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Five semi analytical and numerical simulations for the fractional nonlinear space-time telegraph equation

Advances in Difference Equations volume 2021, Article number: 227 (2021) Cite this article

Abstract

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches.

1 Introduction

Recently, the nonlinearity has been used in distinct fields such as the neural network [2], infectious disease epidemiology [3], plasma physics [4], thermodynamics [5], optic physics [6], population ecology [7], biology and mechanics of fluids [8, 9]. Defining empirically function and parameters in formulating the nonlinear phenomena in nonlinear evolution equations and systems provides reliable data that explain more about dynamic behavior processes and complicated physical ones [10, 11]. Many physicists and mathematicians have focused their attention on formulating some nonlinear phenomena to demonstrate their characterization of undiscovered models [12–14]. Deriving accurate analytical, semi-analytical, approximate techniques has taken more attention of many researchers; consequently, many distinct techniques have been derived, such as [15–20].

Nowadays, based on the integer-order derivative’s failure to show the nonlocal property of the investigated model, many nonlinear phenomena have been formulated in fractional forms [21–23]. Therefore, many fractional derivative operators have been mathematically formulated, such as Atangana–Baleanu fractional operator, the conformal fractional, Caputo fractional, Riemann–Liouville, Caputo–Fabrizio fractional derivative definitions [24–26], to transform the fractional nonlinear partial differential equations to nonlinear ordinary differential ones [27, 28].

In this context, this paper investigates the fractional nonlinear telegraph equation that is used to describe the transmission of the voltage standard [29]. This mathematical model is considered as a primary icon in the electromagnetic waves’ area. The fractional model of this model is given by [30–32]

$$ \mathcal{D}_{t t} ^{2 \alpha } \mathcal{Q}- \mathcal{D}_{x x} ^{2 \alpha } \mathcal{Q}+\mathcal{D}_{t} ^{\alpha } \mathcal{Q}+ b \mathcal{Q}+d \mathcal{Q}^{3}=0, \quad 0 < \alpha \leq 1, $$
(1)

where b, d are arbitrary constants, while \(\mathcal{D}=\mathcal{D}(x,t)\). Employing the conformable fractional wave transformation with the next form \(\mathcal{Q}(x,t)=\mathcal{S}(\mathfrak{T})\), \(\mathfrak{T}= \frac{a x^{\alpha }}{\alpha }-\frac{c t^{\alpha }}{\alpha }\), where a, c are arbitrary constants, converts Eq. (1) into

$$ \bigl(c^{2}-a^{2} \bigr) \mathcal{S}''-c \mathcal{S}'+b \mathcal{S}+d \mathcal{S}^{3}=0. $$
(2)

The structure of the paper for the rest of its sections is ordered as follows: Sect. 2 investigates the approximate solutions of the considered fractional nonlinear model. Section 3 studies our obtained solutions to show the matching between the computational and numerical solutions, which shows the accuracy of our solutions. Moreover, this section aims to show the originality of our solutions by comparing them with other previously obtained solutions. Section 4 gives the conclusion.

2 Approximate analysis

This section studies the approximate solutions of the fractional nonlinear considered model through five recent numerical schemes. This study depends on the obtained solutions in [1] by implementing the sech- tanh expansion (STE), extended sinh-Gorden expansion (ESGE), and extended simplest equation (ESE) methods. Consequently, our investigation not just aims to find the numerical solutions, but it extends to show the matching between exact and numerical solutions, which explains the accuracy of schemes used in the previous paper [1] and our used schemes.

STE method’s solutions

$$ \mathcal{Q}_{1}(x,t)=\frac{1}{2} \sqrt{ \frac{b}{d}} \biggl(\tanh \biggl( \frac{3 b t^{\alpha }+\sqrt{b} \sqrt{9 b-2} x^{\alpha }}{4 \alpha } \biggr)-1 \biggr). $$
(3)

ESGE method’s solutions

$$ \mathcal{Q}_{2}(x,t)= \frac{-1}{2} \sqrt{ \frac{-b }{d}} \biggl( \tanh \biggl( \frac{3 b t^{\alpha }+\sqrt{b} \sqrt{9 b-2} x^{\alpha }}{4 \alpha } \biggr)-1 \biggr). $$
(4)

ESE method’s solution

For \(\lambda =0\), \(\alpha _{*} \mu <0\), we obtain

$$ \mathcal{Q}_{3}(x,t)=a_{{0}}-{ \frac{a_{{0}} c \sqrt{-\alpha \mu } }{6 {\mu } \alpha _{*} ( a-c ) ( a+c ) }} \tanh \biggl(\sqrt{-\alpha _{*} \mu } \biggl( \frac{a x^{\alpha }}{\alpha }-\frac{c t^{\alpha }}{\alpha }\biggr) \mp \frac{\ln (C)}{2} \biggr). $$
(5)

Using the above-mentioned obtained solutions in [1] gives the initial and boundary conditions which are requested for applying the suggested schemes as follows.

