# Numerical Analysis of the Fractional-Order Telegraph Equations

Authors: Omar Fouad Azhar; Muhammad Naeem; Fatemah Mofarreh; Jeevan Kafle
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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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## Body

## 1. Introduction

The telegraph equation is usually applied in signal analysis for electrical signal propagation and transmission reaction-diffusion 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 communication 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].


## 2. 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 1.
The R-L left-sided fractional integral operator of a functionf∈Cv, v≥−1, is given as
(1)Iγfω=1Γγ∫0ωω−ζγ−1fζdζ,γ>0,ω>0,I0fω=fω.Definition 2.
The Caputo sense fractional derivative offω is defined by
(2)Dωγfω=Im−γDmfω=1m−γ∫ω0ω−ζm−γ−1fmζdζ,for m−1<γ≤m, m∈N, ω>0, f∈Cvm, and v≥−1.Definition 3.
The CF fractional derivative offω is given by
(3)Dωγfω=Bγ1−γ∫0ωexp−γω−ζ1−γDfζdζ,where 0<γ<1 and Bγ is a normalization function, where B0=B1=1.Definition 4.
The natural transform ofφt¯ is defined by
(4)Nφt¯=Us,v=∫−∞∞e−st¯φv,t¯dt¯,s,v∈−∞,∞.
Fort¯∈0,∞, the natural transform of φt¯ is defined by
(5)Nφt¯Ht¯=N+=U+s,v=∫−∞∞e−st¯φv,t¯dt¯,s,v∈0,∞,where Ht¯ is the Heaviside function.Definition 5.
The inverse natural transformation ofUs,v is given by
(6)N−1Us,v=φt¯,∀t¯≥0.Lemma 6 (Linearity property).
If the natural transform ofφ1t¯ is φ1s,v and φ2t¯ is φ2s,v, then
(7)Nc1φ1t¯+c2φ2t¯=c1Nφ1t¯+c2Nφ2t¯=c1φ1s,v+c2φ2s,v,where c1 and c2 are constants.Lemma 7 (Inverse property).
If the inverse natural transform ofφ1s,v and φ2s,v is φ1t¯ and φ2t¯, respectively, then
(8)N−1c1φ1s,v+c2φ2s,v=c1N−1φ1s,v+c2N−1φ2s,v=c1φ1t¯+c2φ2t¯,where c1 and c2 are constants.Definition 8.
The natural transform ofDt¯γφt¯ by means of Caputo sense is given as
(9)NDt¯γ=svγNφt¯−1sφ0.Definition 9.
The natural transform ofDt¯γφt¯ by means of CF is defined as
(10)NDt¯γ=11−γ+γv/sNφt¯−1sφ0.


## 3. Methodology

In this section, we present a novel approximate analytical procedure based on the natural transform to the following equation:(11)Dt¯γφξ¯,t¯=Lφξ¯,t¯+Nφξ¯,t¯+hξ¯,t¯,with the initial condition
(12)φξ¯,0=ϕξ¯,where N, L, and hξ¯,t¯ 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
(13)1pγ,v,sNφξ¯,t¯−ϕξ¯s=NMξ¯,t¯,where
(14)pγ,v,s=1−γ+γvs.By taking the inverse natural transform using (6), we rewrite (13) as
(15)φξ¯,t¯=N−1ϕξ¯s+pγ,v,sNMξ¯,t¯.Nφξ¯,t¯ can be decomposed into
(16)Nφξ¯,t¯=∑i=0∞At¯,where At¯ is the Adomian polynomial [2, 4]. We assume that equation (11) has the analytical expansion
(17)φξ¯,t=∑i=0∞φiξ¯,t¯.By substituting equations (16) and (17) into (15), we obtain
(18)∑i=0∞φiξ¯,t¯=N−1ϕξ¯s+pγ,v,sNhξ¯,t¯+N−1pγ,v,sN∑i=0∞Lφiξ¯,t¯+At¯.From (18), we get
(19)φ0CFξ¯,t¯=N−1ϕξ¯s+pγ,v,sNhξ¯,t¯,φ1CFξ¯,t¯=N−1pγ,v,sNLφ0ξ¯,t¯+A0,⋮φl+1CFξ¯,t¯=N−1pγ,v,sNLulξ¯,t¯+Al,l=1,2,3,⋯.By substituting (19) into (17), we get the NDMCF solution of (11) as
(20)φCFξ¯,t¯=φ0CFξ¯,t¯+φ1CFξ¯,t¯+φ2CFξ¯,t¯+⋯.


