Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9, 1471-1483 2025 Publisher: Learning Gate DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 9 July 2025; Revised: 19 August 2025; Accepted: 25 August 2025; Published: 23 September 2025 * Correspondence: meensphd@gmail.com Approximate solutions of transient reaction-diffusion equations for second- order regeneration at spherical microelectrodes via HPM Rajendran Nalini1, Athimoolam Meena1*, Lakshmanan Rajendran2, Mohammad Izadi3 1Department of Mathematics, Saraswathi Narayanan College, Madurai-625022, India; meensphd@gmail.com (A.M.), 2Department of Mathematics, AMET Deemed to be University, Kanathur, Chennai- 603112, India. 3Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran; izadi@uk.ac.ir (M.I.). Abstract: The purpose of this study is to investigate the transient behavior of a second-order regeneration reaction at spherical microelectrodes, where nonlinear reaction–diffusion dynamics govern the system response. The research is designed to capture the interplay between electrochemical electron transfer and the subsequent homogeneous chemical reaction that regenerates the electroactive species, a process of considerable importance in electrochemical sensors and catalytic systems. The methodology combines analytical and numerical approaches: the homotopy perturbation method (HPM) is applied to derive approximate solutions for non-steady-state concentrations and current responses, while numerical simulations are carried out using Scilab to ensure accuracy and reliability. The approach effectively manages the inherent nonlinearity of the governing equations, offering tractable expressions for system behavior. The findings demonstrate strong agreement between the HPM-based analytical solutions and numerical simulations, confirming the validity of the proposed approach. The study concludes that HPM is a robust and efficient tool for analyzing nonlinear electrochemical systems. The practical implications highlight its potential application in the modeling and optimization of electrochemical devices, such as biosensors, microelectrodes, and catalytic fuel cells, where transient dynamics and regeneration mechanisms play a critical role. Keywords: Electrochemical modelling, Homotopy perturbation method, Reaction–diffusion equations, Second-order regeneration kinetics, Spherical microelectrodes. 1. Introduction Transient reaction–diffusion processes at spherical microelectrodes are fundamental to understanding complex electrochemical systems, particularly those involving second-order regeneration reactions. These reactions couple electron transfer with homogeneous chemical transformations, producing nonlinear behavior that influences both concentration profiles and current responses. The spherical geometry offers distinct advantages, such as radial symmetry and enhanced mass transport, making