JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 286529 10.1155/2013/286529 286529 Research Article New Iterative Method Based on Laplace Decomposition Algorithm Widatalla Sabir 1, 2 Liu M. Z. 1 Natesan Srinivasan 1 Department of Mathematics Harbin Institute of Technology Harbin 150001 China hit.edu.cn 2 Department of Physics and Mathematics College of Education, Sinnar University Singa 107 Sudan univ.edu.sd 2013 14 3 2013 2013 26 11 2012 28 01 2013 29 01 2013 2013 Copyright © 2013 Sabir Widatalla and M. Z. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce a new form of Laplace decomposition algorithm (LDA). By this form a new iterative method was achieved in which there is no need to calculate Adomian polynomials, which require so much computational time for higher-order approximations. We have implemented this method for the solutions of different types of nonlinear pantograph equations to support the proposed analysis.

1. Introduction

Since 2001, Laplace decomposition algorithm (LDA) has been one of the reliable mathematical methods for obtaining exact or numerical approximation solutions for a wide range of nonlinear problems.

The Laplace decomposition algorithm was developed by Khuri in  to solve a class of nonlinear differential equations. The basic idea in Laplace decomposition algorithm, which is a combined form of the Laplace transform method with the Adomian decomposition method, was developed to solve nonlinear problems. The disadvantage of the Laplace decomposition algorithm is that the solution procedure for calculation of Adomian polynomials is complex and difficult and takes a lot of computational time for higher-order approximations as pointed out by many researchers .

The Laplace decomposition algorithm plays an important role in modern scientific research for solving various kinds of nonlinear models; for example, Laplace decomposition algorithm was used in  to solve a model for HIV infection of CD4+T cells; LDA was employed in  to solve Abel's second kind singular integral equations. In  it was used to solve boundary Layer equation.

Even though there has been some developments in the LDA , the use of Adomian polynomials has not been abandoned.

The main purpose of this paper is to introduce a new iterative method based on Laplace decomposition algorithm procedure without the need to compute Adomian polynomials and thus reduce the size of calculations needed.

The scheme is tested for some classes of pantograph equations, and the results demonstrate reliability and efficiency of the proposed method.

2. Basic Idea of LDA and the New Technique

To illustrate the basic concept of Laplace decomposition algorithm, we consider the following general nonlinear model: (1)L(m)u=Nu+Ru+g(t), where L(m) is the highest order derivative, R and N are linear and nonlinear operators, respectively, and g(t) is an inhomogeneous term.

Applying the Laplace transform (denoted throughout this paper by ) to both sides of (1) and using given conditions, we obtain (2)[u(t)]=(s)+𝒢(s)+s-m[Nu]+s-m[Ru], where (3)(s)=r=0m-1s-(r+1)u(r)(0),𝒢(s)=s-m[g(t)].

The Laplace decomposition algorithm defines the unknown function u(t) by an infinite series as (4)u(t)=n=0un(t), where the components un(t) will be determined recurrently. Substituting this infinite series into (2) and using the linearity of Laplace transform lead to (5)n=0[un(t)]=(s)+𝒢(s)+s-mn=0[Nun]+s-mn=0[Run].

Also the nonlinear functions Nun are defined by infinite series as follows: (6)Nun=n=0An, where An are the Adomian polynomials , depending only on u0,u1,,un, and defined by (7)An=1n!dndλn[N(i=0nλiui)]λ=0,n=0,1,2,.

Substituting (6) into (5), we get (8)n=0[un(t)]=(s)+𝒢(s)+s-mn=0[An]+s-mn=0[Run].

The Laplace decomposition algorithm presents the recurrence relation as (9)u0(t)=-1[(s)+𝒢(s)],un+1(t)=-1s-m[An]+-1s-m[Run],n=0,1,2,.

Applying the inverse Laplace transform to (8) leads to (10)n=0[un(t)]=-1[(s)+𝒢(s)]+(-1s-m)×n=0An+(-1s-m)n=0Run.

Equation (10) can be written as (11)u0(t)+u1(t)+u2(t)++un(t)+=-1[(s)+𝒢(s)]+(-1s-m)[A0+A1+A2+]+(-1s-m)[R(u0+u1+u2+)].

