https://doi.org/10.31449/inf.v47i10.5260 Informatica 47 (2023) 155–160 155 
Variational Iteration Method for Solving Electrocardiography 
Inverse Problem 
 
Ebtihal Sabah Al-Bayati, Ahmed Farooq Qasem 
Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq 
E-mail: ebtehal.21csp19@student.uomosul.edu.iq , ahmednumerical@uomosul.edu.iq 
Keywords: bidomain model, variational iteration method, electrocardiography inverse problem 
Received: October 7, 2023 
In this paper, the variational iteration method is proposed as a solution to an inverse problem in 
electrocardiography. The aim is to obtain approximate solutions to the model with the lowest error rate. 
This is crucial in determining the patient's heart activity and facilitating rapid medical intervention. The 
main challenges in accurately computing clinically relevant maps of cardiac depolarization stem from 
the ill-posed nature of the continuous problem and the presence of noise in the data. To tackle these 
difficulties, we have developed regularizing iterative algorithms based on domain decomposition 
techniques. These algorithms reformulate the inverse problem into several cases, depending on the 
solution area. This formulation has enabled us to establish a new stopping criterion that is more 
responsive and accurately reflects the behavior of the error on the non-accessible part of the boundary. 
The numerical results obtained through the variational iteration method demonstrate that the proposed 
approach effectively captures the error behavior on the non-accessible boundary. An additional 
advantage of these approaches is their ability to reduce execution time through their parallelized 
versions. Thus, we have successfully demonstrated the effectiveness of these methods in terms of quality 
(accuracy of approximation) and quantitative aspects (computation cost). 
Povzetek: Članek predstavlja variacijsko iteracijsko metodo za reševanje inverznih problemov v 
elektrokardiografiji z nizko stopnjo napake. 
 
1 Introduction 
Due to the importance of mathematical simulation in 
science, several mathematical models were built for the 
electrical activity in the heart, including the bidomian 
model [1,2,3]. The heart muscle is anisotropic, the effects 
of these properties on heart tissue are discussed using the 
bidomain electrophoresis model, it is a microscopic 
model that describes the bioelectric behavior of the heart 
[4], The mathematical model of electrophysiology of the 
heart is a dynamic system that quantitatively describes 
the electrical processes occurring in the tissues of the 
heart [5]. The bidomain model is a system of nonlinear 
and convergent ordinary differential equations (ODEs), 
which used to model the gastric electrophysiology [6]. 
The use of the bidomain model is necessary for the 
correct modeling of the defibrillation response, and the 
error arising from the numerical solution of bidomain 
models is relatively small; various types of boundary 
condition can be imposed that can take into account the 
leakage of current to the surrounding tissues other than 
the myocardium [7]. In order for the model results to be 
useful, it is necessary to obtain accurate input of the 
models [8,9], Therefore, we will try to be precise in 
choosing the initial conditions to get better results and by 
using the Variational Iteration Method (VIM) which 
described and used to give approximate solutions for 
some non-linear problems [10,11,12], This method 
relying on The initial condition and based on Taylor 
series approximation, which applied as center difference, 
front difference, and back difference schemes[13], This 
method has the ability to reduce the volume of 
calculations and easily overcome the difficulty of the 
perturbation method[14]; Later, this method was used 
based on Laplace transformations to get the solution in 
the form of a series [15] , there is no need to linearize or 
treat the nonlinear terms to apply the method on 
nonlinear differential equations [16]. 
2  Bidoman model    
It is a mathematical way to descript the electrical 
activity of heart which developed since the invention of 
the ECG measuring by Einthoren [17].  It is used to 
simulate cardiac electrical tissue computationally and to 
study the stimulation of cardiac tissue and defibrillation 
of the heart. The heart domain denoted by Ω and the 
conductivity tensor of intercellular and extracellular by 

156 Informatica 47 (2023) 155–160 E.S. Al-Bayati et al. 
𝜎 𝑖 
,𝜎 𝑒 , let 𝜈 𝑚 be atransmembrane potential and 𝑢 𝑖 
, 
𝑢 𝑒 be electric potential of  intercellular and extracellular 
respectively , such that  𝜈 𝑚 =𝑢 𝑖 
−𝑢 𝑒 . The bidomain 
model in two dimensional defined as: 
{
 
 
 
 
 
