numerical solution for nonlinear telegraph … solution for nonlinear telegraph equation 245 to...

Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 5, 243 - 257

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/nade.2016.6418

Numerical Solution for Nonlinear

Telegraph Equation by Modified

Adomian Decomposition Method

Hind Al-badrani1, Sharefah Saleh2, H. O. Bakodah3 and M. Al-Mazmumy3

1 Department of Administration Information Systems

College of Business Administration, Taibah University

Al-Madinah Al-Munawarah, Saudi Arabia

2Department of Mathematics, Faculty of Science-AL Salmania Campus

King Abdulaziz University, Jeddah, Saudi Arabia

3Department of Mathematics, Faculty of Science-AL Faisaliah Campus

King Abdulaziz University, Jeddah, Saudi Arabia

Copyright © 2016 Hind Al-badrani et al. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

In this paper, Adomian decomposition method (ADM) and Modified ADM are

used to obtain one soliton solution to the nonlinear telegraph equation. Some

examples are provided to illustrate the method. The results show the simplicity

and the efficiency of the method.

Keywords: Nonlinear telegraph equation, Adomian Decomposition method

(ADM), Modified Adomian Decomposition method (MADM)

1. Introduction

Telegraph equation is commonly used in the study of wave propagation of

electric signals in a cable transmission line and also in wave phenomena. Many

researchers have used various numerical and analytical methods to solve the

telegraph equation [2, 3, 4, 7, 9, 10, 11, 12, 13, 14, 15]. Mohebbi and Dehaghan

[16] studied high order compact solution to solve the telegraph equation. Gao and Chi [8] used unconditionally stable difference scheme for a one-space dimensional

244 Hind Al-badrani et al.

linear hyperbolic equation. Saadatmandi and Dehghan [17] developed a numerical

solution based on Chebyshev Tau method. Yousefi [20] used Legendre multi

wavelet Galerkin method for solving the hyperbolic telegraph equation. Meghan

and Ghesmati [6] developed a numerical approach based on the truly meshless

local weak-strong (MLWS) methods to deal with the second order two-space-

dimensional telegraph equation. In references [1, 5] the authors used Adomian

decomposition method to solve the telegraph equation.

In recent years, it has been shown that the Adomian decomposition method can

solve effectively, easily, and accurately a large class of linear and nonlinear,

ordinary or partial, deterministic or stochastic differential equations. The

approximate solutions converge rapidly to accurate solutions. The method is well

suited to physical problems since it doesn’t require the linearization, perturbation,

and other restrictive methods and assumptions, which may change the problem

being solved, sometimes seriously. In this paper, the Adomian decomposition

method is used to solve the linear and nonlinear telegraph equation. The main

purpose of this paper is to illustrate the advantages and the simplicity of using the

ADM for solving nonlinear telegraph equation. A modified ADM is also present-

ed to solve nonlinear telegraph problem. The results of numerical experiments are

presented, and are compared with analytical solutions to confirm the good

accuracy of the method.

2. The Adomian decomposition method applied to telegraph

equation

Consider the one-dimensional nonlinear telegraph equation of the form

𝑢𝑡𝑡 − 𝑢𝑥𝑥 + 𝑎 𝑢𝑡 +Φ(𝑢) = 𝑓(𝑥, 𝑡) (1)

with the following indicated initial conditions

𝑢(𝑥, 0) = 𝑔1(𝑥) (2)

𝜕𝑢(𝑥,0)

𝜕𝑡= 𝑔2(𝑥) (3)

For solving by Adomian decomposition method we consider operator 𝐿𝑡𝑡 =𝜕2

𝜕𝑡2 therefore we have 𝐿𝑢 = 𝑢𝑥𝑥 − 𝑎 𝑢𝑡 −Φ(𝑢) + 𝑓(𝑥, 𝑡).

