Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 4s (2025) 

 

95 https://internationalpubls.com 

Analytical and Computational Investigation of Non-Steady State 

Reaction and Kinetics at Spherical Ultramicroelectrodes Concerning 

Conducting Polymer Modification using Homotopy Perturbation 

Method 

 

A.Uma1 , R.Swaminathan1* 
1*PG & Research Department of Mathematics, Vidhyaa Giri College of Arts and Science (Affiliated to Alagappa 

University) Puduvayal-630108, TamilNadu, India. 

1*Email: swaminathanmath@gmail.com 

 

Article History: 

Received: 14-09-2024 

Revised: 19-11-2024 

Accepted: 28-11-2024 

Abstract:  

In this work,  mathematical modelling of non-steady state reaction and kinetics at spherical  

ultramicroelectrodes within conducting polymer modification is considered. The main 

objective of this work is to propose a new analytical formulation for the system of nonlinear 

non-steady state reaction diffusion equation in spherical ultramicroelectrodes.  Employing 

Homotopy Perturbation Method, the concentrations of species, mediator and current may all 

be obtained analytically for all conceivable experimental results of the parameter. The 

accumulated analytical outcomes are analyzed with numerical simulations and implemented 

to investigate various parameters. By comparing the analytical solution with numerical 

findings, the accuracy of the method is presented. In order to better understand the system 

dynamics, a numerical simulation of the issue is also provided through Matlab Software. The 

new analytical results contribute to optimizing the consistency of this model. These novel 

approaches produce a compact set of analytical approximations that possess straightforward 

to compute and verify as well. 

Keywords: Nonlinear equation, Non-steady state, reaction diffusion process, Spherical 

ultramicroelectrodes, Homotopy Perturbation Method. 

 

1.Introduction 

In the electrochemical process and the kinetics of rapid reactions, Ultra microelectrodes serve as a 

practical aid for interpreting the system's functioning. Last 10 years, many voltammetric investigations 

have employed UME with tip diameters of the order of a micrometre along with ten percent of 𝜇𝑚1. It 

demonstrates a time-independent current reaction with a spherical or disc shape, which has both 

practical as well as theoretical benefits2-4. Numerous sensing applications might use modified 

microelectrodes within Polymer layer5-6. Fleischmann et al.7 discussed the electrochemical 

characteristics of spherical ultramicroelectrodes. Special structures like fibers and embedded 

reticulated foams  are needed for the moderate commercialization of reactions at microelectrodes. The 

use of the unique benefits of microelectrodes for the synthesis such as the simplicity of setup and the 

enlargement of the fluid region when the support electrolyte is absent. Even so, certain electrode and 

cell designs are needed, such the utilizing electrodes in three dimensions. Rebouilat et al.8 have issued 

an empirical investigation of the equilibrium current response anticipated for a Conducting spherical 

ultramicroelectrodes with polymer modification beneath amperometric circumstances. Albery et al.9  



Communications on Applied Nonlinear Analysis 

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have made significant contributions to the theoretical explanation of facilitated electron transport at 

electroactive polymer films that have been formed on large-scale electrode surfaces. In contrast, the 

later strategies yield identical outcomes but are distinct in specifics.  

Within Conducting Polymer modified ultramicroelectrodes, Anitha et al.10 worked at the solution of 

coupled time-varying nonlinear reaction-diffusion equations. Sentamarai et al.11 and Yogeshwari et 

al.12 determine the substrate concentration and mediated profiles within UME by adopting the 

Variational Iteration Method and Taylor Series Method, respectively.  

The present work intends to utilize HPM to construct an analytical formulation for the concentration 

of mediator and current based on non-steady state reaction-diffusion equations at an electrode surface 

within spherical Ultramicroelectrodes along with Conducting Polymer Modification. 

2.Mathematical formulation 

A mathematical model has determined the relationship between the substrate reaction and diffusion in 

the electro conductive polymer. We will simply give a quick summary because a detailed analysis of 

the underlying assumptions and physical depiction of the issue has already been done by  Fleischmann 

et al.7and  Rebouillat et al.8 . The governing equations are derived while considering the following 

assumptions. 

1. The substrate will diffuse spherically in the thin film, and the mediator and substrate species 

will intent chemically in a bio molecular way. 

2. Consider the deposited film to be a uniform medium. 

3. A partition and diffusion coefficient cause the substrate to divide into layers. 

The equation regarding non-steady state  reaction-diffusion within the polymer layer could be phrased 

in the following manner. 