## 4. Convergence Analysis

We have presented uniqueness and convergence of theNDMCF in this section.Theorem 10.
TheNTDMCF solution of (11) is unique when 0<δ1+δ21−γ+γt¯<1.Proof.
LetF=CJ,. be the Banach space with the norm ϕt=maxt∈Jϕt, ∀ continuous function on J. Let G:F⟶F be a nonlinear mapping, where
(21)φl+1C=φ0C+N−1pγ,v,sNLφlx¯i,t¯+Nφlx¯i,t¯,l≥0.
Suppose that∣Lφ−Lφ∗∣<δ1∣φ−φ∗∣ and ∣Nφ−Nφ∗∣<δ2∣φ−φ∗∣, where δ1 and δ2 are Lipschitz constants and φ≔φζ,t¯ and φ∗≔φ∗ζ,t are two different function values.
(22)Gφ−Gφ∗≤maxt¯∈JN−1pγ,v,sNLφ−Lφ∗+pγ,v,sNNφ−Nφ∗≤maxt¯∈Jδ1N−1pγ,v,sNφ−φ∗+δ2N−1pγ,v,sNφ−φ∗≤maxt¯∈Jδ1+δ2N−1pγ,v,sNφ−φ∗≤δ1+δ2N−1pγ,v,sNφ−φ∗=δ1+δ21−γ+γt¯φ−φ∗.
Gis a contraction of0<δ1+δ21−γ+γt¯<1. The solution of (11) is unique from the Banach fixed-point theorem.☐☐Theorem 11.
TheNTDMCF solution of (11) is convergent.Proof.
Letφm=∑r=0mφrξ¯,t¯. To prove that φm is a Cauchy sequence in F. Consider
(23)φm−φn=maxt¯∈J∑r=n+1mφr≤maxt¯∈JN−1pγ,v,sN∑r=n+1mLφr−1+Nφr−1=maxt¯∈JN−1pγ,v,sN∑r=n+1m−1Lφr+Nφr≤maxt¯∈JN−1pγ,v,sNLφm−1−Lφn−1+Nφm−1−Nφn−1≤δ1maxt¯∈JN−1pγ,v,sNLφm−1−Lφn−1+δ2maxt¯∈JN−1pγ,v,sNNum−1−Nt¯n−1=δ1+δ21−γ+γt¯φm−1−φn−1,n=1,2,3,⋯.
Letm=n+1; then,
(24)φn+1−φn≤δφn−φn−1≤δ2φn−1φn−2≤⋯≤δnφ1−φ0,where δ=δ1+δ21−γ+γt¯. Similarly, we have
(25)φm−φn≤φn+1−φn+φn+2φn+1+⋯+φm−φm−1,δn+δn+1+⋯+δm−1φ1−φ0≤δn1−δm−n1−δφ1.
As0<δ<1, we get 1−δm−n<1. Therefore,
(26)φm−φn≤δn1−δmaxt¯∈Jφ1.
Sinceφ1<∞,φm−φn⟶0 when n⟶∞. Hence, φm is a Cauchy sequence in F; therefore, the series φm is convergent.☐☐