Applying the inverse operator 𝐿𝑡𝑡−1(. ) = ∫ ∫ (. )𝑑𝑡𝑑𝑡

𝑡

0

𝑡

0 to both sides of the above

equation, we get

𝑢(𝑥, 𝑡) = 𝑢(𝑥, 0) +𝜕𝑢(𝑥, 0)

𝜕𝑡𝑡 + ∫ ∫ (𝑢𝑥𝑥 − 𝑎 𝑢𝑡 −Φ(𝑢) + 𝑓(𝑥, 𝑡))𝑑𝑡𝑑𝑡

𝑡

0

𝑡

0

(4)

Substituting (2) and (3) into (4), we have

𝑢(𝑥, 𝑡) = 𝑔1(𝑥) + 𝑔2(𝑥)𝑡 + ∫ ∫ (𝑢𝑥𝑥 − 𝑎 𝑢𝑡 −Φ(𝑢) + 𝑓(𝑥, 𝑡))𝑑𝑡𝑑𝑡𝑡

0

𝑡

0 (5)

Numerical solution for nonlinear telegraph equation 245

To solve equation (5) by Adomian decomposition method, the solution u is

represented by an infinite series given by

𝑢 = ∑ 𝑢𝑛∞𝑛=0 (6)

The components 𝑢𝑛will be determined recursively. However, the nonlinear term

Φ(𝑢)at the right side of (5) will be represented by an infinite series of the

Adomian polynomials 𝐴𝑛 in the form

Φ(𝑢) = ∑𝐴𝑛(𝑢0𝑢1, … . , 𝑢𝑛) (7)

∞

𝑛=0

where 𝐴𝑛 , 𝑛 ≥ 0 are defined by

𝐴𝑛 =1

𝑛!

𝑑𝑛

𝑑 𝜆𝑛[Φ(∑𝜆𝑗𝑢𝑗

𝑛

𝑗=0

)]|

𝜆=0

, 𝑛 = 0,1,2, … (8)

which can be evaluated for all forms of nonlinearity. Substituting (6) and (7) into

(5) yields to

∑𝑢𝑛

∞

𝑛=0

= 𝑔1(𝑥) + 𝑔2(𝑥)𝑡 + ∫ ∫ {(∑𝑢𝑛

∞

𝑛=0

)

𝑥𝑥

− 𝑎 (∑𝑢𝑛

∞

𝑛=0

)

𝑡

− (∑𝐴𝑛

∞

𝑛=0

) + 𝑓(𝑥, 𝑡)}𝑑𝑡𝑑𝑡 (9) 𝑡

0

𝑡

0

so we determine the components 𝑢𝑛(𝑛 ≥ 0) from the following recursive relation

{

u0 = 𝑔

1(𝑥) + 𝑔

2(𝑥)𝑡 +∫ ∫ 𝑓(𝑥, 𝑡) 𝑑𝑡𝑑𝑡

𝑡

0

𝑡

0

= ℎ(𝑥, 𝑡)

𝑢𝑛+1 = ∫ ∫ {𝑢𝑛𝑥𝑥 − 𝑎 (𝑢𝑛)𝑡 − (𝐴𝑛)}𝑑𝑡𝑑𝑡𝑡

0

𝑡

0

, 𝑛 = 0,1,2, ….

(10)

The solution of equation (1) is now determined. However, in practice all series

∑ 𝑢𝑛∞𝑛=0 must be truncated to the series 𝜑𝑛 = ∑ 𝑢𝑖

∞𝑛=0 with lim

𝑛→∞𝜑𝑛 = 𝑢.

3. The Modifications of the Adomian Decomposition Method

In this section, a reliable modification of the Adomain decomposition

method developed by Wazwaz [18, 19] will be deduced. The modified form was

established based on the assumption that the function ℎ(𝑥, 𝑡) can be divided into

two parts namely ℎ0 𝑎𝑛𝑑 ℎ1. Under this assumption we set ℎ = ℎ0 + ℎ1.

Based on this, the modified recursive relation is formulated as follows

{

u0 = ℎ0(𝑥)

u1 = ℎ2(𝑥) + ∫ ∫ {(𝑢0)𝑥𝑥 − 𝑎 (𝑢0)𝑡 − (𝐴0)}𝑑𝑡𝑑𝑡𝑡

0

𝑡

0

(11)

𝑢𝑛+1 = ∫ ∫ {(𝑢𝑛)𝑥𝑥 − 𝑎 (𝑢𝑛)𝑡 − (𝐴𝑛)}𝑑𝑡𝑑𝑡𝑡

0

𝑡

0

, 𝑛 = 1,2,….