   𝔇𝒮1

𝜕2𝑠1

𝜕𝑢2
 + 

2𝔇𝒮1

𝓊
 
𝜕𝑠1

𝜕𝑢
 - 𝜌𝑠1𝑠2 = 

𝜕𝑠1

𝜕𝑡
    (2.1) 

𝔇𝒮2

𝜕2𝑠2

𝜕𝑢2
 + 

2𝔇𝒮2

𝓊
 
𝜕𝑠2

𝜕𝑢
 - 

𝜌

2
𝑠1𝑠2 = 

𝜕𝑠2

𝜕𝑡
    (2.2) 

𝔇ℳ
𝜕2𝑚

𝜕𝑢2
 + 

2𝔇ℳ

𝓊
 
𝜕𝑚

𝜕𝑢
 + 𝜌𝑠1𝑠2 = 

𝜕𝑚

𝜕𝑡
    (2.3) 

Following are the boundary conditions that describe the problem 

At 𝑡 = 0, 𝑠1 = 0;  𝑠2 = 0; 𝑚 = 0.       (2.4) 

At 𝑢 = 0, 
𝑑𝑠1

𝑑𝑢
= 0 ; 

𝑑𝑠2

𝑑𝑢
= 0; 

𝑑𝑚

𝑑𝑢
= 0.       (2.5) 

At 𝑢 = 𝑟, 𝑠1 = 𝕂𝑠1
𝑇;  𝑠2 = 𝕂𝑠2

𝑇; 𝑚 = 𝑚𝑇      (2.6) 

The Net flux is represented as  

 𝒞𝔍 = 𝔇𝒮1
(

𝜕𝑠1

𝜕𝑢
)𝑢=𝑟        (2.7) 



Communications on Applied Nonlinear Analysis 

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The results of the final evaluation should be presented in dimensionless parameters before we progress 

on to a complete mathematical review of the boundary value problem described in the equations (2.1)-

(2.7) 

We introduce the Non-Dimensional Parameters as, 

𝜁 =  
𝑠1

𝕂𝑠1𝑇 ; 𝜂 =
𝑠2

𝕂𝑠2𝑇 ; 𝜃 =  
𝑚

𝑚𝑇 ; 𝑋 =
𝑢

𝑟
 ; 𝑇 =

𝐷𝑡

𝑟2
;  

𝜒𝒮1
=

𝜌𝑚𝑇𝑟2

𝔇𝒮1

 ;𝜒𝒮2
=

𝜌𝑚𝑇𝑟2

𝔇𝒮2

; 𝜒ℳ =
𝜌𝕂𝑠1

∞𝑟2

𝔇ℳ
      (2.8) 

The reaction diffusion parameters 𝜒𝒮1
, 𝜒𝒮2

 and 𝜒ℳ are used to measure the correlation between the 

chemical reaction rate and the amount of charge percolation or substrate diffusion.  

The system of non-steady state nonlinear reaction-diffusion equation can be written as  

 
𝜕2𝜁(𝑋)

𝜕𝑋2
 + 

2

𝑋
 
𝜕𝜁(𝑋)

𝜕𝑋
 - 𝜒𝒮1

𝜁(𝑋)𝜃(𝑋) = 
𝜕𝜁

𝜕𝑇
      (2.9) 

 
𝜕2𝜂(𝑋)

𝜕𝑋2
 + 

2

𝑋
 
𝜕𝜂(𝑋)

𝜕𝑋
 - 

𝜒𝒮2

2
 𝜁(𝑋)𝜃(𝑋) = 

𝜕𝜂

𝜕𝑇
      (2.10) 

𝜕2𝜃(𝑋)

𝜕𝑋2
 + 

2

𝑋
 
𝜕𝜃(𝑋)

𝜕𝑋
 + 𝜒ℳ𝜁(𝑋)𝜃(𝑋) = 

𝜕𝜃

𝜕𝑇
      (2.11) 

The appropriate boundaries were expressed as 

 𝜁(𝑋) = 𝜂(𝑋) = 𝜃(𝑋)= 0  when T = 0     (2.12) 

    
𝜕𝜁(𝑋)

𝜕𝑋
=

𝜕𝜂(𝑋)

𝜕𝑋
=  

𝜕𝜃(𝑋)