## 5. Results

Example 1.
Consider the fractional-order telegraph equation(27)∂2γφ∂t¯2γ+2∂γφ∂t¯γ+φ=∂2φ∂ξ¯2,0<γ≤1,t≥0,with the initial condition
(28)φξ¯,0=eξ¯,φt¯ξ¯,0=−2eξ¯.Now, applying the natural transformation to equation (27), we get
(29)N∂2γφ∂t¯2γ=−N2∂γφ∂t¯γ+φ−∂2φ∂ξ¯2.Using the inverse natural transformation,(30)φξ¯,t¯=N−1φξ¯,0s+μs2φt¯ξ¯,0−μ/s+γ1−μ/sμ/s2N2∂γφ∂t¯γ+φ−∂2φ∂ξ¯2.Using the ADM procedure, we get(31)φ0ξ¯,t¯=N−1φξ¯,0s+μs2φt¯ξ¯,0=N−1eξ¯s−2μeξ¯s2,φ0ξ¯,t¯=eξ¯1−2t¯,φj+1=−N−1μ/s+γ1−μ/sμ/s2N2∂γφj∂t¯γ+φj−∂2φj∂ξ¯2,j=0,1,2,⋯.Forj=0,
(32)φ1ξ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφ0∂t¯γ+φ0−∂2φ0∂ξ¯2=2eξ¯t¯2−2γ+γt¯.The subsequent terms are(33)φ2ξ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφ1∂t¯γ+φ1−∂2φ1∂ξ¯2=−4eξ¯t¯¯233−3γ+γt¯,φ3ξ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφ2∂t¯γ+φ2−∂2φ2∂ξ¯2=2eξ¯t¯¯334−4γ+γt¯,⋮The NDM result for Example1 is
(34)φξ¯,t¯=φ0ξ¯,t¯+φ1ξ¯,t¯+φ2ξ¯,t¯+φ3ξ¯,t¯+φ4ξ¯,t¯⋯,φξ¯,t¯=eξ¯1−2t¯+2eξ¯t¯2−2γ+γt¯−4eξ¯t¯233−3γ+γt¯+2eξ¯t¯334−4γ+γt¯+⋯.Whenγ=1, then the NDM result is
(35)φξ¯,t¯=eξ¯1−2t¯+2t¯22!−2t¯33!+2t¯44!⋯.The exact result is(36)φξ¯,t¯=eξ¯−2t¯.In Figure1, the first and second graphs show the exact and NDM solutions ofφξ¯,t¯ for Example 1 at γ=1. Figure 2 shows the different fractional-order graphs of φξ¯,t¯ at γ=0.8 and γ=0.6, and Figure 3 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 4 shows the two- and three-dimensional graphs of different fractional-order γ.Example 2.
Consider the fractional-order telegraph equation(37)∂2γφ∂ξ¯2γ=∂2φ∂t¯2+∂φ∂t¯+φ,0<γ≤1,t¯≥0,with the initial condition
(38)φ0,t¯=e−t¯,φξ¯0,t¯=e−t¯.Figure 1
The exact and analytical solution figures of Example1 at γ=1.Figure 2
The different fractional-order graphs ofφξ¯,t¯ at γ=0.8 and γ=0.6.Figure 3
The fractional-order graph atγ=0.40.Figure 4
The 3D and 2D different fractional-order graphs ofγ.Now, applying the natural transformation to (37), we get
(39)N∂2γφ∂ξ¯2γ=N∂2φ∂t¯2+∂φ∂t¯+φ.Using the inverse natural transformation,(40)φξ¯,t¯=N−1φ0,t¯s+μs2φξ¯0,t¯+μ/s+γ1−μ/sμ/s2N∂2φ∂t¯2+∂φ∂t¯+φ.Applying the Adomian decomposition method, we get(41)φ0ξ¯,t¯=N−1φ0,t¯s+φξ¯0,t¯s2=N−1e−t¯s+μe−t¯s2,φ0ξ¯,t¯=e−t¯1+ξ¯,φj+1ξ¯,t¯=N−1μ/s+γ1−μ/sμ/s2N∂2φj∂t¯2+∂φj∂t¯+φj,j=0,1,2,⋯.Forj=0,
