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 higherorder approximations. We have implemented this method for the solutions of different types of nonlinear pantograph equations to support the proposed analysis.
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 [
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 [
Even though there has been some developments in the LDA [
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.
To illustrate the basic concept of Laplace decomposition algorithm, we consider the following general nonlinear model:
Applying the Laplace transform (denoted throughout this paper by
The Laplace decomposition algorithm defines the unknown function
Also the nonlinear functions
Substituting (
The Laplace decomposition algorithm presents the recurrence relation as
Applying the inverse Laplace transform to (
Equation (
By taking
Consequently, the exact solution may be obtained by
For the analytic nonlinear operator
By considering,(
Equation (
In this section we will apply our scheme to different types of pantograph equations.
Consider the following nonlinear pantograph differential equation:
The exact solution of this problem
Based on the iteration formula (
Thus, we get
Knowing that the exact solution of this exampleisgiven in [
We see that the approximation solutions obtained by the present method have good agreement with the exact solution of this problem.
In Table
Figure
Comparison of the absolute errors for Example

Exact solution  Standard LDA  Present method 




0.2  1.048708864495296 


0.6  1.137669652809217 


0.8  1.177476957990361 


1.0  1.213898289556670 


2.0  1.339484983470599 


4.0  1.276427846019296 


6.0 



8.0 



10 

1.910 

The error functions
Consider the following nonlinear pantograph integrodifferential equation (PIDE):
For this example we write iteration formula (
and the first
In Table
Figure
Comparison of the absolute errors for Example

Exact solution  DTM [ 
Standard LDA  Present method 





0.2 




0.6 




0.8 




1.0 




2.0 




4.0 




6.0 

7.3079 


8.0 

148.51  3.6889  1.2615 
The error functions
Consider the following nonlinear PIDE:
We obtain the following successive approximations:
Note that the exact solution of this example is
In Table
Figure
Comparison of the absolute errors for Example

Exact solution  DTM [ 
Standard LDA  Present method 





0.2 




0.4 




0.6  1.093271280234305 



0.8  1.780432742793974 



1.0  2.718281828459046 



2.0  14.77811219786130  2.111  1.4168  1.1919 
3.0  60.25661076956300  21.25  14.341  9.6958 
The error functions
Consider a system of multipantograph equations:
We can adapt (
Table
Table
The absolute errors for Example

Exact solution  Present method  






0.2 




0.4 




0.6 




0.8 




1.0 




 





 
0.2 




0.4 




0.6 




0.8 




1.0 




The CPU time analysis of the present method and the standard LDA for obtaining the first three components of Examples
Solution method  The required CPU time [in seconds]  

Example 
Example 
Example 
Example 

Present method  1.1716  2.0121  1.5608  3.0301 
Standard LDA  1.1872  2.4046  1.9201  3.3431 
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.
The authors wish to thank the referees for valuable comments. The research was supported by the NSF of China no. 11071050.