𝜕𝑋
 = 0 when X = 0     (2.13) 

𝜁(𝑋) = 𝜂(𝑋) = 𝜃(𝑋)= 1  when X = 1     (2.14) 

The following equation represents the normalized current response  

  Δ =  −𝐷(
𝜕𝜁(𝑋)

𝜕𝑋
)𝑋=1            (2.15) 

3. Analytical solution of the concentrations using Homotopy Perturbation Method  

Many writers have focused on researching the solution of nonlinear equations during the past few 

decades using a variety of techniques, including the Homotopy perturbation Method14-17, Taylor Series 

Method18-20, Akbari Ganji Method21-28, Variational iteration method29,30.  Homotopy perturbation 

Method is conservative in its efficiency, applicability, and accuracy. The non-steady state nonlinear 

equations (2.9) through (2.15) may be solved using this technique to provide the analytical formulation 

for the concentration of species and mediator. 

We construct the homotopy for (2.9) – (2.11) as,  

(1-p)[ 
𝜕𝜁

𝜕𝑇
 −  

𝜕2𝜁

𝜕𝑋2
 −  

2

𝑋
 

𝜕𝜁

𝜕𝑋
 + 𝜒𝒮1

𝜁𝜃] + p[ 
𝜕𝜁

𝜕𝑇
 −  

𝜕2𝜁

𝜕𝑋2
 − 

2

𝑋
 

𝜕𝜁

𝜕𝑋
 +  𝜒𝒮1

𝜁𝜃] = 0       (3.1) 

Equation (1) - (3) has an analytical solution as,  

 𝜁 = 𝜁0 + 𝑝𝜁1 + 𝑝2𝜁2 +. . . . .  



Communications on Applied Nonlinear Analysis 

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 𝜃 = 𝜃0 + 𝑝𝜃1 + 𝑝2𝜃2 +. . . . .       (3.2) 

Substituting (3.2) in (3.1) and comparing the coefficients of p, we get  

 𝑝0 = 
𝜕𝜁0

𝜕𝑇
 − 

𝜕2𝜁0

𝜕𝑋2
 −  

2

𝑋
 
𝜕𝜁0

𝜕𝑋
−  𝜒𝒮1

𝜁0 = 0     (3.3) 

In Laplace plane, (3.3) can be written as,  

 
𝑑2𝜁0

𝑑𝑋2
+  

2

𝑋
 
𝑑𝜁0

𝑑𝑋
− (s + 𝜒𝒮1

)𝜁0 = 0      (3.4) 

Subject to the boundary conditions, 

 
𝑑𝜁0

𝑑𝑋
(0) = 0 ;  𝜁0(1) =

1

𝑠
       (3.5) 

By reduction of order, we consider the equation  

𝑑2𝜁0

𝑑𝑋2
+  𝑃 

𝑑𝜁0

𝑑𝑋
− Q𝜁0 = 𝑅       (3.6) 

Comparing (3.4) and (3.6),  

 P = 
2

𝑋
 ; Q = −(s + 𝜒𝒮1

) ; R = 0      (3.7) 

Consider 𝜁0 = 𝜃𝜂         (3.8) 

The general solution of (21) represented as  2
𝑑𝜃

𝑑𝑋
 + P𝜃 = 0    (3.9) 

Then 𝜃 =
1

𝑋
          (3.10) 

 Equation (3.6) and (3.7) reduces to 

 𝜂′′ −(s + 𝜒𝒮1
)𝜂 = 0        (3.11) 

Integrating (3.11) twice, we get   

 𝜂 = A𝑒(√s+𝜒𝒮1
)𝑋

+ B𝑒(−√s+𝜒𝒮1
)𝑋

      (3.12) 

Substituting (3.12) and (3.10) in (3.8) and then using boundary conditions, we obtain  

 𝜁0(𝑋, 𝑠) =
1

𝑋
[
sinh (√s+𝜒𝒮1

𝑋)

𝑠.sinh (√s+𝜒𝒮1
)
]       (3.13) 

Employing inverse Laplace transform, the concentration of species 𝑠1 can be represented as, 

  𝜁 (𝑋, 𝑇) =
sinh (√𝜒𝒮1

𝑋)

𝑋sinh (√𝜒𝒮1
)