(42)φ1ξ¯,t¯=N−1μ/s+γ1−μ/sμ/s2N∂2φ0∂t¯2+∂φ0∂t¯+φ0=e−t¯1−γ+ξ¯+γξ¯22.The subsequent terms are(43)φ2ξ¯,t¯=N−1μ/s+γ1−μ/sμ/s2N∂2φ1∂t¯2+∂φ1∂t¯+φ1=e−t¯ξ¯−γξ¯+ξ¯22+γ6ξ¯3,φ3ξ¯,t=N−1μ/s+γ1−μ/sμ/s2N∂2φ2∂t¯2+∂φ2∂t¯+φ2=e−t¯ξ¯22−γξ¯22+γξ¯424+ξ¯36,⋮The NDM solution for Example2 is
(44)φξ¯,t¯=φ0ξ¯,t¯+φ1ξ¯,t¯+φ2ξ¯,t¯+φ3ξ¯,t¯+φ4ξ¯,t¯⋯,φξ¯,t¯=e−t¯2+3ξ¯−γ+γξ¯22−γξ¯+ξ¯22+γ6ξ¯3+ξ¯22−γξ¯22+γξ¯424+ξ¯36+⋯.The exact result is(45)φξ¯,t¯=eξ¯−t¯.In Figure5, the first and second graphs show the exact and NDM solutions ofφξ¯,t¯ for Example 2 at γ=1. Figure 6 shows the different fractional-order graphs of φξ¯,t¯ 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 two- and three-dimensional graphs of different fractional-order γ.Example 3.
Consider the fractional linear telegraph equation(46)∂2γφ∂t¯2γ+3∂γφ∂t¯γ+2φ=∂2φ∂ξ¯2+∂2φ∂ζ¯2,0<γ≤1,t≥0,with the initial condition
(47)φξ¯,ζ¯,0=eξ¯+ζ¯,φt¯ξ¯,ζ¯,0=−3eξ¯+ζ¯.Figure 5
The exact and analytical solution figures of Example2 at γ=1.Figure 6
The different fractional-order graphs ofφξ¯,t¯ at γ=0.8 and γ=0.6.Figure 7
The fractional-order graph atγ=0.40.Figure 8
The 3D and 2D different fractional-order graphs ofγ.Using the natural transformation to equation (46), we get
(48)N∂2γφ∂t¯2γ=−N3∂γφ∂t¯γ+2φ−∂2φ∂ξ¯2−∂2φ∂ζ¯2.Applying the inverse natural transformation,(49)φξ¯,ζ¯,t¯=N−1φξ¯,ζ¯,0s+φt¯ξ¯,ζ¯,0s2−μ/s+γ1−μ/sμ/s2N3∂γφ∂t¯γ+2φ−∂2φ∂ξ¯2−∂2φ∂ζ¯2.Implementing the Adomian decomposition method, we get(50)φ0ξ¯,ζ¯,t¯=N−1φξ¯,ζ¯,0s+μs2φt¯ξ¯,ζ¯,0=N−1eξ¯+ζ¯s−3μeξ¯+ζ¯s2,φ0ξ¯,ζ¯,t¯=eξ¯+ζ¯1−3t¯,φj+1ξ¯,ζ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N3∂γφj∂t¯γ+2φj−∂2φj∂ξ¯2−∂2φj∂ζ¯2,j=0,1,2,⋯.Forj=0,
(51)φ1ξ¯,ζ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N3∂γφ0∂t¯γ+2φ0−∂2φ0∂ξ¯2−∂2φ0∂ζ¯2,φ1ξ¯,ζ¯,t¯=9eξ¯+ζ¯t¯−γt¯+γt¯22.The subsequent terms are(52)φ2ξ¯,ζ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N3∂γφ1∂t¯γ+2φ1−∂2φ1∂ξ¯2−∂2φ1∂ζ¯2=−27eξ¯+ζ¯t¯22−γt¯22+γt¯36,φ3ξ¯,ζ¯,t¯=−N−1μ/s+γ1−μ/sμ/s2N3∂γφ2∂t¯γ+2φ2−∂2φ2∂ξ¯2−∂2φ2∂ζ¯2=81eξ¯+ζ¯t¯36−γt¯36+γt¯424,⋮The NDM result for Example3 is
(53)φξ¯,ζ¯,t¯=φ0ξ¯,ζ¯,t¯+φ1ξ¯,ζ¯,t¯+φ2ξ¯,ζ¯,t¯+φ3ξ¯,ζ¯,t¯+φ4ξ¯,ζ¯,t¯⋯,φξ¯,ζ¯,t¯=eξ¯+ζ¯1−3t¯+9t¯−γt¯+γt¯22−27t¯22−γt¯22+γt¯36+81t¯36−γt¯36+γt¯424⋯.Whenγ=1, then the NDM result is
(54)φξ¯,ζ¯,t¯=eξ¯+ζ¯1−3t¯+3t¯22!−3t¯33!+3t¯44!⋯.The exact result is(55)φξ¯,ζ¯,t¯=eξ¯+ζ¯−3t¯.In Figure9, the first and second graphs show the exact and NDM solutions ofφξ¯,t¯ for Example 3 at γ=1. Figure 10 shows the different fractional-order graphs of φξ¯,t¯ 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(56)∂2γφ∂t¯2γ+2∂γφ∂t¯γ+3φ=∂2φ∂ξ¯2+∂2φ∂ζ¯2+∂2φ∂χ2,0<γ≤1,t¯≥0,with the initial condition