 
𝐴 𝑚 (𝐶 𝑚 
𝜕 𝜈 𝑚 𝜕𝑡
 +𝐼 𝑖𝑜𝑛
(𝜈 𝑚 ,𝑤 ))−𝑑𝑖𝑣 (𝜎 𝑖 
∇𝜈 𝑚 )=𝑑𝑖𝑣 ( 𝜎 𝑖 
∇ 𝑢 𝑒 )   𝑖𝑛 Ω×]0,𝑇 [ 
𝑑𝑖𝑣 (( 𝜎 𝑖 
+𝜎 𝑒 )∇ 𝑢 𝑒 )=−𝑑𝑖𝑣 ( 𝜎 𝑖 
∇𝜈 𝑚 )                 𝑖𝑛 Ω×]0,𝑇 [ 
𝜕 𝑡 𝑤 +𝑔 (𝜈 𝑚 ,𝑤 )=0                                               𝑖𝑛 Ω×]0,𝑇 [
𝜎 𝑖 
𝛻 𝜈 𝑚 .𝑛 =−𝜎 𝑖 
𝛻 𝑢 𝑒 .𝑛                                       𝑜𝑛𝛴 ×]0,𝑇 [
( 𝜎 𝑖 
+𝜎 𝑒 )𝛻 𝑢 𝑒 .𝑛 =− 𝜎 𝑖 
𝛻 𝜈 𝑚 .𝑛                         𝑜𝑛𝛴 ×]0,𝑇 [ 
(1)  
Where 𝐴 𝑚 is defined to be the surface to volume ratio 
of the cell membrane, 𝐶 𝑚 the membrane 
capacitance, 𝐼 𝑖𝑜𝑛 is the current due to the ion exchange. 
𝑔 (𝜈 𝑚 ,𝑤 ) is a function of fields having the same 
dimension of  𝑤 which is the concentrations of different 
chemical, 𝑛 is the outward unit normal on Ω, and 𝛴 =
Г
𝑒𝑛𝑑𝑜 ∪ Г
𝑒𝑝𝑖 , Г
𝑒𝑛𝑑𝑜 internal boundaries in the rated 
endocardia and Г
𝑒𝑝𝑖  an outer boundaries, epicedial 
[18]. 
 
Figure 1: Cardiac field distribution 
 
3 Variational iteration method 
The variational iteration method (VIM) proposed to find 
a numerical solution for linear and nonlinear equations 
by (He ,1998,1999) [10] , then it was developed to solve 
various types of integral equations and ordinary, partial 
and fractional differential equations. 
consider the general equation 
 
 𝐿 𝑢 +𝑁 𝑢 =𝑔 (𝑥 )    (2) 
 
where 𝐿 is linear operator , 𝑁 is the nonlinear 
operator and 𝑔 is inhomogeneous term. Assuming 𝑢 0
(𝑥 ) 
is the initial solution, then the variational iteration 
technique becomes:   
 
𝑢 𝑛 +1
(𝑥 0
)=𝑢 𝑛 (𝑥 0
)+∫ 𝜆 (𝜉 ) (
𝑥 0
0
𝐿 𝑢 𝑛 +𝑁 𝑢 𝑛 −
𝑔 )𝑑𝑥             (3) 
 
With 𝑢 0
 as initial condition ,we can rewrite this 
equation as follow  
 
𝑢 𝑛 +1
(𝑥 )=𝑢 𝑛 (𝑥 )+∫ 𝜆 (𝜉 ) (
𝑥 0
𝐿 𝑢 𝑛 (𝜉 )+𝑁 𝑢 𝑛 (𝜉 )−
𝑔 (𝜉 ))𝑑𝜉        (4) 
 
Where 𝜆 is the general Lagrange multiplier [15] 
𝜆 (𝜉 )=
(−1)
𝑚 (𝑚 −1)!
 (𝜉 −𝑥 )
𝑚 −1
   (5) 
 
𝑚 the highest order of the derivative, the final 
solution is u(𝑥 )= lim
n→∞
𝑢 𝑛 (𝑥 ). 
4 Application 
The system (1) will be resolved using VIM in two cases: 
Case (1): The system (1) can be solved using VIM 
when 𝜆 =1 𝑖𝑛 Ω×]0,𝑇 [ 
 
{
 
 
 
 
 
 
 