+
2𝜋

𝑋
∑ [

𝑛(−1)𝑛+1sin (𝑛𝜋𝑋)𝑒
−(𝑛2𝜋2+𝜒𝒮1

)𝑇

(𝑛2𝜋2+𝜒𝒮1
)

]∞
𝑛=1   (3.14) 

3.1. Relation between the concentration of species 𝒔𝟏 and 𝒔𝟐 

Utilizing equation (2.9) and (2.10), we derive the equation as, 

𝜕2

𝜕𝑋2
(

𝜁𝜒𝒮2

2
− 𝜂𝜒𝒮1

) + 
2

𝑋
 

𝜕

𝜕𝑋
(

𝜁𝜒𝒮2

2
− 𝜂𝜒𝒮1

) −
𝜕

𝜕𝑇
(

𝜁𝜒𝒮2

2
− 𝜂𝜒𝒮1

) = 0   (3.15) 



Communications on Applied Nonlinear Analysis 

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Consider  K = 
𝜁𝜒𝒮2

2
− 𝜂𝜒𝒮1

        (3.16) 

Equation (3.15) can be written as   
𝜕2𝐾

𝜕𝑋2
+  

2

𝑋
 
𝜕𝐾

𝜕𝑋
−

𝜕𝐾

𝜕𝑇
= 0     (3.17) 

Subject to the boundary conditions,  

At T = 0, K = 0 ;          (3.18) 

At X = 0, 
𝜕𝐾

𝜕𝑋
= 0;          (3.19) 

At X= 1, K = 
𝜒𝒮2

2
− 𝜒𝒮1

        (3.20) 

Applying Laplace transform for equation (3.17) and then solving by the use of the boundary conditions 

(3.18)-(3.20), we get  

 K(X, T) = (
𝜒𝒮2

2
− 𝜒𝒮1

)[1 + 2𝜋 ∑ 𝑛(−1)𝑛+1sin (𝑛𝜋𝑋)𝑒−(𝑛2𝜋2𝑇)∞
𝑛=1 ] (3.21) 

By (3.16), we obtain the concentration of species 𝑠2 as expressed as  

𝜂 (𝑋, 𝑇) =
𝜒𝒮2

[ 𝜁 (𝑋,𝑇)]

2𝜒𝒮1

− (
𝜒𝒮2

2𝜒𝒮1

− 1) (1 +
2𝜋

𝑋
∑ [

𝑛(−1)𝑛+1sin (𝑛𝜋𝑋)𝑒−(𝑛2𝜋2𝑇)

𝑛2𝜋2
]∞

𝑛=1 ) (3.22) 

3.2. Relation between the concentration of species 𝒔𝟏 and mediator 

Employing equation (2.9) and (2.11), we express the equation as, 

𝜕2

𝜕𝑋2
(𝜁𝜒ℳ − 𝜃𝜒𝒮1

) + 
2

𝑋
 

𝜕

𝜕𝑋
(𝜁𝜒ℳ − 𝜃𝜒𝒮1

) −
𝜕

𝜕𝑇
(𝜁𝜒ℳ − 𝜃𝜒𝒮1

)  = 0   (3.23) 

Take  L = 𝜁𝜒ℳ − 𝜃𝜒𝒮1
        (3.24) 

Equation (3.15) becomes   
𝜕2𝐿

𝜕𝑋2
+  

2

𝑋
 

𝜕𝐿

𝜕𝑋
−

𝜕𝐿

𝜕𝑇
= 0      (3.25) 

The corresponding  boundary conditions are  

At T = 0, L = 0 ;          (3.26) 

At X = 0, 
𝜕𝐿

𝜕𝑋
= 0;          (3.27) 

At X= 1, L = 𝜒ℳ − 𝜒𝒮1
        (3.28) 

Using Laplace transform in the equation (3.25) and solving by the use of the boundary conditions 

(3.26)-(3.28) and then applying inverse Laplace formula, we get  

 L(X, T) = (𝜒ℳ − 𝜒𝒮1
)[1 + 2𝜋 ∑ 𝑛(−1)𝑛+1sin (𝑛𝜋𝑋)𝑒−(𝑛2𝜋2𝑇)∞

𝑛=1 ] (3.29) 

By (3.24), we get the mediator concentration is expressed as  

𝜃(𝑋, 𝑇) = (
𝜒ℳ

𝜒𝒮1

) (1 +
2𝜋

𝑋
∑ [

𝑛(−1)𝑛+1 sin(𝑛𝜋𝑋)𝑒
−(𝑛2𝜋2𝑇)