(57)φξ¯,ζ¯,χ,0=sinhξ¯sinhζ¯sinhχ,φt¯ξ¯,ζ¯,χ,0=−sinhξ¯sinhζ¯sinhχ.Figure 9
The exact and analytical solution figures of Example3 at γ=1.Figure 10
The different fractional-order graphs ofφξ¯,t¯ at γ=0.8 and γ=0.6.Figure 11
The fractional-order graph atγ=0.40.Figure 12
The 3D and 2D different fractional-order graphs ofγ.Taking the natural transformation of equation (56),
(58)N∂2γφ∂t¯2γ=−N2∂γφ∂t¯γ+3φ−∂2φ∂ξ¯2−∂2φ∂ζ¯2−∂2φ∂χ2.Using the inverse natural transformation,(59)φξ¯,ζ¯,χ,t¯=N−1φξ¯,ζ¯,χ,0s+μs2φt¯ξ¯,ζ¯,χ,0−N−1μ/s+γ1−μ/sμ/s2N2∂γφ∂t¯γ+3φ−∂2φ∂ξ¯2−∂2φ∂ζ¯2−∂2φ∂χ2.Applying the Adomian decomposition method, we get(60)φ0ξ¯,ζ¯,χ,t¯=N−1φξ¯,ζ¯,χ,0s+φt¯ξ¯,ζ¯,χ,0s2=N−1sinhξ¯sinhζ¯sinhχs−sinhξ¯sinhζ¯sinhχs2,φ0ξ¯,ζ¯,χ,t¯=sinhξ¯sinhζ¯sinhχ1−t¯,φj+1ξ¯,ζ¯,χ,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφj∂t¯γ+3φj−∂2φj∂ξ¯2−∂2φj∂ζ¯2−∂2φj∂χ2,j=0,1,2,⋯.Forj=0,
(61)φ1ξ¯,ζ¯,χ,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφ0∂t¯γ+3φ0−∂2φ0∂ξ¯2−∂2φ0∂ζ¯2−∂2φ0∂χ2,φ1ξ¯,ζ¯,χ,t¯=−N−12sinhξ¯sinhζ¯sinhχsγ+2,φ1ξ¯,ζ¯,χ,t¯=−2sinhξ¯sinhζ¯sinhχt¯−γt¯+γt¯22.The subsequent terms are(62)φ2ξ¯,ζ¯,χ,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφ1∂t¯γ+3φ1−∂2φ1∂ξ¯2−∂2φ1∂ζ¯2−∂2φ1∂χ2,φ2ξ¯,ζ¯,χ,t¯=4sinhξ¯sinhζ¯sinhχt¯22−γt¯22+γt¯36,φ3ξ¯,ζ¯,χ,t¯=−N−1μ/s+γ1−μ/sμ/s2N2∂γφ2∂t¯γ+3φ2−∂2φ2∂ξ¯2−∂2φ2∂ζ¯2−∂2φ2∂χ2,φ3ξ¯,ζ¯,χ,t¯=−8sinhξ¯sinhζ¯sinhχt¯36−γt¯36+γt¯424,⋮The NDM result for Example4 is
(63)φξ¯,ζ¯,χ,t¯=φ0ξ¯,ζ¯,χ,t¯+φ1ξ¯,ζ¯,χ,t¯+φ2ξ¯,ζ¯,χ,t¯+φ3ξ¯,ζ¯,χ,t¯+⋯,φξ¯,ζ¯,χ,t¯=sinhξ¯sinhζ¯sinhχ1+t¯−2t−γt¯+γt¯22+4t¯22−γt¯22+γt¯36−8t¯36−γt¯36+γt¯424+⋯.The exact result is(64)φξ¯,ζ¯,χ,t¯=e−2t¯sinhξ¯sinhζ¯sinhχ.In Figure13, the first and second graphs show the exact and NDM solutions ofφξ¯,t¯ for Example 4 at γ=1. Figure 14 shows the different fractional-order graphs of φξ¯,t¯ 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.Figure 13
The exact and analytical solution figures of Example4 at γ=1.Figure 14
The different fractional-order graphs ofφξ¯,t¯ at γ=0.8 and γ=0.6.Figure 15
The 3D different fractional-order graphs ofγ.


## 6. 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.



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