it an ideal platform for probing microscale kinetics. Modeling such systems requires solving nonlinear partial differential equations that capture spatial and temporal variations in concentration. Developing reliable analytical and numerical solutions to these equations is critical for advancing electrochemical theory. Moreover, these studies have wide practical applications in biosensing, electrocatalysis, and microanalytical chemistry, where accurate transient modeling supports improved device performance and optimization. Electrode reaction mechanisms involving second-order homogeneous kinetics have been investigated using both numerical and analytical techniques. Molina and Laborda [1] proposed a generalized numerical scheme based on finite-difference equations to describe diffusion and kinetic processes across different time intervals. Eswari and Rajendran [2] applied the homotopy perturbation mailto:meensphd@gmail.com https://orcid.org/0009-0008-4793-7075 https://orcid.org/0000-0003-2312-6848 https://orcid.org/0000-0002-2003-4553 https://orcid.org/0000-0002-6116-4928 1472 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate method (HPM) to derive approximate analytical expressions for concentration and current in homogeneous catalytic reactions at spherical microelectrodes. Molina, et al. [3] examined the chromate ampero metric behavior of CE processes at spherical microelectrodes under fast chemical conditions, while Delmastro [4] presented the polarographic kinetic current theory for second-order regeneration reactions. In addition, EC processes at ultramicroelectrodes have been extensively studied through computational simulations [5-7] and analytical approximations [8, 9]. Rajendran and co-workers further contributed by reporting accurate analytical solutions for steady-state chronoamperometric currents in EC reactions at disc [10] spheroidal ultramicroelectrodes. Izadi et al. applied numerical techniques to solve the nonlinear equation [11-13]. Despite these advances, accurate analytical approximations for transient concentrations and kinetic currents in second-order regeneration reactions at spherical electrodes remain unavailable. This gap limits predictive modeling of such nonlinear electrochemical systems under time-dependent conditions. To address this challenge, the present study employs the homotopy perturbation method (HPM) to obtain approximate analytical solutions for both concentration distributions and current responses. The approach not only manages the inherent nonlinearity of the governing equations but also provides results that can be compared with numerical simulations for validation. By bridging this gap, the study aims to extend the analytical toolkit available for modeling transient electrochemical processes at spherical electrodes. 