𝑛2𝜋2
]∞

𝑛=1 ) − [
(𝜒ℳ[ 𝜁 (𝑋,𝑇)]

𝜒𝒮1

]  (3.30) 

The normalized flux can be derived as,  



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Δ =  −𝐷(
𝜕𝜁(𝑋)

𝜕𝑋
)𝑋=1   =

−𝐷. 𝑠1
𝑇

𝑟
(1 − √𝜒𝒮1

𝑐𝑜𝑡ℎ√𝜒𝒮1
− 2∑ [𝑒−(𝑛2𝜋2𝑇)]∞

𝑛=1  

     +2𝜋2 ∑ [
𝑛2(−1)𝑛+1𝑒

−(𝑛2𝜋2+𝜒𝒮1
)𝑇

(𝑛2𝜋2+𝜒𝒮1
)

]∞
𝑛=1 )  (3.31) 

4. Specifying cases 

As X→ 0, the concentration of species and mediator closely reaches at the center of the conducting 

polymer and it can be expressed as,  

 𝜁 (0, 𝑇) = 𝜁 (0,∞) + 2𝜋2 ∑ [
𝑛2(−1)𝑛+1𝑒

−(𝑛2𝜋2+𝜒𝒮1
)𝑇

(𝑛2𝜋2+𝜒𝒮1
)

]∞
𝑛=1    (4.1) 

𝜂 (0, 𝑇) =
𝜒𝒮2

[ 𝜁 (0,𝑇)]

2𝜒𝒮1

− (
𝜒𝒮2

2𝜒𝒮1

− 1)(1 + 2∑ [(−1)𝑛+1𝑒−(𝑛2𝜋2𝑇)]∞
𝑛=1 ) (4.2) 

𝜃(0, 𝑇) = (
𝜒ℳ

𝜒𝒮1

) (1 + 2∑ [(−1)𝑛+1𝑒−(𝑛2𝜋2𝑇)]∞
𝑛=1 ) − [

(𝜒ℳ[ 𝜁 (0,𝑇)]

𝜒𝒮1

]  (4.3) 

As T→ 0, the concentration of substrates and mediator closely relative to the boundary of the 

conducting polymer and the analytical expression of the concentrations becomes 

 𝜁 (𝑋, 0) =
sinh (√𝜒𝒮1

𝑋)

𝑋sinh (√𝜒𝒮1
)

+
2𝜋

𝑋
∑ [

𝑛(−1)𝑛+1sin (𝑛𝜋𝑋)

(𝑛2𝜋2+𝜒𝒮1
)

]∞
𝑛=1     (4.4) 

𝜂 (𝑋, 0) =
𝜒𝒮2

[ 𝜁 (𝑋,0)]

2𝜒𝒮1

− (
𝜒𝒮2

2𝜒𝒮1

− 1) (1 +
2𝜋

𝑋
∑ [

𝑛(−1)𝑛+1sin (𝑛𝜋𝑋)

𝑛2𝜋2
]∞

𝑛=1 ) (4.5) 

𝜃(𝑋, 0) = (
𝜒ℳ

𝜒𝒮1

) (1 +
2𝜋

𝑋
∑ [

𝑛(−1)𝑛+1 sin(𝑛𝜋𝑋)

𝑛2𝜋2
]∞

𝑛=1 ) − [
(𝜒ℳ[ 𝜁 (𝑋,0)]

𝜒𝒮1

]  (4.6 

As T→ ∞ in the above non-steady state analytical expression for the concentrations of species 𝑠1, 𝑠2 

and mediator becomes steady state and concentrations can be written as  

  𝜁 (𝑋, 𝑇) =
sinh (√𝜒𝒮1

𝑋)

𝑋sinh (√𝜒𝒮1
)
       (4.7) 

      𝜂 (𝑋, 𝑇) =
𝜒𝒮2

[ 𝜁 (𝑋,𝑇)]

2𝜒𝒮1

− (
𝜒𝒮2

2𝜒𝒮1

− 1)     (4.8) 

  𝜃(𝑋, 𝑇) = (
𝜒ℳ

𝜒𝒮1

) − [
(𝜒ℳ[ 𝜁 (𝑋,𝑇)]

𝜒𝒮1

]     (4.9)  

5. Numerical simulation  

The non-dimensional form of equations (2.9)-(2.11) that relate to boundary conditions (2.12)-(2.14) 

were numerically solved in order to test the accuracy of the HPM solution. Graphical comparisons 

between our analytical findings and numerical results demonstrate the effectiveness of the current 

approach. The new analytical results with a dimensionless concentration of substrate s1 and substrate 

s2 in its numerical representation are compared in Tables 1 and 2. It offers an acceptable agreement for 

each parameter setting that is being compared. The highest typical error of 0.06% in the substrate s1 

and 0.3% in the species s2 separates the prior numerical result from the latest analytical outcome 

derived by using HPM Method. 



