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.


Introduction
The telegraph equation is usually applied in signal analysis for electrical signal propagation and transmission reactiondiffusion modeling in several areas of science. It is also employed in the random one-dimensional movement of bugs along a hedge. The communication system plays a vital part in civilization around the world in the current modern era. A wide range of microwave and radiofrequency communication systems continue to benefit from significant industry attention. These technologies use media of communication to convey the signal from one place to another [1,2].
These media can be divided into two classes, namely, guided and unguided. The signal is sent by the coaxial cable or a transmission line in the controlled medium. The controlled medium can carry high-frequency and current waves, whereas electromagnetic waves in unguided media transmit a signal via radiofrequency and microwave channels, part or whole communication path. An antenna is used to send and receive electromagnetic waves. The challenge of efficient telegraphic transmission is addressed with guided transmission media, notably cable transmission media. A cable com-munication channel is a directed transmission system that depicts a physical system that directly transmits data between two or more sites. To maximize the guided communication system, power and signal losses must be determined or projected, as these losses exist in all scenarios. To quantify these losses and eventually secure maximum output, some equations that can compute these losses must be developed. In practice, these equations appear in the fractional order rather than the integer order [3].
Many authors applied different analytical and numerical methods to solve the telegraph equations, such as Laplace transformation and the homotopy perturbation technique [4]. The q-homotopy analysis transformation technique is applied for the analytical result of the time-fractional telegraph equations. The reproducing kernel technique [5] and variational iteration technique are used to find the solution of the telegraph equations. The approximate solutions provided by q-HAM show convergence toward the actual result of the models [6], and the Adomian technique [7] and differential transformation technique are utilized in the analysis of the fractional-order hyperbolic telegraph equations, respectively. The solutions of the variational iteration technique are precisely the same as those of the Adomian decomposition technique, but the variational iteration technique required less computation [8]. The Haar wavelet [9], the generalized differential transform technique [10], the Legendre spectral Galerkin technique [11], and the linear hyperbolic telegraph equations have been solved using Sinc collocation techniques, and these methods have an exponential rate of convergence, making them particularly helpful for approximate partial differential equation solutions [12].
This article is aimed at applying the natural decomposition method (NDM) to solve telegraph equations. Rawashdeh and Maitama [13], for the first time, use natural transformation with the decomposition method for the nonlinear partial differential equations. NDM does not require prescribed assumptions, linearization, discretization, or perturbation and prevents any roundoff error. Recently, NDM is employed in the fractional-order Fisher equation [14] and fractional-order system of Burgers' equation [15].

Basic Definitions
In this section, we reproduce the definitions of Riemann-Liouville (R-L), Caputo, and Caputo-Fabrizio (CF) fractional derivatives for the benefit of the reader.
Definition 2. The Caputo sense fractional derivative of f ðωÞ is defined by for Definition 3. The CF fractional derivative of f ðωÞ is given by where 0 < γ < 1 and BðγÞ is a normalization function, where Bð0Þ = Bð1Þ = 1.
Definition 4. The natural transform of φð tÞ is defined by For t ∈ ð0, ∞Þ, the natural transform of φð tÞ is defined by where Hð tÞ is the Heaviside function.
Definition 8. The natural transform of D γ t φð tÞ by means of Caputo sense is given as Definition 9. The natural transform of D γ t φð tÞ by means of CF is defined as

Methodology
In this section, we present a novel approximate analytical procedure based on the natural transform to the following equation: with the initial condition 2 Journal of Function Spaces where N , L, and hð ξ, tÞ are nonlinear, linear, and source terms, respectively. Now, we employ NT to equation (11) by considering fractional derivatives using two fractional definitions. By taking the natural transform of equation (11) by means of the CF fractional derivative, we obtain where By taking the inverse natural transform using (6), we rewrite (13) as where A t is the Adomian polynomial [2,4]. We assume that equation (11) has the analytical expansion By substituting equations (16) and (17) into (15), we From (18), we get By substituting (19) into (17), we get the NDM CF solution of (11) as

Convergence Analysis
We have presented uniqueness and convergence of the ND M CF in this section.