 𝑣 𝑚 𝑛 +1
=𝑣 𝑚 𝑛 +𝜆 ∫
(𝐴 𝑚 (𝐶 𝑚 
𝜕 𝑣 𝑚 𝑛 𝜕𝑡
 +𝐼 𝑖𝑜𝑛
(𝑣 𝑚 𝑛 ,𝑤 𝑚 𝑛 ))−𝑑𝑖𝑣 (𝜎 𝑖 
𝛻 𝑣 𝑚 𝑛 )−𝑑𝑖𝑣 ( 𝜎 𝑖 
𝛻 𝑢 𝑒 𝑛 ))𝑑𝑡 
 
𝑡 0
𝑢 𝑒 𝑛 +1
=𝑢 𝑒 𝑛 +(𝑢 𝑒 𝑛 (t,0,y)+
𝜕 (𝑢 𝑒 𝑛 (t,0,y)
𝜕𝑥
 x )−
1
(𝜎 𝑖 +𝜎 𝑒 )
 (𝜎 𝑖 𝑣 𝑚 𝑛 −𝜎 𝑖 𝑣 𝑚 𝑛 (t,0,y)−𝜎 𝑖 𝜕 𝑣 𝑚 𝑛 (t,0,y)
𝜕𝑥
 x)
+
𝜆 (𝜎 𝑖 +𝜎 𝑒 )
 ∫ ∫ (−𝜎 𝑖 
𝜕 2
𝑣 𝑚 𝑛 𝜕𝑦
2
−(𝜎 𝑖 +𝜎 𝑒 ) 
𝜕 2
𝑢 𝑒 𝑛 𝜕𝑦
2
 )𝑑𝑥 𝑑𝑥
𝑥 0
𝑥 0
𝑤 𝑚 𝑛 +1
=𝑤 𝑚 𝑛 +𝜆 ∫
𝜕 𝑤 𝑚 𝑛 𝜕𝑡
+𝑔 (𝑣 𝑚 𝑛 ,𝑤 𝑚 𝑛 ))𝑑𝑡 
𝑡 0
      
         (6) 
 
with the initial conditions [18] 
 
𝑣 𝑚 0
(0,x,y)=𝑣 0
(𝑋 ),𝑤 𝑚 0
(0,x,y)=𝑤 0
(𝑋 )        (7) 
𝐼 𝑖𝑜𝑛 (𝜈 𝑚 ,𝑤 )=
𝑊 𝑇 𝑖𝑛
 𝑉 2
(𝑉 −1)− 
𝑉 𝑇 𝑜𝑢𝑡 , 
  𝑔 (𝜈 𝑚 ,𝑤 )={
𝑤 −1
𝑇 𝑜𝑝𝑒𝑛 𝑖𝑓 𝑣 ≤𝑣 𝑔𝑎𝑡𝑒 𝑤 𝑇 𝑐𝑙𝑜𝑠𝑒𝑑 𝑖𝑓 𝑣 >𝑣 𝑔𝑎𝑡𝑒                  (8) 
 
Where 𝑇 𝑖𝑛
≤𝑇 𝑜𝑢𝑡 ≤𝑇 𝑜𝑝𝑒𝑛 ,𝑇 𝑐𝑙𝑜𝑠𝑒𝑑  and  0<
𝑣 𝑔𝑎𝑡𝑒 <1 , are given positive constants. 
 
Using iterative formula (6) with the initial conditions: 
Z
{
 
 
 
 
𝑢 𝑒 0
=𝑥 6
+3𝑥 4
𝑦 2
+3𝑥 2
𝑦 4
+𝑦 6
−𝑥 2
𝑦 3
−3𝑥 4
−6𝑥 2
𝑦 2
−3𝑦 4
+3𝑥 2
+3𝑦 2
−1
𝑢 𝑖 0
=2𝑥 6
+6𝑥 4
𝑦 2
+6𝑥 2
𝑦 4
+2𝑦 6
−2𝑥 2
𝑦 3
−6𝑥 4
−12𝑥 2
𝑦 2
−6𝑦 4
+6𝑥 2
+6𝑦 2
−2
𝑣 𝑚 0
=𝑥 6
+3𝑥 4
𝑦 2
+3𝑥 2
𝑦 4
+𝑦 6
−𝑥 2
𝑦 3
−3𝑥 4
−6𝑥 2
𝑦 2
−3𝑦 4
+3𝑥 2
+3𝑦 2
−1
𝑤 𝑚 0
=
1
(𝑣 𝑚 0
)
2
 (𝑣 𝑚 0
−1)
(9) 
 
Then the first iterative formula 
 
{
 
 
 
 
 
