Telegraph Equation Research Articles

High resolution numerical algorithm of order (2, 4, 4) for 2D quasi-linear hyperbolic equations based on half-step discretization on an unequal grid

This study suggests a novel three-level implicit half-step scheme of order two and four in time and space direction respectively, utilizing an unequal grid for the approximation of 2D quasi-linear hyperbolic partial differential equations on an unequal grid. The stability analysis of the model Telegraphic equation for unequal grid has been discussed and it has been demonstrated that the presented method is unconditionally stable for the Telegraphic equation. The alternating direction implicit (ADI) techniques are discussed for linear difference equations on a dissimilar domain. The developed method investigated several well-known physical problems on an unequal mesh to demonstrate its accuracy and efficiency.

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

AbstractThe ordinary diffusion models in continuous media are derived from basic principles: Conservation of energy and momentum. The classical telegraph equation is one of the earliest and most commonly used equation to describe the transmission of electromagnetic waves. In fractal and disordered media, to capture the complex diffusion feature, we consider rather fractional models. Indeed, the mean square displacement in anomalous media is no longer proportional to the time but rather proportional to a power of time whose exponent may be between zero and one or between one and two. This is the case for the present fractional telegraph equation in the presence of non‐negligible voltage wave. Here, the well‐posedness and stability of a telegraph problem with two time‐fractional derivatives is discussed. In addition to being a noninteger problem, the equation in the model is subject to a nonlinear source as well as a nonlinear lower order fractional derivative term. First, the well‐posedness in an appropriate underlying space is established by the use of resolvent operators. Then, it is proved that, despite the presence of these nonlinearities, solutions are Mittag–Leffler stable. This confirms (and extends) the fact that the lower order term plays the role of a damper in the fractional case. To this end, we combine the energy method with some properties of the Caputo fractional derivative.

Analytical Solution of Telegraph Equations by Variational Iteration Method

In this paper, the variational iteration method (VIM) is utilized for the solution of telegraph equations, which emerge in the propagation of electrical signals along a broadcast line. The suggested method is progressively effective and simpler to deal with as contrast with the decomposition method. Numerical outcomes demonstrate the effectiveness and exactness of the proposed VIM.

A Novel Solution Approach for Time-Fractional Hyperbolic Telegraph Differential Equation with Caputo Time Differentiation

In the current analysis, a specific efficient and applicable novel solution approach, based on a fractional power series technique and Laplace transform operator, is considered to predict certain accurate approximate solutions (ASs) for a time-fractional hyperbolic telegraph equation by aid of time-fractional derivatives in a Caputo sense. The solutions are obtained in a fractional Maclurian series formula by solving the original problem in the Laplace space aided by a limit concept having fewer small iterations than the classical fractional power series technique. To confirm applicability and feasibility of the proposed approach, three appropriate initial value problems are considered. Consequently, some simulations of gained outcomes are numerically and graphically implemented to support the effect of the fractional-order parameter on the geometric behavior of the obtained solutions. In addition, graphical representations are also fulfilled to verify the convergence analysis of the fractional series solutions of the classical solution. The proposed technique is therefore proposed to be a straightforward, accurate and powerful approach for handling varied time-fractional models in various physical phenomena.

Dynamic characteristics of a hyperbolic reaction–diffusion predator–prey system with self‐diffusion and nonidentical inertia

Hyperbolic reaction–diffusion (HRD) systems have emerged as a better descriptor of the macroscopic spatial interaction models used for studying pattern formation in chemical and biological systems. In contrast to the parabolic reaction–diffusion (PRD) models, the spatial disturbances in an HRD system travel through space with finite velocity. This paper considers the simplest two‐species HRD model with different inertia based on the telegraph equation. The underlying reaction terms are considered as a predator–prey interaction with type III response function and prey refuge. We prescribe the analytical conditions for the existence of diffusion‐driven instabilities of the considered system. Simulation results further verify the theoretical results. A connection between the refuge parameter and inertial time is discussed in generating different spatiotemporal patterns. Extension of the two‐species PRD system to HRD system with diagonal diffusion matrix causes a diffusion‐driven wave instability, which is never possible for its parabolic counterpart. Furthermore, the patterns under pure wave instability are dependent on the initial population density.

Semi-analytical approximation of time-fractional telegraph equation via natural transform in Caputo derivative

Abstract In the present research study, time-fractional hyperbolic telegraph equations are solved iteratively using natural transform in one, two, and three dimensions. The fractional derivative is considered in the Caputo sense. These equations serve as a model for the wave theory process of signal processing and transmission of electric impulses. To evaluate the validity and effectiveness of the suggested strategy, a graphical comparison of approximated and exact findings is performed. Convergence analysis of the approximations utilising L ∞ {L}_{\infty } has been done using tables. The suggested approach may successfully and without errors solve a wide variety of ordinary differential equations, partial differential equations (PDEs), fractional PDEs, and fractional hyperbolic telegraph equations.

