equation of fractional order (PIDEFO). The new model of the PIDEFO is defined in terms of the operator presented recently. The operator takes form of the Riemann‐Liouville and Caputo derivatives in special case. The generalized model of the PIDEFO reduces to (1) linear space‐time fractional reaction‐diffusion equation (FRDE), (2) space‐time fractional diffusion equations (STFDE), and (3) time fractional telegraph equations (TFTE) in particular cases. The considered model is solved with collocation method using Jacobi polynomials. For theoretical accuracy, we prove that the Jacobi approximation converges to the exact solution uniformly. Two test examples are considered to verify the accuracy of the proposed method. Further, we present two applications of the proposed model and solve it using the presented method. The obtained numerical results are compared with the existing results. Finally, the numerical stability of the presented method is analyzed under the effect of the random noise in the force term.

Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential

Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential

Fractional boundary value problems

We study some functionals associated with a process driven by a fractional boundary value problem (FBVP for short). By FBVP we mean a Cauchy problem with boundary condition written in terms of a fractional equation, that is an equation involving time-fractional derivative in the sense of Caputo. We focus on lifetimes and additive functionals characterizing the boundary conditions. We show that the corresponding additive functionals are related to the fractional telegraph equations. Moreover, the fractional order of the derivative gives a unified condition including the elastic and the sticky cases among the others.

Haar Wavelets Method for Time Fractional Riesz Space Telegraph Equation with Separable Solution

In this paper, time fractional Riesz space telegraph partial differential equation is proposed. By applying the variable separation condition, the main telegraph equation, which consists of two variables, is reduced to an ordinary differential equation of single variable. This simplifies the problem computationally. Then the well-known Haar wavelets method is developed to derive the approximate solution of the reduced equation which includes low cost and fast calculations. The error bounds for function approximation are established. To illustrate the reliability and capability of the method, some examples are provided. The results show that the proposed algorithm is very simple and effective.

Superdiffusion in self-reinforcing run-and-tumble model with rests.

This paper introduces a run-and-tumble model with self-reinforcing directionality and rests. We derive a single governing hyperbolic partial differential equationfor the probability density of random-walk position, from which we obtain the second moment in the long-time limit. We find the criteria for the transition between superdiffusion and diffusion caused by the addition of a rest state. The emergence of superdiffusion depends on both the parameter representing the strength of self-reinforcement and the ratio between mean running and resting times. The mean running time must be at least 2/3 of the mean resting time for superdiffusion to be possible. Monte Carlo simulations validate this theoretical result. This work demonstrates the possibility of extending the telegrapher's (or Cattaneo) equationby adding self-reinforcing directionality so that superdiffusion occurs even when rests are introduced.

Full discretization to an hyperbolic equation with nonlocal coefficient

We present full discretization of the telegraph equation with nonlocal coeffecient using Rothe-nite element method. For solving the equation numerically we use the Newton Raphson method, but the nonlocal term causes diffeculties because the Jacobien matrix is full. To remedy these diffeculties we apply the technique used by Sudhakar [4]. The optimal a priori error estimates for both semi discrete and fully discrete schemes are derived in V and H1 and a numerical experiment is described to support our theoretical result.

Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

Localization of plane waves in the stochastic telegrapher's equation.

From the exact solution of the stochastic telegrapher's equation, Fourier plane-wave-like modes are introduced. Then the time evolution of the plane-wave modes are analyzed when the absorption of energy in the telegrapher's equation has strong time fluctuations. We demonstrate that fluctuations in the loss of energy introduce a localized gap with a size that depends on the correlation timescale of the fluctuations. We prove that for a large time correlation the gap is strongly reduced, which means that there is delocalization in the plane-wave modes with respect to the plane waves in the ordinary telegrapher's equation. This result is of relevance in the study of the transport of electromagnetic waves in a conducting medium, and sheds light on the functional role of the fluctuations in the loss of energy in the telegrapher's dynamics.

New results of positive doubly periodic solutions to telegraph equations

<abstract><p>The paper is devoted to obtain new results of positive doubly periodic solutions to telegraph equations. One of the interesting features in our proof is that we give a new attempt to solve telegraph equation by using the theory of Hilbert's metric. Then we apply the eigenvalue theory to analyze the existence, multiplicity, nonexistence and asymptotic behavior of positive doubly periodic solutions. We also study a corresponding eigenvalue problem in a more general case.</p></abstract>

Modeling Transmission Lines Using a Hybrid Knowledge-Based and Data-Driven Approach

This article presents a novel hybrid knowledge-based and data-driven scheme to determine the governing partial differential equation (PDE) for the transmission line (TL). The two-dimensional (2-D) current and voltage distributions along the TL are used as the input data. The hypothetical functions, including the candidate terms that may appear in the telegraphic equations, are built based on prior knowledge of the TL system of interest. Through the spatial and temporal derivatives performed on 2-D current and voltage data, the governing PDEs of TLs are solely represented by the linear algebraic equations. The ridge regression is employed to ascertain the actual PDEs of TLs via extracting the active terms from hypothetical functions. The accuracy and effectiveness of this approach are demonstrated through three benchmarked examples, including the lossy, nonuniform, and nonlinear TLs. The results verify that the proposed scheme can inverse the per-unit-length (p.-u.-l.) parameters and identify the governing PDEs. Our work offers a helpful technique to establish connections between observation and the theoretical TL model.