 
𝑢 𝑒 1
=ℎ+
𝑟𝑡
𝑠 𝑢 𝑖 1
=2ℎ+2
𝑟𝑡
𝑠 𝑣 𝑚 1
=h+
𝑟𝑡
𝑠 𝑤 𝑚 1
=
1
ℎ+(
𝑟𝑡
𝑠 )
2
(ℎ−1+
𝑟𝑡
𝑠 )
   (10) 
 
where               
   
{
 
 
 
 
𝑟 =𝑥 6
𝑇 𝑖𝑛
+3𝑥 4
𝑦 2
𝑇 𝑖𝑛
+3𝑥 2
𝑦 4
𝑇 𝑖𝑛
+𝑦 6
𝑇 𝑖𝑛
−𝑥 2
𝑦 3
𝑇 𝑖𝑛
−3𝑥 4
𝑇 𝑖𝑛
−6𝑥 2
𝑦 2
𝑇 𝑖𝑛
−3𝑦 4
𝑇 𝑖𝑛
+3𝑥 2
𝑇 𝑖𝑛
+3𝑦 2
𝑇 𝑖𝑛
+𝑇 𝑜𝑢𝑡 𝑇 𝑖𝑛
−𝑇 𝑖𝑛
−𝑇 𝑜𝑢𝑡
𝑠 =𝑇 𝑖 𝑛 𝐶 𝑚 𝑇 𝑜𝑢𝑡
ℎ=𝑥 6
+3𝑥 4
𝑦 2
+3𝑥 2
𝑦 4
+𝑦 6
−𝑥 2
𝑦 3
−3𝑥 4
−6𝑥 2
𝑦 2
−3𝑦 4
+3𝑥 2
+3𝑦 2
−1
j=26.4t−13.2𝑥 2
𝑦𝑡 +79.2𝑥 4
𝑡 +79.2𝑦 4
𝑡 +158.4𝑥 2
𝑦 2
𝑡 −105.6𝑦 2
𝑡 −4.4𝑦 3
𝑡 −105.6𝑥 2
𝑡 
 
and at the boundary conditions  
 
{
 
 
 
 
[
2ℎ 2ℎ−12(2𝑦 2
+6𝑥 2
+
1
3
𝑦 3
−𝑦 4
−6𝑥 2
𝑦 2
−5𝑥 4
−1)𝑡 …
2ℎ 2ℎ−12(6𝑦 2
+2𝑥 2
+𝑥 2
𝑦 −5𝑦 4
−6𝑥 2
𝑦 2
−𝑥 4
−1)𝑡 …
]
[
ℎ
2
⁄ (
ℎ
2
⁄ )+(−66𝑥 4
−79.2𝑥 2
𝑦 2
−13.2𝑦 4
+4.4𝑦 3
+79.2𝑥 2
+26.4𝑦 2
−13.2)𝑡 …
ℎ
2
⁄ (
ℎ
2
⁄ )+(−13.2𝑥 4
−79.2𝑥 2
𝑦 2
−66𝑦 4
+13.2𝑥 2
𝑦 +26.4𝑥 2
+79.2𝑦 2
−13.2)𝑡 …
]
  
 

Variational Iteration Method for Solving Electrocardiograph… Informatica 47 (2023) 155–160 157 
The general form 
  
{
 
 
 
 
𝑉 𝑉𝐼𝑀
=𝑥 6
+3𝑥 4
𝑦 2
+3𝑥 2
𝑦 4
+𝑦 6
−𝑥 2
𝑦 3
−3𝑥 4
−6𝑥 2
𝑦 2
−3𝑦 4
+3𝑥 2
+3𝑦 2
−1+ℎ+
𝑟𝑡
𝑠 +⋯
𝑢 𝑉𝐼𝑀
=𝑥 6
+3𝑥 4
𝑦 2
+3𝑥 2
𝑦 4
+𝑦 6
−𝑥 2
𝑦 3
−3𝑥 4
−6𝑥 2
𝑦 2
−3𝑦 4
+3𝑥 2
+3𝑦 2
−1+ℎ+𝑗 +⋯
𝑤 𝑉𝐼𝑀
=
1
ℎ
2
+(ℎ−1)
+
1
ℎ+(
𝑟𝑡
𝑠 )
2
(ℎ−1+
𝑟𝑡
𝑠 )
+⋯
         
(11) 
 