2. Mathematical Formulation The catalytic reaction scheme for second-order regeneration reaction at spherical electrodes can be written as follows: 𝑂 + 𝑛𝑒 ⇄ 𝑅 2𝑅 𝑘 → 𝑂 + 𝑍 The reaction procedure for the second-order regeneration process is shown in Figure 1. Figure 1. The reaction scheme for second-order regeneration reaction. 1473 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate The nonlinear initial–boundary value problem describing a second-order regeneration reaction at spherical electrodes can be expressed in dimensional form as [4]: 𝜕𝐶0(𝑟,𝑡) 𝜕𝑡 = 𝐷0 [ 𝜕2𝐶0(𝑟,𝑡) 𝜕𝑟2 + 2 𝑟 𝜕𝐶0(𝑟,𝑡) 𝜕𝑟 ] + 𝑘𝐶𝑅 2(𝑟,𝑡) 2 , (1) 𝜕𝐶𝑅(𝑟,𝑡) 𝜕𝑡 = 𝐷𝑅 [ 𝜕2𝐶𝑅(𝑟,𝑡) 𝜕𝑟2 + 2 𝑟 𝜕𝐶𝑅(𝑟,𝑡) 𝜕𝑟 ] − 𝑘𝐶𝑅 2(𝑟, 𝑡). (2) The corresponding initial and boundary conditions are: At 𝑡 = 0 , 𝑟 ≥ 𝑟𝑜, 𝐶0 = 𝐶0 ∗, and 𝐶𝑅 = 0. (3) At 𝑟 = 𝑟𝑜, 𝐶0 = 0, and 𝐷0 𝜕𝐶0(𝑟𝑜,𝑡) 𝜕𝑟 = −𝐷𝑅 𝜕𝐶𝑅(𝑟𝑜,𝑡) 𝜕𝑟 . (4) At 𝑟 = ∞ , 𝐶0 = 𝐶0 ∗, and 𝐶𝑅 = 0. (5) The current is defined as: 𝑖(𝑡) 𝑛𝐹𝐴 = 𝐷𝑅 𝜕𝐶𝑅(𝑟𝑜,𝑡) 𝜕𝑟 . (6) Introducing the transformation ψ(r, t) = 2𝐶0(𝑟, 𝑡) + 𝐶𝑅(𝑟, 𝑡) and assuming equal diffusion coefficients 𝐷0 = 𝐷𝑅 = 𝐷equations (1)–(2) reduce to: 𝜕𝜓(𝑟,𝑡) 𝜕𝑡 = 𝐷 [ 𝜕2𝜓(𝑟,𝑡) 𝜕𝑟2 + 2 𝑟 𝜕𝜓(𝑟,𝑡) 𝜕𝑟 ], (7) 𝜕𝐶𝑅(𝑟,𝑡) 𝜕𝑡 = 𝐷 [ 𝜕2𝐶𝑅(𝑟,𝑡) 𝜕𝑟2 + 2 𝑟 𝜕𝐶𝑅(𝑟,𝑡) 𝜕𝑟 ] − 𝑘𝐶𝑅 2(𝑟, 𝑡). (8) The revised initial and boundary conditions become: At 𝑡 = 0, 𝑟 ≥ 𝑟𝑜, ψ(𝑟, 0) = 2co ∗, and 𝐶𝑅(𝑟, 0) = 0. (9) At 𝑟 = 𝑟𝑜, ψ(𝑟𝑜,𝑡) = 𝐶𝑅(𝑟𝑜, 𝑡), and 𝜕𝜓(𝑟𝑜,𝑡) 𝜕𝑟 = − 𝜕𝐶𝑅(𝑟𝑜,𝑡) 𝜕𝑟 . (10) At𝑟 = ∞, ψ(∞, 𝑡) = 2co ∗, and 𝐶𝑅(∞, 𝑡) = 0. (11) The dimensional current expression becomes [ 𝑖(𝑡) 𝑛𝐹𝐴𝐷 ] = [ 𝜕𝐶𝑅 𝜕𝑟 ] 𝑟= 𝑟0 . (12) Introducing the dimensionless variables: 𝜌 = 𝑟 𝑟𝑜 , 𝑇 = 𝐷𝑡 𝑟𝑜 2 , 𝑈 = ψ 2co ∗ , 𝑉 = 𝐶𝑅 2co ∗, 𝑚 = (2co ∗𝑘)(𝑟𝑜)2 𝐷 . (13) the governing equations (7)–(8) reduce to the following non-dimensional form: 𝜕𝑈(𝜌,𝑇) 𝜕𝑇 = 𝜕2𝑈(𝜌,𝑇) 𝜕𝜌2 + 2 𝜌 𝜕𝑈(𝜌,𝑇) 𝜕𝜌 , (14) 𝜕𝑉(𝜌,𝑇) 𝜕𝑇 = 𝜕2𝑉(𝜌,𝑇) 𝜕𝜌2 + 2 𝜌 𝜕𝑉(𝜌,𝑇) 𝜕𝜌 − 𝑚𝑉2(𝜌, 𝑇), (15) with initial and boundary conditions: 𝑈(𝜌, 0) = 1 , 𝑉(𝜌, 0) = 0. (16) 𝑈(1, 𝑇) = 𝑈(1, 𝑇)and 𝑈′(1, 𝑇) = −𝑉′(1, 𝑇). (17) 𝑈(∞, 𝑇) = 1, 𝑉(∞, 𝑇) = 0. (18) Finally, the dimensionless current is expressed as: [ 𝑖(𝑡) 𝑛𝐹𝐴𝐷 ] = [ 𝜕𝑉(𝜌,𝑇) 𝜕𝜌 ] 𝜌=1 . (19) The nonlinear reaction–diffusion equations modeling second-order regeneration at spherical microelectrodes have wide scientific and engineering applications. They aid in optimizing biosensors, simulating enzymatic processes in biofuel cells, guiding drug delivery system design, and improving electrochemical interfaces in batteries and super capacitors. This framework highlights the multidisciplinary importance of coupled diffusion–reaction dynamics near curved reactive surfaces. 