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Table 1. Comparison among the new analytical results with numerical results for the species s1 concentration for different reaction diffusion 

parameter values. 

Species s1 Concentration 

T= 1 and  𝜒𝒮1
 = 1 T= 1 and  𝜒𝒮1

 = 5 T= 1 and  𝜒𝒮1
 = 10 

T Numerical 

result 

Eqn. (2.9) 

Analytical 

result using 

HPM 

Eqn. (3.14) 

% of 

variation 

between 

(2.9) and 

(3.14) 

Numerical 

result 

Eqn. (2.9) 

Analytical 

result using 

HPM 

Eqn. (3.14) 

% of 

variation 

between 

(2.9) and 

(3.14) 

Numerical 

result 

Eqn. (2.9) 

Analytical 

result using 

HPM 

Eqn. (3.14) 

% of 

variation 

between 

(2.9) and 

(3.14) 

0.1 0.8580 0.8523 0.0057 0.4992 0.4875 0.0117 0.2809 0.2727 0.0082 

0.2 0.8626 0.8566 0.006 0.5098 0.4997 0.0101 0.2948 0.2864 0.0084 

0.3 0.8691 0.8637 0.0054 0.5295 0.5205 0.009 0.3160 0.3103 0.0057 

0.4 0.8785 0.8738 0.0047 0.5584 0.5505 0.0079 0.3506 0.3456 0.005 

0.5 0.8908 0.8868 0.004 0.5973 0.5907 0.0066 0.3989 0.3947 0.0042 

0.6 0.9060 0.9029 0.0031 0.6473 0.6421 0.0052 0.4639 0.4607 0.0032 

0.7 0.9244 0.9221 0.0023 0.7102 0.7065 0.0037 0.5497 0.5475 0.0022 

0.8 0.9460 0.9446 0.0014 0.7881 0.7859 0.0022 0.6622 0.6611 0.0011 

0.9 0.9712 0.9705 0.0007 0.8836 0.8826 0.001 0.8088 0.8085 0.0003 

Average Error 

% 

 0.0037   0.0064   0.0043 

 

 

 

 

Table 2. Deviation of New analytical solution with Numerical findings of   concentration of species s2 for various values of reaction diffusion 

parameters 



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Species s2 Concentration 

T= 1 ,  𝜒𝒮1
 = 1 , 𝜒𝒮2

 = 0.1 T= 1 ,  𝜒𝒮1
 = 1 , 𝜒𝒮2

 = 1 T= 1 ,  𝜒𝒮1
 = 1 , 𝜒𝒮2

 = 4 

T Numerical 

result 

Eqn. (2.10) 

Analytical 

result using 

HPM 

Eqn. (3.22) 

% of 

variation 

between 

(2.10) & 

(3.22) 

Numerical 

result 

Eqn. (2.10) 

Analytical 

result using 

HPM 

Eqn. (3.22) 

% of 

variation 

between 

(2.10) & 

(3.22) 

Numerical 

result 

Eqn. (2.10) 

Analytical 

result using 

HPM 

Eqn. (3.22) 

% of 

variation 

between 

(2.10) & 

(3.22) 

0.1 0.9926 0.9926 0.0000 0.9294 0.9261 0.0033 0.7528 0.7046 0.0482 

0.2 0.9929 0.9928 0.0001 0.9316 0.9283 0.0033 0.7595 0.7132 0.0463 

0.3 0.9932 0.9932 0.0000 0.9350 0.9318 0.0032 0.7709 0.7274 0.0435 

0.4 0.9937 0.9936 0.0001 0.9404 0.9369 0.0035 0.7871 0.7476 0.0395 

0.5 0.9944 0.9943 0.0001 0.9461 0.9434 0.0027 0.8059 0.7736 0.0323 

0.6 0.9952 0.9951 0.0001 0.9538 0.9515 0.0023 0.8348 0.8058 0.029 

0.7 0.9962 0.9961 0.0001 0.9632 0.9611 0.0021 0.8637 0.8442 0.0195 

0.8 0.9973 0.9972 0.0001 0.9742 0.9723 0.0019 0.9019 0.8892 0.0127 

0.9 0.9987 0.9986 0.0001 0.9869 0.9853 0.0016 0.9470 0.9410 0.006 

Average Error 

% 

 0.0000   0.0027   0.3078 

 