By taking vn=i=0nui and (11), the following procedure can be constructed: (12)v0=-1[(s)+𝒢(s)]vn+1=v0+(-1s-m)i=0n[Ai]+(-1s-m)(R(vn)),n=0,1,2,.

Consequently, the exact solution may be obtained by (13)u(t)=limnvn=limni=0nui.

For the analytic nonlinear operator N, we can write (14)limnN(i=0nui)=limni=0nAi.

By considering,(12) and (14) can be reconstructed as (15)v^0=-1[(s)+𝒢(s)]v^n+1=v^0+(-1s-m)(N(v^n))+(-1s-m)(R(v^n)),n=0,1,2,.

Equation (15) is a new iteration method based on LDA. The advantage of this scheme is that there is no need to calculate Adomian polynomials.

3. Test Problems

In this section we will apply our scheme to different types of pantograph equations.

Example 1 (see [<xref ref-type="bibr" rid="B1">13</xref>]).

Consider the following nonlinear pantograph differential equation: (16)u(t)=14u(t)+u(t2)(1-u(t2)),u(0)=1.

The exact solution of this problem u(t)=1/2+1/2cos(2t/4)+2/2sin(2t/4).

Based on the iteration formula (15), we get (17)v0=1,vn+1=v0+(-1s-1)(14vn(t))+(-1s-1)(vn(t2)(1-vn(t2))).

Thus, we get (18)v1=1+t4,v2=1+t4-t232-t3192,v3=1+t4-t232-t3192+t43072+t549152-t6589824-t716515072,v4=1+t4-t232-t3192+t43072+t530720-1.801E-6t6-9.461E-8t7+3.548E-9t8+small term.

Knowing that the exact solution of this exampleisgiven in , (19)u(t)=12+12cos(24t)+22sin(24t)=1+t4-t232-t3192+t43072+t530720-1.356E-6t6-9.688E-8t7+3.028E-9t8+.

We see that the approximation solutions obtained by the present method have good agreement with the exact solution of this problem.

In Table 1 the absolute errors of the present method and standard LDA for n=5 are compared.

Figure 1 compares the numerical errors En(ti)=|u(ti)-un(ti)| for n=1,2,3, and 4 obtained by (a) the present method and (b) the standard LDA. This plot indicates that the series solution obtained by the present method converges faster than the standard LDA.

Comparison of the absolute errors for Example 1.

t Exact solution Standard LDA Present method
n = 5 n = 5
0.2 1.048708864495296 8.803 E - 11 1.900 E - 13
0.6 1.137669652809217 6.594 E - 8 4.123 E - 10
0.8 1.177476957990361 3.753 E - 7 3.077 E - 9
1.0 1.213898289556670 1.450 E - 6 1.461 E - 8
2.0 1.339484983470599 9.834 E - 5 1.838 E - 6
4.0 1.276427846019296 6.905 E - 3 2.299 E - 4
6.0 8.410651664398781 E - 1 8.393 E - 2 3.960 E - 3
8.0 2.421580540539403 E - 1 4.911 E - 1 3.122 E - 2
10 2.331110590497953 E - 1 1.910 1.643 E - 1

The error functions En(ti) for Example 1: (a) present method and (b) standard LDA.

Example 2.

Consider the following nonlinear pantograph integrodifferential equation (PIDE): (20)u′′(t)=35u(56t)-3u(16t)-t+0t[u(12x)u(13x)+2u2(12x)]dx,u(0)=0,u(0)=1.

For this example we write iteration formula (15) as (21)v^0=t-t36,v^n+1=v0+(-1s-2)×(35vn(5t/6)-3vn(t6))+(-1s-2)×0t[vn(x2)vn(x3)+2vn2(x2)]dx,n=0,1,2,,

and the first n terms are (22)v1=t-13!t3+15!t5-67272160t7+3115676416t9,v2=t-13!t3+15!t5-17!t7+6716325395793920t9+,v3=t-13!t3+15!t5-17!t7+19t9-2307198620058050560t11+vn=t-13!t3+15!t5-17!t7+19t9-+(-1)n(2n+1)!t(2n+1), which gives the exact solution by u(t)=limnvn=sin(t).

In Table 2 we compare the absolute errors of the present method for n=3 and the standard LDA for n=3 and the differential transform method described in  with nine terms.

Figure 2 displays the numerical errors obtained by the present method and the standard LDA.