1474 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate 3. Result and Discussion 3.1. Solutions of Transient Concentrations and Current Via HPM Numerous applied problems can be resolved by solving systems of nonlinear equations. However, significant progress has been made in developing accurate approximate analytical techniques over the last three decades. Among the most popular approaches are the variation iteration method [14], the hyperbolic and Pade approximation [15], the AGM method [16], the Adomian polynomials [17], and TSM method [18]. One of the popular techniques for resolving nonlinear differential equations is HPM [19, 20]. The HPM method was first proposed by He [21];He [22];He [23] and He [24]. Khan et al. applied the HPM method to solve the nonlinear boundary value problems [25, 26]. The nonlinear equation was also solved using the numerical scheme proposed by Srivastava and Izadi [27] and Izadi, et al. [13]. Recently, Rajendran and co-workers solved many nonlinear issues in physical and electrochemical sciences using the HPM method [28-35]. The approximate solutions for Eq. (14) and Eq. (15) using the homotopy perturbation approach are derived as follows (Appendix-A): 𝑈(𝜌, 𝑇) = 1 − 1 2𝜌 (1 − 𝑒𝑟𝑓 ( 𝜌−1 2√𝑇 )), (20) 𝑉(𝜌, 𝑇) = 1 2𝜌 (1 − 𝑒𝑟𝑓 ( 𝜌−1 2√𝑇 )) + 𝑚√𝑇 𝜌√𝜋 − 𝑚 ( 1 𝜌2 ( 𝑇 4 + √𝑇 √𝜋 )) + 𝑚 4𝜌 [ (1−𝜌)2 2 + 𝑇 + (𝑇 + (1−𝜌)2 2 ) (𝑒𝑟𝑓 ( (1−𝜌) 2√𝑇 )) + √𝑇(1−𝜌)𝑒 −(1−𝜌)2 4𝑇 √𝜋 ].(21) The nondimensional current takes the following form: I(𝑇) = [ 𝑖(𝑡) 𝑛𝐹𝐴𝐷 ] = [ 𝜕𝑉(𝜌,𝑇) 𝜕𝜌 ] 𝜌=1 =|[ 𝑚𝑇 4 + 𝑚𝑇−1 2√𝑇√𝜋 − 1 2 ]|. (22) When m=0 we get I(𝑇) = [ 𝑖(𝑡) 𝑛𝐹𝐴𝐷 ] = [ 𝜕𝑉(𝜌,𝑇) 𝜕𝜌 ] 𝜌=1 = 1 2 [1 + 1 √𝜋𝑇 ]. (23) This expression describes the limiting diffusion current typically observed under semi-infinite spherical diffusion when a potential step is applied. 3.2. Validation of the Model Numerical techniques are utilized to solve the nonlinear differential equations (14) and (15). This equation is solved using the SCILAB software pdex4 function, which resolves initial-boundary value issues for PDE. Table 1 and Tables 2a and 2b compare its numerical result with the HPM method of the dimensionless concentration of 𝑈 𝑎𝑛𝑑 𝑉 for various values of dimensionless time 𝑇, respectively, for fixing the parameter value 𝑚. This Tables clearly shows that𝑈 ≈ 1 𝑎𝑛𝑑 𝑉 ≈ 0when 𝑇 ≥ 10 and 𝜌 ≥ 5 and for higher values, the concentration approaches constant value, indicating that the species reaches equilibrium at higher spatial position. The dimensionless concentration values stabilize as 𝑇 increases, which means that the system reaches a steady-state. 1475 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate Table 1. Comparing the analytical and numerical results for U(ρ,T) for different values of time T. Concentration of U(𝝆, 𝑻) 𝝆 T=0.1 T=1 T=3 T=5 T=10 Numerial Analytical RJM Eq. (20) Deviation Numerical Analytical RJM Eq. (20) Deviation Numerical Analytical RJM Eq. (20) Deviation Numerical Analytical RJM Eq. (20) Deviation Numerical Analytical RJM Eq. (20) Deviation 1 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 2 0.9932 0.9937 0.0005 0.8796 0.8801 0.0005 0.8289 0.8292 0.0003 0.8123 0.8120 0.0003 0.7943 0.7942 0.0001 3 1.0000 0.9999 0.0001 0.9734 0.9738 0.0004 0.9306 0.9301 0.0005 0.9125 0.9122 0.0003 0.8909 0.8909 0.0000 4 1.0000 1.0000 0.0000 