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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
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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)
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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,….
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246 Hind Al-badrani et al.
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).
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Numerical solution for nonlinear telegraph equation 247
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
x 𝑢𝑎 𝑢𝑒 |𝑢𝑒 − 𝑢𝑎|
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
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248 Hind Al-badrani et al.
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.
I. Standard Adomian decomposition method
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).
x 𝑢𝑎 𝑢𝑒 |𝑢𝑒 − 𝑢𝑎|
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
decomposition compare with exact solution at t=0.1
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Numerical solution for nonlinear telegraph equation 249
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.
II. Reliable Adomian decomposition method
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
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250 Hind Al-badrani et al.
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
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, see( Table(6) and (Fig.6)).
x 𝑢𝑎 𝑢𝑒 |𝑢𝑒 − 𝑢𝑎|
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.
x |𝑢𝑒 − 𝑢𝑎| at 𝑛 = 1
|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
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Numerical solution for nonlinear telegraph equation 251
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.
I. Standard Adomian decomposition method
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)).
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252 Hind Al-badrani et al.
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.
x |𝑢𝑒 − 𝑢𝑎| at 𝑛 = 1
|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
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Numerical solution for nonlinear telegraph equation 253
Figure8: The graph of exact solution and approximate ADM solution at n=1, n=3, n=5.
II. Reliable Adomian decomposition method
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
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254 Hind Al-badrani et al.
Figure9: The graph of exact solution and approximate by the modified Adomian
decomposition method at t=0.1, t=0.3, t=0.5.
Figure10: The graph of exact solution and approximate by the modified Adomian
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
x |𝑢𝑒 − 𝑢𝑎| at 𝑛 = 1
|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
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Numerical solution for nonlinear telegraph equation 255
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