Comparison of the absolute errors for Example 2.

t Exact solution DTM  Standard LDA Present method
N = 9 n = 3 n = 3
0.2 1.986693307950612 E - 1 5.2735 E - 16 7.3960 E - 18 6.3679 E - 18
0.6 5.646424733950354 E - 1 9.0678 E - 11 3.8301 E - 13 2.2537 E - 13
0.8 7.173560908995228 E - 1 2.1431 E - 9 8.2825 E - 12 6.0382 E - 12
1.0 8.414709848078965 E - 1 2.4892 E - 8 8.4781 E - 11 1.0616 E - 11
2.0 9.092974268256817 E - 1 5.0015 E - 5 1.5123 E - 8 2.4971 E - 7
4.0 - 7.568024953079282 E - 1 9.5074 E - 2 1.2344 E - 3 5.5720 E - 4
6.0 - 2.794154981989259 E - 1 7.3079 1.8828 E - 1 5.2101 E - 2
8.0 9.893582466233818 E - 1 148.51 3.6889 1.2615

The error functions En(ti) for Example 2: (a) present method and (b) standard LDA.

Example 3.

Consider the following nonlinear PIDE: (23)u(t)+(12t-2)u(t)-20tu2(12x)dx=1,u(0)=0, which has the exact solution u(t)=tet. The iteration form of (15) for this example is (24)v^0=t,v^n+1=v0-(-1s-1)(t2-2)vn+2(-1s-1)×(0tvn2(x2)dx),n=0,1,2,.

We obtain the following successive approximations: (25)v1=t+t2-t36+t424,v2=t+t2+t32-t46+7t5120+O(6),v3=t+t2+t32+t46-11t5120+t624+O(7),v4=t+t2+t32+t46+t524-13t6360+103t75040+O(8).

Note that the exact solution of this example is (26)u(t)=tet=t+t2+t32+t46+t524+t6120+t7720+.

In Table 3 we compare the absolute errors of the present method for n=4 and the standard LDA for n=4 and the differential transform method described in  with four terms.

Figure 3 displays the numerical errors obtained by the present method and the standard LDA.

Comparison of the absolute errors for Example 3.

t Exact solution DTM  Standard LDA Present method
N = 4 n = 4 n = 4
0.2 2.442805516320340 E - 1 1.388 E - 5 2.6127 E - 6 2.6093 E - 6
0.4 5.967298790565082 E - 1 4.632 E - 4 1.5387 E - 4 1.5303 E - 4
0.6 1.093271280234305 3.671 E - 3 1.6166 E - 3 1.5961 E - 3
0.8 1.780432742793974 1.617 E - 2 8.4028 E - 3 8.2072 E - 3
1.0 2.718281828459046 5.162 E - 2 2.9756 E - 2 2.8645 E - 2
2.0 14.77811219786130 2.111 1.4168 1.1919
3.0 60.25661076956300 21.25 14.341 9.6958

The error functions En(ti) for Example 3: (a) present method and (b) standard LDA.

Example 4.

Consider a system of multipantograph equations: (27)u1(t)=-u1(t)-e-tcos(t2)u2(t2)-2e-(3/4)tcos(t2)sin(t4)u1(t4),u2(t)=etu12(t2)-u22(t2),u1(0)=1,u2(0)=0.

We can adapt (15) to solve this system as follows: (28)v10(t)=1,v20(t)=0,v1(j+1)=v10-(-1s-1)×(v1j(t)+e-tcos(t2)v2j(t2)+2e-3t/4cos(t2)sin(t4)v1j(t4)),v2(j+1)=v20+(-1s-1)×(etv1j2(t2)-v2j2(t2)),j=0,1,2,.

Table 4 gives the absolute errors Evij=|ui-vij|,i=1,2,j=1,2,3 of the present method. The table clearly indicates that when we increase the truncation limit n, we have less error.

Table 5 summarizes the CPU times needed to obtain the first three components of the series solutions pertaining to the four above-mentioned examples by the present method and the standard LDA. The CPU time analysis was conducted on a personal computer with a 3.77 GHz processor and 4 GB of RAM using MATLAB 7.10.