The exact solution for system (1) with initial conditions 
(9) and the values 𝑇 𝑜𝑝𝑒𝑛 =1 ; 𝑇 𝑐𝑙𝑜𝑠𝑒𝑑 =1 ; 𝑇 𝑖𝑛
=1 ; 
𝑇 𝑜𝑢𝑡 =1 ;  𝐶 𝑚 =1; 𝜎 𝑖 
=1 ;𝜎 𝑒 
=0.2  is 
𝑉𝑚 =((𝑥 2
+𝑦 2
−1)
3
−𝑥 2
𝑦 3
)𝑒 𝑡   (12) 
 
Table 1:  Demonstrates comparison absolute error of 
numerical solution using VIM with the exact solution 
when  𝑛 =3,𝑥 ∈[−1.5,1.5];𝑦 ∈[−1.5,1.5];𝑡 =0.25. 
𝑽 𝒆𝒙𝒂 𝒄 𝒕 𝑽 𝑽𝑰𝑴 𝑬𝒓𝒓𝒐𝒓𝒆 𝑽𝑰𝑴 
55.77659477 55.77296875 3.626020×10
−3
 
19.00253201 19.00459321 2.061200×10
−3
 
4.944311443 4.948546790 4.235347×10
−3
 
0.9159855499 0.9208438900 4.858340×10
−3
 
0.1857372348 0.1907085100 4.971275×10
−3
 
0.1036097736 0.09859375000 5.016024×10
−3
 
0.6066703134 0.6015764900 5.093823×10
−3
 
1.040166975 1.035006110 5.160865×10
−3
 
1.040189078 1.035028210 5.160868×10
−3
 
0.6120414440 0.6069467900 5.094654×10
−3
 
 
 
 
 
 
 
 
 
Figure 2:  Solution path for system (1) using VIM 
 
We notice from the table (1) and figure (2), that the 
numerical solution approaches the exact solution based 
on the parameters found using system (11), where the 
absolute error rate reaches 10
−3
 with few iterations n=3, 
Whereas the interval that is approved is [−1.5,1.5]x[−1.5, 
1.5]. 
 
Case 2: If we Segmentation the area of heart into an 
internal and an external area only, after substituting the 
second equation into the first and third equations of the 
system (1), we get the following system 𝑖𝑛 Ω×]0,𝑇 [ : 
 
{
𝐴 𝑚 (𝐶 𝑚 
𝜕 𝜈 𝑚 𝜕𝑡
 +𝐼 𝑖𝑜𝑛
(𝜈 𝑚 ,𝑤 ))+𝑑𝑖𝑣 ( 𝜎 𝑒 
𝛻 𝑢 𝑒 )=0   𝑖𝑛 Ω×]0,𝑇 [
𝜕𝑤
𝜕𝑡
+𝑔 (𝜈 𝑚 ,𝑤 )=0                      𝑖𝑛 Ω×]0,𝑇 [
               
(13) 
 
and on the boundaries of the heart in figure (1), the 
system (1) will be as: 
 
{
𝐴 𝑚 (𝐶 𝑚 
𝜕 𝜈 𝑚 𝜕𝑡
 +𝐼 𝑖𝑜𝑛 (𝜈 𝑚 ,𝑤 ))=0         𝑜𝑛 𝛴 ×]0,𝑇 [
𝜕𝑤
𝜕𝑡
+𝑔 (𝜈 𝑚 ,𝑤 )=0                                   𝑜𝑛 𝛴 ×]0,𝑇 [
                     
(14) 
 
The systems (13) and (14) can be solved via VIM with 
the initial conditions (7) and: 
𝐼 𝑖𝑜𝑛 (𝜈 𝑚 ,𝑤 )=𝑔 (𝜈 𝑚 ,𝑤 )=𝜈 𝑚 −𝑤  (15) 
Then 
 
{
 
 
 
 
𝑣 𝑚 𝑛 +1
=𝑣 𝑚 𝑛 +𝜆 ∫
(𝐴 𝑚 (𝐶 𝑚 
𝜕 𝑣 𝑚 𝑛 𝜕𝑡
 +𝐼 𝑖𝑜𝑛
(𝑣 𝑚 𝑛 ,𝑤 𝑚 𝑛 )) −𝑑𝑖𝑣 (𝜎 𝑒 
𝛻 𝑢 𝑒 𝑛 ))𝑑𝑡 
 
𝑡 0
𝑖𝑛 Ω×]0,𝑇 [
𝑤 𝑚 𝑛 +1
=𝑤 𝑚 𝑛 +𝜆 ∫
𝜕 𝑤 𝑚 𝑛 𝜕𝑡
+𝑔 (𝑣 𝑚 𝑛 ,𝑤 𝑚 𝑛 ))𝑑𝑡 
𝑡 0
                                                   𝑖𝑛 Ω×]0,𝑇 [
 