0.9956 0.9958 0.0002 0.9721 0.9724 0.0003 0.9575 0.9572 0.0003 0.9373 0.9372 0.0001 5 1.0000 1.0000 0.0000 0.9995 0.9995 0.0000 0.9896 0.9898 0.0002 0.9796 0.9794 0.0002 0.9631 0.9629 0.0002 6 1.0000 1.0000 0.0000 1.0000 0.9999 0.0001 0.9965 0.9966 0.0001 0.9907 0.9905 0.0002 0.9783 0.9780 0.0003 7 1.0000 1.0000 0.0000 1.0000 0.9999 0.0001 0.9989 0.9989 0.0000 0.9960 0.9959 0.0001 0.9877 0.9872 0.0005 8 1.0000 1.0000 0.0000 1.0000 0.9999 0.0001 0.9997 0.9997 0.0000 0.9984 0.9983 0.0001 0.9935 0.9927 0.0008 9 1.0000 1.0000 0.0000 1.0000 0.9999 0.0001 0.9999 0.9999 0.0000 0.9995 0.9994 0.0001 0.9973 0.9959 0.0014 10 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000 0.9999 0.0001 1.0000 0.9998 0.0002 1.0000 0.9978 0.0022 Average Deviation 0.0001 Average Deviation 0.0001 Average Deviation 0.0002 Average Deviation 0.0002 Average Deviation 0.0006 Table 2a. Comparing the analytical and numerical concentration of specie V(ρ,T) for different values of time T and for a fixed value of m=0.1. Concentration of V(𝝆, 𝑻) (m = 0.1) 𝝆 T=0.1 T=1 T=3 T=5 T=10 Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation 1 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 2 0.0068 0.0102 0.0034 0.1196 0.1312 0.0116 0.1689 0.1951 0.0262 0.1845 0.2249 0.0404 0.2010 0.2742 0.0732 3 0.0000 0.0037 0.0037 0.0264 0.0365 0.0101 0.0684 0.0880 0.0196 0.0856 0.1155 0.0299 0.1055 0.1596 0.0541 4 0.0000 0.0032 0.0032 0.0043 0.0133 0.0090 0.0275 0.0429 0.0154 0.0416 0.0639 0.0223 0.0605 0.0994 0.0389 5 0.0000 0.0028 0.0028 0.0005 0.0085 0.0080 0.0103 0.0234 0.0131 0.0191 0.0378 0.0187 0.0356 0.0651 0.0295 6 0.0000 0.0024 0.0024 0.0000 0.0072 0.0072 0.0035 0.0151 0.0116 0.0092 0.0243 0.0151 0.0209 0.0446 0.0237 7 0.0000 0.0021 0.0021 0.0000 0.0064 0.0064 0.0011 0.0115 0.0104 0.0031 0.0173 0.0142 0.0119 0.0320 0.0201 8 0.0000 0.0019 0.0019 0.0000 0.0058 0.0058 0.0003 0.0098 0.0095 0.0016 0.0136 0.0120 0.0063 0.0242 0.0179 9 0.0000 0.0017 0.0017 0.0000 0.0053 0.0053 0.0000 0.0088 0.0088 0.0005 0.0116 0.0111 0.0026 0.0192 0.0166 10 0.0000 0.0016 0.0016 0.0000 0.0048 0.0048 0.0000 0.0081 0.0081 0.0000 0.0103 0.0103 0.0000 0.0161 0.0161 Average Deviation 0.0023 Average Deviation 0.0068 Average Deviation 0.0123 Average Deviation 0.0174 Average Deviation 0.0290 1476 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate Table 2b. Comparing the analytical and numerical concentration of specie 𝑉(𝜌, 𝑇) for different values of time T and for a fixed value of m=0.01. Concentration of V(𝝆, 𝑻) (m = 0.01) 𝝆 T=0.1 T=1 T=3 T=5 T=10 Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation Numerical Analytical RJM Eq. (21) Deviation 1 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 0.5000 0.5000 0.0000 2 0.0067 0.0067 0.0000 0.1203 0.1210 0.0007 0.1709 0.1732 0.0023 0.1873 0.1917 0.0044 0.2052 0.2126 0.0074 3 0.0000 0.0004 0.0004 0.0267 0.0272 0.0005 0.0693 0.0709 0.0016 0.0873 0.0906 