The choice of ℎ0 and ℎ1, such that 𝑢𝑛 contains the minimal number of terms, has

a strong influence in accelerating the convergence of the solution. The

modification demonstrate a rapid convergence of the series solution if compared

with standard (ADM) and it may give the exact solution for nonlinear equations

by using two iterations only without using the so-called Adomain polynomials.

4. Numerical results

In this section we present numerical results to test the efficiency for solving the

nonlinear telegraph equation using Adomian decomposition method and its

modification.

Example 1

Consider the nonlinear Telegraph equation

𝑢𝑡𝑡 − 𝑢𝑥𝑥 + 2𝑢𝑡 − 𝑢2 = 𝑒−2𝑡𝑐𝑜𝑠ℎ2(𝑥) − 2𝑒−𝑡 cosh(𝑥) (12)

Subject to 𝑢(𝑥, 0) = cosh(𝑥) , 𝑢𝑡(𝑥, 0) = −cosh (𝑥), with the exact solution

𝑢(𝑥, 𝑡) = 𝑒−𝑡 cosh(𝑥). The Adomian polynomials 𝐴𝑛 for the nonlinear term can

be evaluated by using the relation (8) assuming the nonlinear function in the form

Φ(𝑢) = 𝑢2. Therefore the Adomian polynomials are given by

𝐴0 = 𝑢02

𝐴1 = 2𝑢0𝑢1

𝐴2 = 2𝑢0𝑢2 + 𝑢12

𝐴3 = 2𝑢0𝑢3 + 2𝑢1𝑢2 . Similarly, other polynomials can be generated.

I. Standard Adomian decomposition method

Applying Adomian decomposition method we obtain the recursive relation

𝑢0(𝑥, 𝑡) = cosh(𝑥) − 𝑡. cosh(𝑥) −1

4𝑐𝑜𝑠ℎ2(𝑥) + 2. cosh(𝑥) +

1

2. 𝑐𝑜𝑠ℎ2(𝑥). 𝑡

−2. cosh(𝑥) . 𝑡 +1

4𝑒−2𝑡. 𝑐𝑜𝑠ℎ2(𝑥) − 2𝑒−𝑡. cosh(𝑥) (13)

𝑢𝑘+1(𝑥, 𝑡) = 𝐿𝑡𝑡−1( (𝑢𝑘)𝑥𝑥 − 2(𝑢𝑘)𝑡 + 𝐴𝑘), 𝑘 ≥ 0

The results are given in Table (1) and the profile of this case is shown in Figure (1).


Figure1: The solution function 𝑢(𝑥, 𝑡) evaluated by the Adomian decomposition

compare with exact solution at t=0.1

II. Reliable Adomian decomposition method

Using modified Adomian decomposition method, the recursive relation is

𝑢0(𝑥, 𝑡) = 3 cosh(𝑥) − 3𝑡𝑐𝑜𝑠ℎ(𝑥) −1

4cosh2(x)

𝑢1(𝑥, 𝑡) =1

2𝑐𝑜𝑠ℎ2(𝑥)𝑡 +

1

4𝑒−2𝑡𝑐𝑜𝑠ℎ2(𝑥) − 2𝑒−𝑡 𝑐𝑜𝑠ℎ(𝑥)

+𝐿𝑡𝑡−1(𝑢0𝑥𝑥 − 2𝑢0𝑡 + 𝐴0) (14)

uk+1(𝑥, 𝑡) = 𝐿𝑡𝑡−1(𝑢𝑘𝑥𝑥 − 2𝑢𝑘𝑡 + 𝐴𝑘), 𝑘 ≥ 1

The results are given in Table (2) and the profile of this case is shown in Figure (2).