Theorem 10.
The NTDM CF solution of (11) is unique when
Let m = n + 1; then, where As 0 < δ < 1, we get 1 − δ m−n < 1. Therefore, Since  Journal of Function Spaces φ m is a Cauchy sequence in F; therefore, the series φ m is convergent. ☐ ☐

Results
Example 1. Consider the fractional-order telegraph equation with the initial condition Now, applying the natural transformation to equation (27), we get Using the inverse natural transformation, Using the ADM procedure, we get For j = 0, The subsequent terms are The NDM result for Example 1 is When γ = 1, then the NDM result is

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The exact result is In Figure 1, the first and second graphs show the exact and NDM solutions ofφð ξ, tÞ for Example 1 at γ = 1. Figure 2 shows the different fractional-order graphs of φð ξ, tÞ at γ = 0:8 and γ = 0:6, and Figure 3 shows the fractionalorder graph at γ = 0:40. The graphs show the close relation between the exact and actual solutions with each other. Figure 4 shows the two-and three-dimensional graphs of different fractional-order γ.

Example 2. Consider the fractional-order telegraph equation
with the initial condition Now, applying the natural transformation to (37), we get Using the inverse natural transformation,   Journal of Function Spaces Applying the Adomian decomposition method, we get For j = 0, The subsequent terms are The NDM solution for Example 2 is The exact result is In Figure 5, the first and second graphs show the exact and NDM solutions ofφð ξ, tÞ for Example 2 at γ = 1. Figure 6 shows the different fractional-order graphs of φð ξ, tÞ at γ = 0:8 and γ = 0:6, and Figure 7 shows the fractional-order graph at γ = 0:40. The graphs show the close relation between the exact and actual solutions with each other. Figure 8 shows the twoand three-dimensional graphs of different fractional-order γ.

Example 3. Consider the fractional linear telegraph equation
Using the natural transformation to equation (46), we get Applying the inverse natural transformation, Implementing the Adomian decomposition method, we get

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For j = 0, The subsequent terms are The NDM result for Example 3 is ! " When γ = 1, then the NDM result is The exact result is In Figure 9, the first and second graphs show the exact and NDM solutions ofφð ξ, tÞ for Example 3 at γ = 1. Figure 10 shows the different fractional-order graphs of φð ξ , tÞ at γ = 0:8 and γ = 0:6, and Figure 11 shows the fractional-order graph at γ = 0:40. The graphs show the close relation between the exact and actual solutions with each other. Figure 12 shows the two-and three-dimensional graphs of different fractional-order γ.

Example 4. Consider the fractional linear telegraph equation
with the initial condition

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Taking the natural transformation of equation (56), Using the inverse natural transformation, Applying the Adomian decomposition method, we get For j = 0, The subsequent terms are

12
Journal of Function Spaces The NDM result for Example 4 is The exact result is φ ξ, ζ, χ, t = e −2 t sinh ξ sinh ζ sinh χ ð Þ: ð64Þ In Figure 13, the first and second graphs show the exact and NDM solutions ofφð ξ, tÞ for Example 4 at γ = 1. Figure 14 shows the different fractional-order graphs of φð ξ , tÞ at γ = 0:8 and γ = 0:6. The graphs show the close relation between the exact and actual solutions with each other. Figure 15 shows the three-dimensional graphs of different fractional-order γ.

Conclusions
In this article, we have investigated the telegraph equations through natural transformation with the Caputo-Fabrizio derivative. It is also shown that the fractional-order results were convergent to the actual result in the examples, as the fractional order approached the integer order. The implementation of the natural decomposition method in the illustrative problems has also confirmed that the fractional-order mathematical models can analyze any experimental data in a better manner compared to the integer-order models. Furthermore, by using different fractional orders, we could find a way to create appropriate mathematical models for any empirical data and thus understand practical implications. The natural decomposition method is simple in its principles; also, the natural decomposition method effectively solves linear and nonlinear fractional differential equations. It can be proved a promising technique for a large variety of such equations arising in mathematical physics. In the future, the natural transform decomposition technique modified with the help of different fractional operators such as Atangana-Baleanu and Yang-Abdel-Cattani operators is the most reliable method for solving different fractional-order linear and nonlinear partial differential equations.

Data Availability
The numerical data used to support the findings of this study are included within the article.