The absolute errors for Example 4.

t Exact solution Present method
u 1 = e - t cos ( t ) E v 11 E v 12 E v 13
0.2 8.024106 E - 1 1.144 E - 2 4.432 E - 4 1.900 E - 5
0.4 6.174056 E - 1 4.990 E - 2 4.274 E - 3 3.656 E - 4
0.6 4.529538 E - 1 4.185 E - 1 1.643 E - 2 2.119 E - 3
0.8 3.130505 E - 1 2.171 E - 1 4.274 E - 2 7.420 E - 3
1.0 1.987661 E - 1 3.437 E - 1 8.925 E - 2 1.960 E - 2

u 2 = sin ( t ) E v 21 E v 22 E v 23

0.2 1.986693 E - 1 2.273 E - 2 5.174 E - 4 1.670 E - 5
0.4 3.894183 E - 1 1.024 E - 1 5.840 E - 3 1.790 E - 4
0.6 5.646425 E - 1 2.575 E - 1 2.630 E - 2 3.282 E - 4
0.8 7.173561 E - 1 5.082 E - 1 8.022 E - 2 1.276 E - 3
1.0 8.414710 E - 1 8.768 E - 1 1.965 E - 1 1.015 E - 2

The CPU time analysis of the present method and the standard LDA for obtaining the first three components of Examples 14.

Solution method The required CPU time [in seconds]
Example 1 Example 2 Example 3 Example 4
Present method 1.1716 2.0121 1.5608 3.0301
Standard LDA 1.1872 2.4046 1.9201 3.3431
4. Conclusion

In this work, we have presented a new iteration method based on the Laplace decomposition algorithm.

The advantage of the new method is that it does not require Adomian polynomials and thus reduces the calculation size.

The new iterative method has been employed to solve different classes of nonlinear pantograph equations, in which the results obtained are in close agreement with the exact solutions.

The convergence of this method is the subject of ongoing research.

Acknowledgments

The authors wish to thank the referees for valuable comments. The research was supported by the NSF of China no. 11071050.

Widatalla S. Iterative methods for solving nonlinear pantograph equations [Ph.D. thesis] 2013 Harbin Institute of Technology Khuri S. A. A Laplace decomposition algorithm applied to a class of nonlinear differential equations Journal of Applied Mathematics 2001 1 4 141 155 10.1155/S1110757X01000183 MR1884973 ZBL0996.65068 Mishra H. K. Nagar A. K. He-Laplace method for linear and nonlinear partial differential equations Journal of Applied Mathematics 2012 2012 16 180315 MR2948169 ZBL1251.65146 10.1155/2012/180315 Saadatmandi A. Dehghan M. Variational iteration method for solving a generalized pantograph equation Computers & Mathematics with Applications 2009 58 11-12 2190 2196 10.1016/j.camwa.2009.03.017 MR2557348 ZBL1189.65172 Ghorbani A. Beyond Adomian polynomials: he polynomials Chaos, Solitons and Fractals 2009 39 3 1486 1492 10.1016/j.chaos.2007.06.034 MR2512946 ZBL1197.65061 Ongun M. Y. The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4+T cells Mathematical and Computer Modelling 2011 53 5-6 597 603 10.1016/j.mcm.2010.09.009 MR2769431 ZBL1217.65164 Khan M. Gondal M. A. A reliable treatment of Abel's second kind singular integral equations Applied Mathematics Letters 2012 25 11 1666 1670 10.1016/j.aml.2012.01.034 MR2957732 ZBL1253.65202 Khan Y. An effective modification of the Laplace decomposition method for nonlinear equations International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 1373 1376 Khan Y. Faraz N. Application of modified Laplace decomposition method for solving boundary layer equation Journal of King Saud University 2011 23 115 119 Hussain M. Khan M. Modified Laplace decomposition method Applied Mathematical Sciences 2010 4 33-36 1769 1783 MR2653179 ZBL1208.35006 Widatalla S. Koroma M. A. Approximation algorithm for a system of pantograph equations Journal of Applied Mathematics 2012 2012 9 714681 10.1155/2012/714681 MR2904532 ZBL1244.65122 Adomian G. Solving Frontier Problems of Physics: The Decomposition Method 1994 Boston, Mass, USA Kluwer Academic Publishers xiv+352 MR1282283 Iserles A. On nonlinear delay differential equations Transactions of the American Mathematical Society 1994 344 1 441 477 10.2307/2154725 MR1225574 ZBL0804.34065