x 𝑢𝑎 𝑢𝑒 |𝑢𝑒 − 𝑢𝑎|

0.00 0.91366178 0.90483742 0.00882436

0. 10 0.91825217 0.90936538 0.00888679

0. 20 0.93215657 0.92299457 0.00916200

0. 30 0.95545619 0.94586140 0.00959479

0.40 0.98848184 0.97819473 0.01028710

0.50 1.03152442 1.02031817 0.01120625

Table1: The solution function 𝑢(𝑥, 𝑡) evaluated by the

Adomian decomposition compare with exact solution at t=0.1


0.00 0.91451972 0.90483742 0.00968230

0. 10 0.91914260 0.90936538 0.00977722

0. 20 0.93305989 0.92299457 0.01006532

0. 30 0.95641958 0.94586140 0.01055818

0.40 0.98946965 0.97819473 0.01127492

0.50 1.03256015 1.02031817 0.01224198

Table 2: The solution function 𝑢(𝑥, 𝑡) evaluated by the modified

Adomian decomposition


Figure2: The solution function 𝑢(𝑥, 𝑡) evaluated by the modified Adomian

decomposition compare with exact solution at t=0.1

Example 2 Consider the nonlinear Telegraph equation

𝑢𝑥𝑥 = 𝑢𝑡𝑡 + 2 𝑢𝑡 + 𝑢2 − 𝑒2𝑥−4𝑡 + 𝑒𝑥−2𝑡 (15)

Subject to 𝑢(𝑥, 0) = 𝑒𝑥, 𝑢𝑡(𝑥, 0) = −2𝑒𝑥, with the exact solution 𝑢(𝑥, 𝑡) =𝑒𝑥−2𝑡.


Applying decomposition method we get the recursive relation

𝑢0(𝑥, 𝑡) =5

4𝑒𝑥 −

5

2𝑒𝑥𝑡 −

1

16𝑒2𝑥 +

1

4𝑒2𝑥𝑡 +

1

16𝑒2𝑥−4𝑡 −

1

4𝑒𝑥−2𝑡

𝑢𝑘+1(𝑥, 𝑡) = 𝐿𝑡𝑡−1((𝑢𝑘)𝑥𝑥 − 2(𝑢𝑘)𝑡 − 𝐴𝑘), 𝑘 ≥ 0

The results are given in Table (3) and the profile is shown in figure (3).


0.00 0.81872970 0.81873075 0.00000267

0. 20 0.90482522 0.90483742 0.00000313

0. 40 1.00000885 1.00000000 0.00000366

0. 60 1.10515894 1.10517092 0.00000416

0.80 1.22141194 1.22140276 0.00000466

1.00 1.34984402 1.34985881 0.00000471

Table 3: The solution function 𝑢(𝑥, 𝑡) evaluated by the Adomian



Figure3: The solution function 𝑢(𝑥, 𝑡) evaluated by the Adomian decomposition

compare with exact solution at t=0.1

Remark

It can be seen that the absolute error are somewhat small as the number of the

components Adomian series is increasing Table(4) and (Fig.4).

Figure 4: The graph of exact solution and approximate ADM solution at n=1, n=3, n=5.


The recursive relation is

𝑢0(𝑥, 𝑡) =5

4𝑒𝑥 −

5

2𝑒𝑥𝑡 −

1

16𝑒2𝑥

𝑢1(𝑥, 𝑡) =1

4𝑒2𝑥𝑡 +

1

16𝑒2𝑥−4𝑡 −

1

4𝑒𝑥−2𝑡 + 𝐿𝑡𝑡

−1(𝑢0𝑥𝑥 − 2𝑢0𝑡 − 𝐴0) (16)

𝑢𝑘+1(𝑥, 𝑡) = 𝐿𝑡𝑡−1(𝑢𝑘𝑥𝑥 − 2𝑢𝑘𝑡 − 𝐴𝑘), 𝑘 ≥ 1

x |𝑢𝑒 − 𝑢𝑎| at 𝑛 = 1

|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 3

|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 5

0.00 0.00130805 0.00000267 0.00000641

0. 10 0.00141470 0.00000313 0.00001362

0. 20 0.00152551 0.00000366 0.00004038

0. 30 0.00163928 0.00000416 0.00002152

0.40 0.00175429 0.00000466 0.00000568

0.50 0.00186816 0.00000471 0.00000694

Table(4): The absolute error at n=1, n=3, n=5


The result are given in Table (5) and the profile is shown in figure (5).