0.0033 0.1087 0.1142 0.0055 4 0.0000 0.0003 0.0003 0.0044 0.0051 0.0007 0.0278 0.0291 0.0013 0.0425 0.0449 0.0024 0.0625 0.0665 0.0040 5 0.0000 0.0003 0.0003 0.0005 0.0013 0.0008 0.0104 0.0116 0.0012 0.0203 0.0223 0.0020 0.0368 0.0399 0.0031 6 0.0000 0.0002 0.0002 0.0000 0.0008 0.0008 0.0035 0.0046 0.0011 0.0093 0.0109 0.0016 0.0216 0.0242 0.0026 7 0.0000 0.0002 0.0002 0.0000 0.0006 0.0006 0.0011 0.0021 0.0010 0.0040 0.0054 0.0014 0.0123 0.0148 0.0025 8 0.0000 0.0002 0.0002 0.0000 0.0006 0.0006 0.0003 0.0012 0.0009 0.0016 0.0029 0.0013 0.0065 0.0090 0.0025 9 0.0000 0.0002 0.0002 0.0000 0.0005 0.0005 0.0000 0.0009 0.0009 0.0005 0.0017 0.0012 0.0028 0.0056 0.0028 10 0.0000 0.0001 0.0001 0.0000 0.0005 0.0005 0.0000 0.0008 0.0008 0.0000 0.0012 0.0012 0.0000 0.0036 0.0036 Average Deviation 0.0002 Average Deviation 0.0006 Average Deviation 0.0011 Average Deviation 0.0019 Average Deviation 0.0034 1477 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate 3.3. Parametric Analysis of Concentration and Current Figure 2. (a,b). (a). Comparison of dimensionless concentration 𝑈 were calculated using Eq. (14) and Eq. (20). (b) shows the steady state of dimensionless concentration 𝑈. In Figure 2(a) at small time the concentration rapidly increases with distance, reaching a nearly steady-state. As T increases (i.e. 𝑇 = 1,3,5,10), the concentration profile smoothens and extends further into the domain. Also, clear that as time increases, the concentration value rises and concentration value is the same (i.e. 𝑈 = 1) when 𝜌 ≥ 10 for all values of time T. The system appears to describe a diffusion-like process where concentration gradually reaches equilibrium. The boundary conditions likely enforce a high concentration when 2 ≤ 𝜌 ≤ 4 and also Fig. 2(b) shows the steady-state concentration 𝑈 at 𝑇 ≥ 100.From the equation (20) and Fig.2(a,b) it is observed that the concentration of U is minimum at 𝜌 =1 and minimum value is 0.5.The concentration of U has been estimated to be maximal when ρ = 10, with a peak value of one. 1478 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate Figure 3. (a-c) Comparison of concentration 𝑉 using Eq. (15) for various values of T (a) when 𝑚 = 0.1 (b) when 𝑚 = 0.01 (c) various values of m (dimensionless parameter). In Figure 3(a) and 3(b) plots the 𝑉(𝜌, 𝑇) for a range of distance 𝜌 for various value of dimensionless time 𝑇when parameter 𝑚 = 0.1 and 𝑚 = 0.01respectively. Here the value of 𝑉(𝜌, 𝑇) is high when 𝑇 increases and its value is equal (i.e. 