Numerical Analysis of the Fractional-Order Telegraph Equations

This paper studied the fractional-order telegraph equations via the natural transform decomposition method with nonsingular kernel derivatives. The fractional result considered in the Caputo-Fabrizio derivative is Caputo sense. Currently, the communication system plays a vital role in a global society. High-frequency telecommunications continuously receive significant attention in the industry due to a slew of radiofrequency and microwave communication networks. These technologies use transmission media to move information-carrying signals from one location to another. We used natural transformation on fractional telegraph equations followed by inverse natural transformation to achieve the solution of the equation. To validate the technique, we have considered a few problems and compared them with the exact solutions.

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An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations

Mathematics ◽

10.3390/math7050426 ◽

2019 ◽

Vol 7 (5) ◽

pp. 426 ◽

Cited By ~ 6

Author(s):

Hassan Khan ◽

Rasool Shah ◽

Poom Kumam ◽

Dumitru Baleanu ◽

Muhammad Arif

Keyword(s):

Fractional Order ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Analytical Techniques ◽

Telegraph Equation ◽

High Rate ◽

Analytical Technique ◽

Decomposition Procedure ◽

Adomian Decomposition ◽

Telegraph Equations

In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation.

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A New Computational Approach for Solving Fractional Order Telegraph Equations

Journal of Scientific Research ◽

10.3329/jsr.v13i3.50659 ◽

2021 ◽

Vol 13 (3) ◽

pp. 715-732

Author(s):

A. Devi ◽

M. Jakhar

Keyword(s):

Fractional Order ◽

Decomposition Method ◽

Series Solution ◽

Adomian Decomposition Method ◽

Approximate Solutions ◽

Computational Approach ◽

Order Problem ◽

Adomian Decomposition ◽

Telegraph Equations ◽

Derivatives Of

In this work, a modified decomposition method namely Sumudu-Adomian Decomposition Method (SADM) is implemented to find the exact and approximate solutions of fractional order telegraph equations. The derivatives of fractional-order are expressed in terms of caputo operator. Some numerical examples are illustrated to examine the efficiency of the proposed technique. Solutions of fractional order telegraph equations are obtained in the form of a series solution. It is observed that the solutions of fractional order telegraph equations converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested method.

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Numerical Analysis, Circuit Simulation, and Control Synchronization of Fractional-Order Unified Chaotic System

Mathematics ◽

10.3390/math7111077 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1077 ◽

Cited By ~ 2

Author(s):

Li ◽

Zhang ◽

Yang

Keyword(s):

Numerical Analysis ◽

Fractional Order ◽

Chaotic System ◽

Decomposition Method ◽

Design Theory ◽

Control Method ◽

Circuit Simulation ◽

Adomian Decomposition Method ◽

Unified Chaotic System ◽

Adomian Decomposition

The traditional method of solving fractional chaotic system has the problem of low precision and is computationally cumbersome. In this paper, different fractional-order calculus solutions, the Adams prediction–correction method, the Adomian decomposition method and the improved Adomian decomposition method, are applied to the numerical analysis of the fractional-order unified chaotic system. The result shows that different methods have higher precision, smaller computational complexity, and shorter running time, in which the improved Adomian decomposition method works best. Then, based on the fractional-order chaotic circuit design theory, the circuit diagram of fractional-order unified chaotic system is designed. The result shows that the circuit simulation diagram of fractional-order unified chaotic system is basically consistent with the phase space diagram obtained from the numerical solution of the system, which verifies the existence of the fractional-order unified chaotic system of 0.9-order. Finally, the active control method is used to control and synchronize in the fractional-order unified chaotic system, and the experiment result shows that the method can achieve synchronization in a shorter time and has a better control performance.

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A Decomposition Method for a Fractional-Order Multi-Dimensional Telegraph Equation via the Elzaki Transform

Symmetry ◽

10.3390/sym13010008 ◽

2020 ◽

Vol 13 (1) ◽

pp. 8

Author(s):

Nehad Ali Shah ◽

Ioannis Dassios ◽

Jae Dong Chung

Keyword(s):

Fractional Order ◽

Decomposition Method ◽

Approximate Analytical Solution ◽

Telegraph Equation ◽

Form Solution ◽

Derivative Operator ◽

Analytical Strategy ◽

Fast Convergence Rate ◽

Computational Work ◽

Telegraph Equations

In this article, the Elzaki decomposition method is used to evaluate the solution of fractional-order telegraph equations. The approximate analytical solution is obtained within the Caputo derivative operator. The examples are provided as a solution to illustrate the feasibility of the proposed methodology. The result of the proposed method and the exact solution is shown and analyzed with figures help. The analytical strategy generates the series form solution, with less computational work and a fast convergence rate to the exact solutions. The obtained results have shown a useful and straightforward procedure to analyze the problems in related areas of science and technology.