Figure 5: The solution function 𝑢(𝑥, 𝑡) evaluated by the modified Adomian


Remark. It can be seen that the absolute error are somewhat small as the number

of the components Adomian series is increasing, see( Table(6) and (Fig.6)).


0.00 0.81873504 0.81873075 0.00000429

0. 20 1.00000258 1.00000000 0.00000258

0. 40 1.22140320 1.22140276 0.00000044

0. 60 1.49182327 1.49182470 0.00000143

0.80 1.82211736 1.82211880 0.00000144

1.00 2.22554407 2.22554093 0.00000314

Table 5: The solution function 𝑢(𝑥, 𝑡) evaluated by the modified

Adomian decomposition compare with exact solution at t=0.1,n=3.


|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 3

|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 5

0. 20 0.00486103 0.00000258 0.00000137

0.40 0.00520801 0.00000044 0.00000256

0.60 0.00550472 0.00000143 0.00000178

0.80 0.00585503 0.00000144 0.00003186

1.00 0.00657238 0.00000314 0.00002165

Table 6: The error between exact and approximate solution 𝑢(𝑥, 𝑡) evaluated

by the modified Adomian decomposition with Different component


Figure6: The graph of exact solution and approximate by the modified Adomian

decomposition method at t=0.1 and n=1, n=3, n=5.

Example 3

In this example, the nonlinear Telegraph equation is considered

𝑢𝑡𝑡 + 2𝑢𝑡 = 𝑢𝑥𝑥 + 𝑢3 − 𝑢 (17)

Subject to 𝑢(𝑥, 0) =1

2+1

2tanh (

𝑥

8+ 5) , 𝑢𝑡(𝑥, 0) =

3

16−

3

16tanh (

𝑥

8+ 5)

2

,

which has an exact solution 𝑢(𝑥, 𝑡) =1

2+1

2tanh (

𝑥

8+3𝑡

8+ 5)

.

The Adomian polynomials 𝐴𝑛 for the nonlinear term can be evaluated by using

the relation (8) assuming that the nonlinear function is Φ(𝑢) = 𝑢3. Therefore the

coefficients of the Adomian polynomials are given by

𝐴0 = 𝑢03

𝐴1 = 3𝑢02𝑢1

𝐴2 = 3𝑢0𝑢12 + 3𝑢0

2𝑢2

𝐴3 = 𝑢13 + 6𝑢0𝑢1𝑢2 + 3𝑢0

2𝑢3. Other polynomials can be generated in a similar

manner.


The recursive relation in this case as following

𝑢0(𝑥, 𝑡) =1

2+1

2tanh (

𝑥

8+ 5) +

3

16𝑡 −

3

16tanh (

𝑥

8+ 5)

2

𝑡

𝑢𝑘+1(𝑥, 𝑡) = 𝐿𝑡𝑡−1(( 𝑢𝑛)𝑥𝑥 − 2(𝑢𝑛)𝑡 + 𝐴𝑛 − 𝑢𝑛), 𝑘 ≥ 0 (18)

The result of this problem at 𝑡 = 0.1, 0.3 𝑎𝑛𝑑 0.5 are given in Table (7) and the profile

is shown in Figure (7). Also, it can be seen that the absolute error are small as the

number of the components Adomian series is increasing, see(Table(8) and

(Fig.8)).


Figure7: The graph of exact solution and approximate ADM solution at t=0.1, t=0.3,

t=0.5.

