The variational iteration method and Adomian decomposition method are applied to solve the FitzHughNagumo (FN) equations. The two algorithms are illustrated by studying an initial value problem. The obtained results show that only few terms are required to deduce approximated solutions which are found to be accurate and efficient.
The pioneering work of Hodgkin and Huxley [
The exact solution of this system is given by:
A numerical scheme for FN equations [
We introduce the main points of each of the two methods, where details can be found in [
The VIM is the general Lagrange method
where
The above analysis yields the following theorem.
The VIM solution of the partial differential equation (
Applying the inverse operator
We next decompose the unknown function
The above analysis yields the following theorem
The ADM solution of the partial differential equation (
We solve the FN equations using the two methods VIM and ADM.
Consider the FN equations in the form
Then the VIM formulae take the forms
Hence, the Lagrange multipliers are
Substituting these values of Lagrange multipliers into the functional correction (
We start with initial approximations as follows
The VIM produces the solutions
Consider the FN equations in the following form:
The ADM assumes that the unknown functions
such that
Then the first iterations are
and so on.
The ADM yields the solutions
We discuss the solutions of the FN equations using the two considered VIM and ADM methods.
Solve the FN equations (
Comparison between the exact and approximate (VIM) solutions for the FN equations at time






−7.561  0.000865955  0.000865955  0.0000831702  0.0000831702 
−3.561  0.3  0.3  0.0288134  0.0288134 
−0.561  0.662823  0.662823  0.28156  0.28156 
1.439  0.130074  0.130074  0.414489  0.414489 
3.439  −0.25639  −0.25639  0.382524  0.382524 
8.439  −0.215232  −0.215232  0.193024  0.193024 
16.439  −0.0616494  −0.0616494  16.439  16.439 
22.439  −0.023095  −0.023095  0.0197795  0.0197795 
48.439  −0.000325199  −0.000325199  0.000278481  0.000278481 
The approximated solutions for
Consider the same problems and use the ADM with the same initial conditions and use the technique discussed in Section
Comparison between the exact solutions and approximation solutions (ADM) for FN equations at time = 5.





 








































48.439  −0.000325199  −0.000325199  0.000278481  0.000278481 
The approximation solutions
The results listed in Table
The maximum errors of our suggested methods VIM and ADM.
Time  VIM  ADM  

Max. errors for 
Max. errors for 
Max. errors for 
Max. errors for 

2.0 




4.0 




6.0 




Now we show a comparison between our schemes and other methods as shown in Table
Comparison between VIM, ADM, and other methods by maximum errors.
Method 



Finite difference  
CN 

0.189 
Hopscotch [ 

0.0506 
Collocation method  
Quadratic [ 

0.138 
Cubic [ 

0.12 
VIM 

0.000316341 
ADM 

0.000316341 
It is clear that the suggested methods for solving FN equation are the best methods than all other methods. Also all other methods give the solution as a discrete solution but our methods give the solution as a function
In this paper the solutions for the FN equations using VIM and ADM methods have been generated. All numerical results obtained using few terms of the VIM and ADM show very good agreement with the exact solutions. Comparing our results with those of previous several methods shows that the considered techniques are more reliable, powerful, and promising mathematical tools. We believe that the accuracy of the VIM and ADM recommend it to be much wider applicability and also we find that the VIM more accurate than ADM.
This paper has been supported by the research support program, King Khalid University, Saudi Arabia, Grand no. KKUCOM11001.