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Analytical Solution of Fractional-Order Hyperbolic Telegraph Equation, Using Natural Transform Decomposition Method

Electronics ◽

10.3390/electronics8091015 ◽

2019 ◽

Vol 8 (9) ◽

pp. 1015 ◽

Cited By ~ 9

Author(s):

Hassan Khan ◽

Rasool Shah ◽

Dumitru Baleanu ◽

Poom Kumam ◽

Muhammad Arif

Keyword(s):

Applied Sciences ◽

Partial Differential Equations ◽

Differential Equations ◽

Fractional Order ◽

Decomposition Method ◽

Telegraph Equation ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Present Technique ◽

Telegraph Equations

In the current paper, fractional-order hyperbolic telegraph equations are considered for analytical solutions, using the decomposition method based on natural transformation. The fractional derivative is defined by the Caputo operator. The present technique is implemented for both fractional- and integer-order equations, showing that the current technique is an accurate analytical instrument for the solution of partial differential equations of fractional-order arising in all branches of applied sciences. For this purpose, several examples related to hyperbolic telegraph models are presented to explain the procedure of the suggested method. It is noted that the procedure of the present technique is simple, straightforward, accurate, and found to be a better mathematical technique to solve non-linear fractional partial differential equations.

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Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

Fractal and Fractional ◽

10.3390/fractalfract5010008 ◽

2021 ◽

Vol 5 (1) ◽

pp. 8

Author(s):

Cundi Han ◽

Yiming Chen ◽

Da-Yan Liu ◽

Driss Boutat

Keyword(s):

Numerical Analysis ◽

Fractional Order ◽

Governing Equation ◽

Numerical Solutions ◽

Bernstein Polynomials ◽

Matrix Product ◽

Algebraic Equations ◽

Rotating Beam ◽

The Matrix ◽

The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

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Theoretical and numerical analysis of novel COVID-19 via fractional order mathematical model

Results in Physics ◽

10.1016/j.rinp.2020.103676 ◽

2021 ◽

Vol 20 ◽

pp. 103676

Author(s):

Amjad Ali ◽

Muhammad Yasin Khan ◽

Muhammad Sinan ◽

F.M. Allehiany ◽

Emad E. Mahmoud ◽

...

Keyword(s):

Mathematical Model ◽

Numerical Analysis ◽

Fractional Order ◽

Theoretical And Numerical Analysis

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A Fractional-Order Element (FOE)-Based Approach to Wireless Power Transmission for Frequency Reduction and Output Power Quality Improvement

Electronics ◽

10.3390/electronics8091029 ◽

2019 ◽

Vol 8 (9) ◽

pp. 1029 ◽

Cited By ~ 3

Author(s):

Guidong Zhang ◽

Zuhong Ou ◽

Lili Qu

Keyword(s):

Fractional Order ◽

High Frequency ◽

Power Efficiency ◽

Power Transmission ◽

Switching Frequency ◽

Wireless Power ◽

Wireless Power Transmission ◽

High Switching Frequency ◽

Power Quality Improvement ◽

Power Inductors

A wireless power transmission (WPT) requires high switching frequency to achieve energy transmission; however, existing switching devices cannot satisfy the requirements of high-frequency switching, and the efficiency of current WPT is too low. Compared with the traditional power inductors and capacitors, fractional-order elements (FOEs) in WPT can realize necessary functions though requiring a lower switching frequency, which leads to a more favorable high-frequency switching performance with a higher efficiency. In this study, a generalized fractional-order WPT (FO-WPT) is established, followed by a comprehensive analysis on its WPT performance and power efficiency. Through extensive simulations of typical FO wireless power domino-resonators (FO-WPDRS), the functionality of the proposed FO-WPT for medium and long-range WPT is demonstrated. The numerical results show that the proposed FOE-based WPT solution has a higher power efficiency and lower switching frequency than conventional methods.

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Improvement of Risk Assessment Using Numerical Analysis for an Offshore Plant Dipole Antenna

Symmetry ◽

10.3390/sym10120681 ◽

2018 ◽

Vol 10 (12) ◽

pp. 681

Author(s):

Yun-Jeong Cho ◽

Kichang Im ◽

Dongkoo Shon ◽

Daehoon Park ◽

Jong-Myon Kim

Keyword(s):

Risk Assessment ◽

Numerical Analysis ◽

High Frequency ◽

Dipole Antenna ◽

Effective Field ◽

Initial Assessment ◽

International Electrotechnical Commission ◽

Test Procedures ◽

High Frequency Structure Simulator ◽

Site Test

This paper proposes a numerical analysis method for improving risk assessment of radio frequency (RF) hazards. To compare the results of conventional code analysis, the values required for dipole antenna risk assessment, which is widely used in offshore plants based on the British standards (BS) guide, are calculated using the proposed numerical analysis. Based on the BS (published document CENELEC technical report (PD CLC/TR) 50427:2004 and international electrotechnical commission (IEC) 60079 for an offshore plant dipole antenna, an initial assessment, a full assessment, and on-site test procedures are performed to determine if there is a potential risk of high-frequency ignition. Alternatively, numerical analysis is performed using the Ansys high frequency structure simulator (HFSS) tool to compare results based on the BS guide. The proposed method computes the effective field strength and power for the antenna without any special consideration of the structure to simplify the calculation. Experimental results show that the proposed numerical analysis outperforms the risk assessment based on the BS guide in accuracy of the evaluation.

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