Table 8: The error between exact and approximate solution 𝑢(𝑥, 𝑡) evaluated by the

Adomian decomposition with Different component

x |𝑢𝑒 − 𝑢𝑎| at 𝑡 = 0.1

|𝑢𝑒 − 𝑢𝑎| at 𝑡 = 0.3

|𝑢𝑒 − 𝑢𝑎| at 𝑡 = 0.5

0.00 0.00000062 0.00000476 0.00001147

0. 10 0.00000061 0.00000465 0.00001119

0. 20 0.00000059 0.00000453 0.00001091

0. 30 0.00000058 0.00000442 0.00001064

0.40 0.00000056 0.00000431 0.00001038

0.50 0.00000055 0.00000420 0.00001012

Table 7: The error between exact and approximate solution

𝑢(𝑥, 𝑡) evaluated by the Adomian decomposition with Different

time.


|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 3

|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 5

0.00 0.00000067 0.00000062 0.00000062

0. 10 0.00000066 0.00000061 0.00000061

0. 20 0.00000064 0.00000059 0.00000059

0. 30 0.00000062 0.00000058 0.00000058

0.40 0.00000061 0.00000056 0.00000056

0.50 0.00000059 0.00000055 0.00000055


Figure8: The graph of exact solution and approximate ADM solution at n=1, n=3, n=5.


The recursive relation is

𝑢0(𝑥, 𝑡) =1

2+1

2tanh (

x

8+ 5)

𝑢1(𝑥, 𝑡) =3

16𝑡 −

3

16tanh (

𝑥

8+ 5)

2

𝑡 + 𝐿𝑡𝑡−1(𝑢0𝑥𝑥 − 2𝑢0𝑡 + 𝐴0 − 𝑢0) (19)

𝑢𝑘+1(𝑥, 𝑡) = 𝐿𝑡𝑡−1(𝑢𝑘𝑥𝑥 − 2𝑢𝑘𝑡 + 𝐴𝑘 − 𝑢𝑘), 𝑘 ≥ 1

The result of this problem at 𝑡 = 0.1, 0.3 𝑎𝑛𝑑 0.5 and n=5 are given in Table (9) and

the profile is shown in Figure (9). Also, it can be seen that the absolute error are

small as the number of the components Adomian series is increasing, see(

Table(10) and (Fig.10)).

x |𝑢𝑒 − 𝑢𝑎| at 𝑡 = 0.1

|𝑢𝑒 − 𝑢𝑎| at 𝑡 = 0.3

|𝑢𝑒 − 𝑢𝑎| at 𝑡 = 0.5

0.00 0.00000062 0.00000476 0.00001144

0. 10 0.00000061 0.00000465 0.00001116

0. 20 0.00000059 0.00000453 0.00001088

0. 30 0.00000058 0.00000442 0.00001061

0.40 0.00000056 0.00000431 0.00001035

0.50 0.00000055 0.00000420 0.00001010

Table 9: The error between exact and approximate solution 𝑢(𝑥, 𝑡) evaluated by the modified Adomian decomposition method with

Different time



decomposition method at t=0.1, t=0.3, t=0.5.


decomposition method at n=1, n=3, n=5.

5. Conclusion

The main objective of this study is to find an approximate solution of one-

dimensional linear and nonlinear Telegraph equation. This problem has been

solved by means of the Adomian decomposition method. In order to increase the

accuracy of the approach, higher components of Adomian series solution should

be taken into account. Some typical examples have been demonstrated in order to


|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 3

|𝑢𝑒 − 𝑢𝑎| at 𝑛 = 5

0.00 0.00000034 0.00000062 0.00000062

0. 10 0.00000034 0.00000061 0.00000061

0. 20 0.00000033 0.00000059 0.00000059

0. 30 0.00000032 0.00000058 0.00000058

0.40 0.00000031 0.00000056 0.00000056

0.50 0.00000030 0.00000055 0.00000055

Table 10 The error between exact and approximate solution 𝑢(𝑥, 𝑡) evaluated by the modified Adomian decomposition with Different

component


illustrate the efficiency and accuracy of the present method. The results show that

the method is seen to be a very reliable alternative and intuitively believed to be a

powerful mathematical tool for finding approximate solutions of linear/nonlinear

telegraph equations. The series solutions obtained by this method do not require

linearization or perturbation. This paper can be used as a standard paradigm for

other applications.

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Received: April 23, 2016; Published: May 30, 2016


http://dx.doi.org/10.1016/s0096-3003(00)00060-6


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