Recent Developments of Numerical Methods for Analyzing Telegraph Equations

Recent Developments of Numerical Methods for Analyzing Telegraph Equations

An Error Analysis of Implicit Finite Difference Method with Mamadu-Njoseh Basis Functions for Time Fractional Telegraph Equation

In this paper, we proposed and analyzed the error estimate of an implicit finite difference method with Mamadu-Njoseh as basis functions for time fractional telegraph equation. To enhance the efficacy of the method we first transform the Caputo type fractional derivative into Riemann-Liouville derivatives. The error analysis of the method is stated and proven. Also, the optimal results for scalars unknown in norm were derived for the two-dimensional case. Numerical illustrations are presented to test the reliability of the analytical and computed results. The resulting numerical evidence shows that the proposed method convergences more rapidly than the standard finite difference method. MAPLE 18 is used for all mathematical procedures in this paper.

From Semi-Markov Random Evolutions to Scattering Transport and Superdiffusion

We here study random evolutions on Banach spaces, driven by a class of semi-Markov processes. The expectation (in the sense of Bochner) of such evolutions is shown to solve some abstract Cauchy problems. Further, the abstract telegraph (damped wave) equation is generalized to the case of semi-Markov perturbations. A special attention is devoted to semi-Markov models of scattering transport processes which can be represented through these evolutions. In particular, we consider random flights with infinite mean flight times which turn out to be governed by a semi-Markov generalization of a linear Boltzmann equation; their scaling limit is proved to converge to superdiffusive transport processes.

An iterative approach using Sawi transform for fractional telegraph equation in diversified dimensions

Abstract In the present study, 1D, 2D, and 3D fractional hyperbolic telegraph equations in Caputo sense have been solved using an iterative method using Sawi transform. These equations serve as a model for signal analysis of electrical impulse transmission and propagation. Along with a table of Sawi transform of some popular functions, some helpful results on Sawi transform are provided. To demonstrate the effectiveness of the suggested method, five examples in 1D, one example in 2D, and one example in 3D are solved using the proposed scheme. Error analysis comparing approximate and exact solutions using graphs and tables has been provided. The proposed scheme is robust, effective, and easy to implement and can be implemented on variety of fractional partial differential equations to obtain precise series approximations.

Application of the Double Fuzzy Sawi Transform for Solving a Telegraph Equation

The main purpose of this study is to introduce a new double fuzzy transform called the double fuzzy Sawi transform. A proof of some basic properties of the single fuzzy Sawi transform and the double fuzzy Sawi transform are provided. These new results are implemented to obtain the exact solution of a non-homogeneous linear fuzzy telegraph equation under a generalized Hukuhara partial differentiability. In addition, by using the symmetric triangular fuzzy numbers, numerical examples are given to demonstrate the validity and superiority of the double fuzzy Sawi transform in solving the fuzzy linear telegraph equation.

Application of the Homotopy Perturbation Method for Differential Equations

In this paper and in the first part of it, homotopy perturbation method is applied to solve second order differential equation with non-constant coefficients. The method yields solutions in convergent series forms with easily computable terms (the convergence of this series is demonstrated in this paper). The result shows that this method is very convenient and can be applied to large class of problems. As for the second part, we found a solution of Telegraph equation using the Laplace transform and Stehfest algorithm method. Next, we used method of Homotopy perturbation. Finally, we give some examples for illustration.

A comparative study for the numerical approximation of 1D and 2D hyperbolic telegraph equations with UAT and UAH tension B-spline DQM

Abstract Two numerical regimes for the one- and two-dimensional hyperbolic telegraph equations are contrasted in this article. The first implemented regime is uniform algebraic trigonometric tension B-spline DQM, while the second implemented regime is uniform algebraic hyperbolic tension B-spline DQM. The resulting system of ODEs is solved by the SSP RK43 method after the aforementioned equations are spatially discretized. To assess the success of chosen tactics, a comparison of errors is shown. The graphs can be seen, and it is asserted that the precise and numerical results are in agreement with one another. Analyses of convergence and stability are also given. It should be highlighted that there is a dearth of study on 1D and 2D hyperbolic telegraph equations. This aim of this study is to efficiently create results with fewer mistakes. These techniques would surely be useful for other higher-order nonlinear complex natured partial differential equations, including fractional equations, integro equations, and partial-integro equations.

Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

New exact solutions to space–time fractional telegraph equation with conformable derivative

In this paper, we extract variety of new exact traveling wave solutions of space–time fractional nonlinear Telegraph equation for transmission lines by using improved generalized Riccati equation mapping (IGREM) method. The aforementioned equation has been solved for the first time using conformable fractional derivative. The nonlinear Telegraph equation plays crucial role in the mathematical study of voltage and current on an electrical transmission line. In an effort to motivate the researchers, we have used very efficacious technique called IGREM method to find a variety of soliton solutions, in which appropriate conformable fractional wave transformation converts nonlinear partial fractional differential equation into nonlinear ordinary equation which is then solved to find the exact solutions of equation. Hence, we succeed in constructing rich solutions that include periodic wave, singular periodic wave solutions, dark, combined dark–bright and rational solutions. To investigate wave profiles of various solutions, 2D and 3D graphs have been explained. The originality of this work has been proved by comparing the obtained results with previously present results in literature. This method has substantial advantage over other methods that it can be used to solve higher order nonlinear PDEs, and this method allows to solve these equations more conveniently with the aid of mathematical software.

Telegraph equation in polar coordinates: Unbounded domain with moving time-harmonic source

The telegraph equation with moving time-harmonic source with angular frequency ω is considered in polar coordinates (r,φ). Two problems are studied: the source moving on a straight line with constant velocity v and the source traveling on a circumference of a circle of radius R with a constant orbital frequency Ω. The solutions are obtained using the integral transforms technique. Two limiting cases of the telegraph equation are also analysed: the Fourier heat conduction equation and the linear wave equation. The singularity of the solution to the wave equation at a point r=R+vt, φ=0 in the case of the source moving on a straight line is described. The relationship between the orbital frequency Ω and the polar coordinates at the wave front has been analyzed. The results of numerical simulation are presented graphically for a wide spectrum of nondimensional parameters.

A convergent exponential B-spline collocation method for a time-fractional telegraph equation

A convergent exponential B-spline collocation method for a time-fractional telegraph equation

On the solution of generalized time-fractional telegraphic equation

In this article, we have introduced a nonlinear extension of the generalized time-fractional telegraph equation (TFTE). Further, an analytical solution to this equation has been obtained by using the Laplace homotopy perturbation method (LHPM). Some known results are also discussed as special cases which supports the strength and viability of the work.

Influence of the catalyst layer thickness on the determination of the OER activity of Fe3O4@CoFe2O4 core-shell nanoparticles

Transition metal oxides-based catalytic layers often present a complex 3D porous architecture affecting the evaluation of their intrinsic electrocatalytic activity. In this work the oxygen evolution reaction activity of core-shell Fe3O4@CoFe2O4 nanoparticles combining a conductive magnetite core and a catalytically active cobalt ferrite shell was studied at different loading and thickness of the catalytic layer. It was observed that their apparent activity is decreasing and that the Tafel slopes are becoming convex when the loading increases. The activity decay could be attributed to the significant resistance to charge transport in the thick porous catalyst layer. This resistance could be estimated by fitting the electrochemical impedance spectra using the transmission line model. The influence of the layer thickness on the experimental current-potential curves and on their Tafel slopes could be simulated using a simple model based on the Telegrapher's equations. It is concluded that in order to measure accurately the activity and Tafel slopes of an electrocatalyst, thin layers must be used, notably for catalyst layers that are not highly conductive.

On the Countering of Free Vibrations by Forcing: Part II—Damped Oscillations and Decaying Forcing

The present two-part paper concerns the active vibration suppression for the simplest damped continuous system, namely the transverse oscillations of an elastic string, with constant tension and mass density per unit length and friction force proportional to the velocity, described by the telegraph or wave-diffusion equation, in two complementary parts. The initial part I considers non-resonant and resonant forcing, by concentrated point forces or continuous force distributions independent of time, with phase shift between the forced and free oscillations, in the absence of damping, in which case the forced telegraph equation reduces to the forced classical wave equation. The present and final part II uses the forced wave-diffusion equation to model the effect of damping, both as amplitude decay and phase shift in time, for non-resonant and resonant forcing by a single point force, with constant magnitude or magnitude decaying exponentially in time at an arbitrary rate. Assuming a finite elastic string fixed at both ends, the free oscillations are (i) sinusoidal modes in space-time with exponential decay in time due to damping. The non-resonant forced oscillations at an applied frequency distinct from a natural frequency are also (ii) sinusoidal in space-time, with constant amplitude and a phase shift such that the work of the applied force balances the dissipation. For resonant forcing at an applied frequency equal to a natural frequency, the sinusoidal oscillations in space-time have (iii) a constant amplitude and a phase shift of π/2. In both cases, the (ii) non-resonant or (iii) resonant forcing dominates the decaying free oscillations after some time. Even by optimizing the forcing to minimize the total energy of oscillation, it remains below the energy of the free oscillation alone, but only for a short time—generally a fraction of the period. A more effective method of countering the damped free oscillations is to use forcing with amplitude decaying exponentially in time; by suitable choice of the forcing decay relative to the free damping, the total energy of oscillation over all time can be reduced to no more than 1/16th of the energy of the free oscillation.