(16) 

158 Informatica 47 (2023) 155–160 E.S. Al-Bayati et al. 
At the boundary conditions, the system (14) becomes: 
 
{
𝑣 𝑚 𝑛 +1
=𝑣 𝑚 𝑛 +𝜆 ∫
(𝐴 𝑚 (𝐶 𝑚 
𝜕 𝑣 𝑚 𝑛 𝜕𝑡
 +𝐼 𝑖𝑜𝑛
(𝑣 𝑚 𝑛 ,𝑤 𝑚 𝑛 )))𝑑𝑡 
 
𝑡 0
𝑖𝑛 𝛴 ×]0,𝑇 [
𝑤 𝑚 𝑛 +1
=𝑤 𝑚 𝑛 +𝜆 ∫
𝜕 𝑤 𝑚 𝑛 𝜕𝑡
+𝑔 (𝑣 𝑚 𝑛 ,𝑤 𝑚 𝑛 ))𝑑𝑡 
𝑡 0
                       𝑖𝑛 𝛴 ×]0,𝑇 [
 
(17) 
 
The exact solution for system (1) with initial conditions 
(9) and the values 𝑇 𝑜𝑝𝑒𝑛 =1 ; 𝑇 𝑐𝑙𝑜𝑠𝑒𝑑 =1 ; 𝑇 𝑖𝑛
=1 ; 
𝑇 𝑜𝑢𝑡 =1 ;  𝐶 𝑚 =1; 𝜎 𝑖 
=1 ;𝜎 𝑒 
=0  is 
𝑉𝑚 =((𝑥 2
+𝑦 2
−1)
3
−𝑥 2
𝑦 3
)𝑒 𝑡  (18) 
 
At the boundaries of the heart, the boundary conditions 
towards 𝑥 are used when 𝑥 =𝑥 0
 with 𝑦 =𝑦 0
, 𝑦 = 𝑦 𝐿 in 
system (17), while for the boundaries of the heart 
towards 𝑦 , then the boundary conditions towards 𝑦 are 
used when 𝑦 =𝑦 0
 with 𝑥 =𝑥 0
, 𝑥 = 𝑥 𝐿 in system (17). 
 
Table 2:   Demonstrates comparison absolute error of 
numerical solution using VIM with the exact solution 
when  𝑛 =3,𝑥 ∈[−1.5,1.5];𝑦 ∈[−1.5,1.5];𝑡 =0.25. 
𝑽 𝒆𝒙𝒂𝒄𝒕 𝑽 𝑽𝑰𝑴 𝑬𝒓𝒓𝒐𝒓𝒆 𝑽𝑰𝑴 
45.66601344 45.66601758 3.941900×10
−4
 
15.55795734 15.55795875 1.343000×10
−4
 
4.048059831 4.048060198 3.494200×10
−5
 
0.7499455391 0.7499456071 6.473700×10
−6
 
0.1520687861 0.1520688000 1.312600×10
−6
 
0.0848285079 0.0848285156 7.322400×10
−7
 
0.4966996426 0.4966996875 4.287600×10
−6
 
0.8516166907 0.8516167679 7.351300×10
−6
 
0.8516347875 0.8516348647 7.351300×10
−6
 
0.5010971524 0.5010971978 4.325500×10
−6
 
 
 
 
 
 
Figure 3:  Solutions path for system (1) at the 
boundary conditions using VIM. 
 
We note from the table (2), the approximation solution is 
significantly improved when dividing the domain 
solution area of the heart, bringing the absolute error to 
10
−6
 . 
 
5 Conclusion 
Using simulation to form a system of differential 
equations and solve them numerically offers the potential 
to reduce the error rate in disease diagnosis and avoid 
excessive treatment dosages. This approach is widely 
applicable across different fields, including 
electrocardiogram simulation. The objective is to develop 
a non-invasive imaging technique that utilizes electrical 
activation patterns on the surface of the heart to provide 
meaningful diagnostic information. The variational 
iteration method provides a solution to inverse Cauchy 
problems and can be implemented within a general 
framework. Its utilization is based on robust domain 
decomposition techniques, which have proven to be 
effective in various applications. Building on this 
foundation, we aim to further explore the application of 
other domain decomposition methods to solve the 
specific inverse problem at hand.   
 
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