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6. Results and discussion  

The equations (3.14), (3.22) and (3.30) provides a newly developed analytical formulation of the 

concentration of species s1, species s2 and mediator in simple closed form respectively. The reaction 

rate constants and time affects the species concentrations . Simple new analytical formulae (3.31) 

describe how much normalized current is present.  

Figure 1.  Analytical and numerical evaluations of the solutions for various values of reaction diffusion 

parameter (a) 𝜒𝒮1
= 1, 3, 5, 10 and T = 1 (b) 𝜒𝒮2

= 1, 3, 5, 10 and for fixed T = 1, 𝜒𝒮1
= 1                            

(c) 𝜒ℳ = 3, 5, 7, 10 and for fixed T = 1, 𝜒𝒮1
= 1.  

  

 

The standardized species s1 concentration is shown in Figure 1(a) for various amounts of the diffusion 

parameter 𝜒𝒮1
. The dotted line indicates Analytical results and solid line indicate Numerical results. 

This graph demonstrates that for each value of 𝜒ℳ and  𝜒𝒮1
  that are lower or equal to 1, 𝜁 is 

approximately comparable to 1. As the concentration of substrate S1 goes down, 𝜒𝒮1
 increases. The 

concentration of species s1 reach the peak value at the large amount of the non-dimensional distance 

in the range of  X≥ 0.9.   



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Figure 1(b) shows that the concentration of substrate S2 for numerous values of  𝜒𝒮2
 depends on the 

constant value of  𝜒𝒮1
.The concentration slowly decreases whenever the diffusion parameter  𝜒𝒮2

 

increases. At the time T = 1, the concentration of species s2 reaches the constant state for very small 

amount of reaction rate constant 𝜒𝒮2
≤ 1.  

 The series of normalized concentration profiles for a mediator is present in Figure 1(c) for various 

values of the 𝜒ℳ and 𝜒𝒮1
 reaction diffusion parameters. All values of 𝜒ℳ and 𝜒𝒮1

 that are both less 

than or equal to 1 can be deduced that it is substantially equal to 1. As 𝜃 increases either 𝜒ℳ increases. 

   

Figure 2. Comparison of the solutions both analytically and numerically for different values of  (a) 

𝜒𝒮2
=  5, 10, 20, 30, 50 and for fixed T = 1, 𝜒𝒮1

= 10 (b) 𝜒ℳ = 3, 5, 10, 15 and for fixed T = 1, 𝜒𝒮1
=

10. The numerical solution is shown by solid line and the analytical finding is depicted by the dotted 

line. 

Figure 2(a) which states that  𝜂 quickly falls down when the diffusion parameter increases as well as 

the for large value of 𝜒𝒮1
≥ 10. The concentration slowly falls down and reaches the steady state for 

very large value of reaction rate constant 𝜒𝒮2
≥ 50. That is, the concentration of species s2 is inversely 

proportional to the reaction rate constant 𝜒𝒮2
. As the very large amount of non-dimensional distance 

at X = 1, the concentration attains its maximum value and the concentration falls for X≤ 1. According 

to the range X≤0.1, the mediator concentration is uniform. That is the inclined curve turned into the 

straight line. 

  Figure 2(b)  delivered that the concentration of mediator rises for all large values of diffusion 

parameter 𝜒ℳ. For the maximal value of reaction rate constant, the concentration is perpendicular to 

the dimensionless distance X.  The concentration of mediator approaches the stable state for the 

reaction rate constant 𝜒ℳ ≥ 10. The influence of the parameter 𝜒ℳ which is directionally proportional 

to the concentration of mediator.  The analytical and numerical values are coincide for increasing 

values of the reaction rate constant 𝜒ℳ.  