2.1 The AD method

Applying the AD method Eq. (2) for evaluating the semi-analytical solutions gives

$$\begin{aligned}& \mathcal{Q}_{0}(\mathfrak{T})=\frac{\mathfrak{T}}{6}+1, \end{aligned}$$
(6)
$$\begin{aligned}& \mathcal{Q}_{1}(\mathfrak{T})=\frac{\mathfrak{T}^{5}}{699{,}840}+ \frac{\mathfrak{T}^{4}}{23{,}328}-\frac{\mathfrak{T}^{3}}{5832}- \frac{\mathfrak{T}^{2}}{27}- \frac{\mathfrak{T}}{3}, \end{aligned}$$
(7)
$$\begin{aligned}& \begin{aligned} \mathcal{Q}_{2}(\mathfrak{T})={}& \frac{\mathfrak{T}^{10}}{734{,}664{,}038{,}400}+ \frac{\mathfrak{T}^{9}}{12{,}244{,}400{,}640}+ \frac{11 \mathfrak{T}^{8}}{9{,}523{,}422{,}720} \\ &{}- \frac{19 \mathfrak{T}^{7}}{264{,}539{,}520}- \frac{7 \mathfrak{T}^{6}}{2{,}361{,}960}- \frac{289 \mathfrak{T}^{5}}{6{,}298{,}560}- \frac{7 \mathfrak{T}^{4}}{23{,}328} \\ &{}+\frac{7 \mathfrak{T}^{3}}{2916}+\frac{7 \mathfrak{T}^{2}}{162}, \end{aligned} \end{aligned}$$
(8)
$$\begin{aligned}& \begin{aligned} \mathcal{Q}_{3}(\mathfrak{T})={}& \frac{\mathfrak{T}^{13}}{24{,}755{,}239{,}437{,}926{,}400}+ \frac{\mathfrak{T}^{12}}{317{,}374{,}864{,}588{,}800} \\ &{}+ \frac{\mathfrak{T}^{11}}{58{,}185{,}391{,}841{,}280}- \frac{29 \mathfrak{T}^{10}}{4{,}407{,}984{,}230{,}400} \\ &{}-\frac{13{,}289 \mathfrak{T}^{9}}{55{,}540{,}601{,}303{,}040}+ \frac{193 \mathfrak{T}^{8}}{146{,}932{,}807{,}680}+ \frac{709 \mathfrak{T}^{7}}{2{,}857{,}026{,}816} \\ &{}+ \frac{19 \mathfrak{T}^{6}}{3{,}542{,}940}+ \frac{251 \mathfrak{T}^{5}}{6{,}298{,}560} \\ &{}-\frac{\mathfrak{T}^{4}}{3888}-\frac{19 \mathfrak{T}^{3}}{2916}- \frac{\mathfrak{T}^{2}}{81}. \end{aligned} \end{aligned}$$
(9)

Hence, the approximate solution of the considered model is given by

$$\begin{aligned} \mathcal{Q}_{\text{Approximate}}(x) =&\frac{\mathfrak{T}^{13}}{24{,}755{,}239{,}437{,}926{,}400}+ \frac{\mathfrak{T}^{12}}{317{,}374{,}864{,}588{,}800} \\ &{}+ \frac{\mathfrak{T}^{11}}{58{,}185{,}391{,}841{,}280}- \frac{23 \mathfrak{T}^{10}}{4{,}407{,}984{,}230{,}400} \\ &{}-\frac{8753 \mathfrak{T}^{9}}{55{,}540{,}601{,}303{,}040}+ \frac{2539 \mathfrak{T}^{8}}{1{,}028{,}529{,}653{,}760} \\ &{}+ \frac{2519 \mathfrak{T}^{7}}{14{,}285{,}134{,}080}+ \frac{17 \mathfrak{T}^{6}}{7{,}085{,}880}- \frac{29 \mathfrak{T}^{5}}{6{,}298{,}560} \\ &{}-\frac{\mathfrak{T}^{4}}{1944}-\frac{25 \mathfrak{T}^{3}}{5832}- \frac{\mathfrak{T}^{2}}{162}- \frac{\mathfrak{T}}{6}+1+\cdots . \end{aligned}$$
(10)

Calculating the computational, semi-analytical, and absolute errors with different values of \(\mathfrak{T}\) gives Table 1.

Table 1 Numerical values of the solutions through the AD method
Full size table

2.2 The EK method

Using the EK method for finding the semi-analytical solutions of Eq. (2) gives

$$\begin{aligned}& \mathcal{Q}_{0}(\mathfrak{T})=\frac{\mathfrak{T}}{6}+1, \end{aligned}$$
(11)
$$\begin{aligned}& \mathcal{Q}_{1}(\mathfrak{T})=\frac{\mathfrak{T}^{5}}{699{,}840}+ \frac{\mathfrak{T}^{4}}{23{,}328}-\frac{\mathfrak{T}^{3}}{5832}- \frac{\mathfrak{T}^{2}}{27}- \frac{\mathfrak{T}}{3}, \end{aligned}$$
(12)
$$\begin{aligned}& \begin{aligned} \mathcal{Q}_{2}(\mathfrak{T})={}& \frac{\mathfrak{T}^{17}}{15{,}103{,}590{,}515{,}900{,}153{,}856{,}000}+ \frac{\mathfrak{T}^{16}}{148{,}074{,}416{,}822{,}550{,}528{,}000} \\ &{}+ \frac{13 \mathfrak{T}^{15}}{64{,}782{,}557{,}359{,}865{,}856{,}000} \\ &{}-\frac{67 \mathfrak{T}^{14}}{9{,}357{,}480{,}507{,}536{,}179{,}200}- \frac{613 \mathfrak{T}^{13}}{1{,}002{,}587{,}197{,}236{,}019{,}200} \\ &{}- \frac{2659 \mathfrak{T}^{12}}{282{,}781{,}004{,}348{,}620{,}800} \\ &{}+\frac{28{,}573 \mathfrak{T}^{11}}{70{,}695{,}251{,}087{,}155{,}200}+ \frac{1163 \mathfrak{T}^{10}}{71{,}409{,}344{,}532{,}480} \\ &{}+ \frac{85 \mathfrak{T}^{9}}{1{,}586{,}874{,}322{,}944}- \frac{7307 \mathfrak{T}^{8}}{1{,}028{,}529{,}653{,}760} \\ &{}-\frac{1483 \mathfrak{T}^{7}}{14{,}285{,}134{,}080}+ \frac{\mathfrak{T}^{6}}{3{,}149{,}280} + \frac{217 \mathfrak{T}^{5}}{18{,}895{,}680} \\ &{}+ \frac{\mathfrak{T}^{4}}{209{,}952}- \frac{11 \mathfrak{T}^{3}}{2916}-\frac{\mathfrak{T}^{2}}{18}. \end{aligned} \end{aligned}$$
(13)

Hence, the approximate solution of the considered model is given by

$$\begin{aligned} \mathcal{Q}_{\text{Approximate}}(x) =&\frac{\mathfrak{T}^{17}}{15{,}103{,}590{,}515{,}900{,}153{,}856{,}000} \\ &{}+ \frac{\mathfrak{T}^{16}}{148{,}074{,}416{,}822{,}550{,}528{,}000} \\ &{}+ \frac{13 \mathfrak{T}^{15}}{64{,}782{,}557{,}359{,}865{,}856{,}000} \\ &{}-\frac{67 \mathfrak{T}^{14}}{9{,}357{,}480{,}507{,}536{,}179{,}200}- \frac{613 \mathfrak{T}^{13}}{1{,}002{,}587{,}197{,}236{,}019{,}200} \\ &{}- \frac{2659 \mathfrak{T}^{12}}{282{,}781{,}004{,}348{,}620{,}800} \\ &{}+\frac{28{,}573 \mathfrak{T}^{11}}{70{,}695{,}251{,}087{,}155{,}200}+ \frac{1163 \mathfrak{T}^{10}}{71{,}409{,}344{,}532{,}480} \\ &{}+ \frac{85 \mathfrak{T}^{9}}{1{,}586{,}874{,}322{,}944}- \frac{7307 \mathfrak{T}^{8}}{1{,}028{,}529{,}653{,}760} \\ &{}-\frac{1483 \mathfrak{T}^{7}}{14{,}285{,}134{,}080}+ \frac{\mathfrak{T}^{6}}{3{,}149{,}280}+\frac{61 \mathfrak{T}^{5}}{4{,}723{,}920}+ \frac{5 \mathfrak{T}^{4}}{104{,}976} \\ &{}-\frac{23 \mathfrak{T}^{3}}{5832}- \frac{5 \mathfrak{T}^{2}}{54}-\frac{\mathfrak{T}}{6}+1+\cdots . \end{aligned}$$
(14)

Calculating the computational, semi-analytical, and absolute errors with different values of \(\mathfrak{T}\) gives Table 2.