 



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Figure 3. The analytical solutions compare to the  numerical findings for different values of  (a) 𝜒𝒮2
=

 10, 20, 50, 100 and for fixed T = 1, 𝜒𝒮1
= 30 (b) 𝜒ℳ = 5, 10, 20, 30 and for fixed T = 1, 𝜒𝒮1

= 30. The 

solid line represents Numerical findings and dotted line indicates Analytical findings. 

Figure 3(a) illustrates the intricate relationship between the species s1 concentration and the reaction 

rate constant 𝜒𝒮2
 and for fixed 𝜒𝒮1

= 30. This representation shows that when the nondimensional 

distance increases, the concentration drops with a decreasing parameter 𝜒𝒮2
. The variation of the 

concentration profile also rises with an increase in total concentration. As a result, when 𝜒𝒮2
 is al 

lowest, the species concentration approaches zero. The relationship between the species diffusion 

parameter 𝜒𝒮2
 and the concentration is inversely correlated whereas the species diffusion coefficient is 

directly correlated.  

As seen in figure 3(b), the fluctuation of mediator concentration for various system characteristics is 

approximated using Eqn.(3.30) and the results are compared to numerical data. The rate of progress at 

which mediator is extracted from the film drops when the diffusion parameter for mediator 𝜒ℳ 

improves over the layer interface. It can be deduced that 𝜒ℳ is in reverse proportion to mediator 

concentration, meaning that when the diffusion parameter rises, the substrate diffusion coefficient 

declines or the layer thickness grows. 

  

Figure 4. Graph concentration profiles of species s1, s2 and mediator versus dimensionless time T.  

 Figure 4(a) and 4(b) represent the comparison of concentration of species s1, s2 and mediator 

with the various values of non-dimensional time. The concentration of  s1 and s2 falls down as the 



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dimensionless time increases. As well as the mediator concentration grows up for all greater amount 

of time. It depicts that the concentration of mediator inversely proportional to the species concentration 

depends on the increasing value time. 

 

Figure 5. Plot of Three-dimensional substrate concentration s1 versus dimensionless distance X and 

various values of diffusion parameter 𝜒𝒮1
. 

Figure 5 indicates the three-dimensional representation of concentration of  species for various 

diffusion parameter versus dimensionless distance. It evident that the dimensionless diffusion 

parameter increases, the concentration of species gradually decreases depends on the distance X. 

7. Conclusion  

The system of nonlinear reaction diffusion equations in the spherical ultramicroelectrodes alongside 

conducting polymer modification  at non-steady state  have been determined analytically in the present 

study. Homotopy Perturbation Method is used to achieve the closed analytical formulation of 

concentration of species, mediator and current.  The non-steady state current response is provided in 

intuitively by a novel analytical expression. The kinetic properties of the spherical ultramicroelectrodes 

will be discovered by the excellent analytical outcomes. These analytical  findings allow one to 

qualitatively evaluate the characteristics of spherical ultramicroelectrodes with polymer modification. 

Concerning other analytical procedures, this method is clear-cut, has a straightforward solution, and 

produces precise results. This technique can solve other boundary value issues in the physical and 

chemical sciences without difficulty. 

Nomenclature 

𝔇𝒮1
 Diffusion coefficient of species s1  𝜇𝑚2/𝑠 𝜒𝒮1

 Dimensionless diffusion parameter  for s1 

𝔇𝒮2
 Diffusion coefficient of species s2  𝜇𝑚2/𝑠 𝜒𝒮2

 Dimensionless diffusion parameter  for s2 

𝔇ℳ Diffusion coefficient of mediator 𝜇𝑚2/𝑠 𝜒ℳ Dimensionless reaction parameter  

 s1 Concentration of species 𝜇𝑚 𝜁 Dimensionless Concentration species s1 

 s2 Concentration of species 𝜇𝑚 𝜂 Dimensionless concentration of species  

s2 

𝑚 Concentration of oxidized  mediator 𝜇𝑚 𝜃 Dimensionless concentration of mediator  

𝜌 Biomolecular rate constants 𝑚𝑠 𝑋 Dimensionless distance  

𝑢 Distance from the electrode 𝜇𝑚 T Dimensionless Time 



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𝑚𝑇 Total concentration of mediator & 

Substrate 

𝑟 Layer thickness 

𝑠𝑇 Bulk concentration of substrate 𝜇𝑚 𝕂 Partition coefficient  

𝒞𝔍 Net flux Δ Dimensionless normalized current 

 

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