Table 2 Numerical values of the solutions through the EK method
Full size table

2.3 The B-spline methods

Implementing the B-spline family schemes for finding the numerical solutions of Eq. (2) leads to the following shown values of exact, numerical, and absolute values of error for the considered model. Tables 3, 4, and 5 show the results obtained by employing the CBS, ECBS, and ExCBS schemes, respectively.

Table 3 Numerical values of the solutions through the CBS method
Full size table
Table 4 Numerical values of the solutions through the ECBS method
Full size table
Table 5 Numerical values of the solutions through the ExCBS method
Full size table

3 Results and discussion

This section shows our solutions and discusses the tables and figures in our paper.

  • The numerical solutions of the fractional nonlinear telegraph equation are investigated through five recent schemes based on the obtained analytical solutions in [1]. The analytical results have been used to construct the requested conditions required for applying these numerical schemes.

  • The above-shown Tables 1–5 and Figs. 1–5 offer the matching between exact and numerical solutions, which shows the accuracy of used analytical and numerical techniques. Additionally, all these tables and figures demonstrate the superiority of the ESE method over the other used analytical schemes in [1]. Moreover, they show the CBS method’s power as it gives the most accurate solutions over the above-used numerical schemes 6.

    Figure 1
    figure 1

    Two-dimensional sketches between the analytical and approximate solutions based on the AD method for illustrating the matching between solutions

    Full size image
    Figure 2
    figure 2

    Two-dimensional sketches between the analytical and approximate solutions based on the EK method for illustrating the matching between solutions

    Full size image
    Figure 3
    figure 3

    Two-dimensional sketches between the analytical and approximate solutions based on the CBS scheme for illustrating the matching between solutions

    Full size image
    Figure 4
    figure 4

    Two-dimensional sketches between the analytical and approximate solutions based on the ECBS method for illustrating the matching between solutions

    Full size image
    Figure 5
    figure 5

    Two-dimensional sketches between the analytical and approximate solutions based on the ExCBS method for illustrating the matching between solutions

    Full size image
    Figure 6
    figure 6

    Two-dimensional sketches for the above-used schemes

    Full size image

  • Comparing our obtained numerical results with those obtained in [33], which used the trigonometric quintic B-spline scheme to investigate the numerical solutions of the same model, shows the accuracy of our solutions, which coincide with their solutions, where both of us offer the accuracy of the ESE computational method over other used computational schemes in [1].

4 Conclusion

This manuscript has investigated numerical solutions of the fractional nonlinear telegraph equation by employing five numerical techniques. Abundant numerical results have been obtained, while matching exact and numerical solutions has been explained by some distinct figures in two dimensions. The superiority of the ESE analytical and CBS numerical schemes has been demonstrated. The originality of our paper has been explained by comparing our solutions with previously constructed solutions.

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors are thankful to the Taif University. Taif University researchers are supported by project number (TURSP-2020/160), Taif University, Taif, Saudi Arabia.

Funding

This paper was funded by “Taif University Researchers Supporting Project number (TURSP-2020/160), Taif University, Taif, Saudi Arabia”.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Jiangsu University, 212013, Zhenjiang, China

    Mostafa M. A. Khater

  2. Department of Mathematics, Obour Institutes, 11828, Cairo, Egypt

    Mostafa M. A. Khater

  3. Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, South Korea

    Choonkil Park

  4. Department of Mathematics, Daejin University, Kunggi, 11159, Korea

    Jung Rye Lee

  5. Department of Mathematics, Faculty of Science, Taif University, P. O. Box 11099, Taif, 21944, Saudi Arabia

    Mohamed S. Mohamed

  6. School of Management & Economics, Jiangsu University of Science and Technology, 212100, Zhenjiang, China

    Raghda A. M. Attia

  7. Department of Basic Science, Higher Technological Institute 10th of Ramadan City, El Sharqia, 44634, Egypt

    Raghda A. M. Attia

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  1. Mostafa M. A. Khater
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The authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Correspondence to Mostafa M. A. Khater or Choonkil Park.

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Khater, M.M.A., Park, C., Lee, J.R. et al. Five semi analytical and numerical simulations for the fractional nonlinear space-time telegraph equation. Adv Differ Equ 2021, 227 (2021). https://doi.org/10.1186/s13662-021-03387-9

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Keywords

  • Fractional nonlinear space–time telegraph equation
  • Approximate solutions