𝑉 = 0) for all values of time 𝑇 at 𝜌 ≥ 10. If larger 𝑚 (𝑚 = 0.1) leads to a slower concentration decay while smaller 𝑚 (𝑚 = 0.01) results in a more rapid drop in 1479 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate concentration. According to equation (21) and figure 2(a,b), the concentration of V reaches its maximum at ρ=0 and reaches its maximum value of 0.5. Figure 4. Dimensionless current were calculated using Eq. (22) for the various values of parameter 𝑚. Figure 4 represents the variation of current over time T for various values of m. The dimensionless current starts from a lower value and increases as time progresses. As T increases, the system becomes more stable, causing the current to change. A higher m likely means faster transport or diffusion, leading to a more pronounced increase in current. A comparison between the analytical and simulation results for different values of m is provided in Table 3. The highest deviation between the analytical predictions and numerical simulations, observed across Tables 1 to 3, is 0.4071. Table 3. Comparison of the analytical results of the current with simulation result. Dimensionless current I 𝑇 m=0.1 m=0.5 m=1 Numerical Analytical RJM Eq. (22) Deviation Numerical Analytical RJM Eq. (22) Deviation Numerical Analytical RJM Eq. (22) Deviation 5 0.0004 0.0002 0.0002 0.8299 0.3143 0.5156 2.2199 1.2548 0.9651 10 0.0073 0.0051 0.0022 1.4792 1.1069 0.3723 3.5194 2.8031 0.7163 20 0.0997 0.0631 0.0366 2.7788 2.5678 0.2110 6.1178 5.6988 0.4190 30 0.3594 0.3530 0.0064 4.0775 3.9712 0.1063 8.7151 8.4940 0.2211 40 0.6189 0.6338 0.0149 5.3755 5.3477 0.0278 11.3113 11.2391 0.0722 50 0.8783 0.9096 0.0313 6.6730 6.7077 0.0347 13.9065 13.9553 0.0488 Average Deviation 0.0153 Average Deviation 0.2113 Average Deviation 0.4071 1480 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate 4. Future Scope The homotopy perturbation method (HPM) offers a powerful framework for obtaining accurate analytical approximations to nonlinear boundary value problems with minimal computational effort. Its versatility extends to second-order homogeneous kinetics [36] and biological wave phenomena [37]. Similar nonlinear initial–boundary value problems arise in diverse fields, including heat propagation in absorptive media, thermal energy transfer in plasmas, and dead-core formation processes [38]. The efficiency of HPM in simplifying nonlinear systems makes it particularly valuable for real-time simulations and for bridging empirical models with theoretical frameworks, thereby advancing applications across science and engineering. 5. Conclusion This study developed and analyzed nonlinear transient reaction–diffusion equations describing a second-order regeneration process at spherical microelectrodes with equal diffusion coefficients. Using the homotopy perturbation method (HPM), convergent analytical solutions were obtained, effectively capturing nonlinear dynamics under transient conditions. The method successfully addressed nonlinearities from second-order kinetics and Butler–Volmer type boundary conditions, proving robust for curved geometries. These approximations provide insights into concentration profiles and fluxes, reducing dependence on numerical simulations. Beyond theory, the approach offers practical implications for optimizing biosensors, enzymatic biofuel cells, and electrochemical energy devices, where accurate modeling of coupled diffusion–reaction processes is crucial for performance improvement. Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. This study followed all ethical practices during writing. Copyright: © 2025 by the authors. This open-access article is distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). References [1] A. Molina and E. 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Solution of equations (14) and (15). The equations (14) and (15) can be written as follows: 𝜕2𝑈 𝜕𝜌2 + 2 𝜌 𝜕𝑈 𝜕𝜌 − 𝜕𝑈 𝜕𝑇 = 0, (A1) 𝜕2𝑉 𝜕𝜌2 + 2 𝜌 𝜕𝑉 𝜕𝜌 − 𝜕𝑉 𝜕𝑇 − 𝑚𝑉2 = 0. (A2) Dimensionless initial and boundary conditions are as follows: At 𝑇 = 0, 𝑈 = 1, 𝑉 = 0. (A3) At 𝜌 = 1, 𝑈 = 𝑉, and 𝑈′ = −𝑉′. (A4) At 𝜌 = ∞, 𝑈(∞) = 1, 𝑉(∞) = 0. (A5) The homotopy for the nonlinear equation (A2) can be constructed as follows: (1 − 𝑝) [ 𝜕2𝑉 𝜕𝜌2 + 2 𝜌 𝜕𝑉 𝜕𝜌 − 𝜕𝑉 𝜕𝑇 ] + 𝑝 [ 𝜕2𝑉 𝜕𝜌2 + 2 𝜌 𝜕𝑉 𝜕𝜌 − 𝜕𝑉 𝜕𝑇 − 𝑚𝑉2] = 0. (A6) Now assume that the solution of Eqn.(A2) is 𝑉 = 𝑉0 + 𝑝𝑉1 + 𝑝2 𝑉2 + ⋯. (A7) By inserting Eq. (A7) into Eq. (A6) and matching powers of p, the following equations are obtained. 𝑝0 ∶ 𝜕2𝑉0 𝜕𝜌2 + 2 𝜌 𝜕𝑉0 𝜕𝜌 − 𝜕𝑉0 𝜕𝑇 = 0, (A8) 𝑝1 ∶ 𝜕2𝑉1 𝜕𝜌2 + 2 𝜌 𝜕𝑉1 𝜕𝜌 − 𝜕𝑉1 𝜕𝑇 − 𝑚𝑉0 2 = 0. (A9) The conditions for (A8) are as follows: 𝑇 = 0, 𝑈 = 1, 𝑉0 = 0, (A10) 𝜌 = 1, 𝑈(1) = 𝑉0(1), 𝑈′(1) = −𝑉0 ′(1), (A11) 𝜌 = ∞, 𝑈(∞) = 1, 𝑉0(∞) = 0. (A12) The following are dimensionless conditions for (A9): 𝑇 = 0, 𝑉1 = 0, (A13) 𝜌 = 1, 𝑉1(1) = 0, 𝑉1 ′ (1) = 0, (A14) 𝜌 = ∞, 𝑉1(∞) = 0. (A15) Taking Laplace transformation with respect to 𝑇 to the Eqs. (A1) and (A8) we get 1483 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 9: 1471-1483, 2025 DOI: 10.55214/2576-8484.v9i9.10155 © 2025 by the authors; licensee Learning Gate 𝜕2�̅� 𝜕𝜌2 + 2 𝜌 𝜕�̅� 𝜕𝜌 − 𝑠�̅� + 1 = 0, (A16) 𝜕2𝑉0̅̅ ̅ 𝜕𝜌2 + 2 𝜌 𝜕𝑉0̅̅ ̅ 𝜕𝜌 − 𝑠𝑉0 ̅̅̅ = 0, (A17) where s is the Laplace variable and the over bar signifies a quantity that has undergone Laplace transformation. On solving Eqs. (A16) and (A17) with the initial and boundary conditions,we get the following results. �̅�(𝜌, 𝑠) = 1 𝑠 − 𝑒−𝜌√𝑠 2𝜌𝑠𝑒−√𝑠 , (A18) 𝑉0 ̅̅̅(𝜌, 𝑠) = 𝑒−𝜌√𝑠 2𝜌𝑠𝑒−√𝑠 . (A19) Using inverse Laplace transform, the final results can be obtained as follows: 𝑈(𝜌, 𝑇) = 1 − 1 2𝜌 (1 − 𝑒𝑟𝑓 ( 𝜌−1 2√𝑇 )), (A20) 𝑉0(𝜌, 𝑇) = 1 2𝜌 (1 − 𝑒𝑟𝑓 ( 𝜌−1 2√𝑇 )). (A21) Taking Laplace transformation with respect to 𝑇 to the Eqs. (A9) we get 𝜕2𝑉1̅̅ ̅ 𝜕𝜌2 + 2 𝜌 𝜕𝑉1̅̅ ̅ 𝜕𝜌 − 𝑠𝑉1̅ − 𝑚 4𝜌2 ( 1 𝑠 − 2(𝜌−1) √𝑠 ) = 0. (A22) On solving Eqs. (A22) using reduction of order , we get the following results. 𝑉1̅(𝜌, 𝑆) = 𝑚𝑒√𝑠(1−𝜌) 4𝑠2𝜌 + 𝑚 2𝜌𝑠 3 2 − 𝑚(1+2√𝑠) 4𝑠2 (𝜌−2 + 2𝜌−4 𝑠 + 24𝜌−6 𝑠2 + ⋯ ) ≈ 𝑚𝑒√𝑠(1−𝜌) 4𝑠2𝜌 + 𝑚 2𝜌𝑠 3 2 − 𝑚(1+2√𝑠) 4𝑠2 (𝜌−2). (A23) Using inverse Laplace transform, the final results can be obtained as follows: 𝑉1(𝜌, 𝑇) = 𝑚 4𝜌 [(𝑇 + (1−𝜌)2 2 ) (𝑒𝑟𝑓 ( (1−𝜌) 2√𝑇 )) + √𝑇(1−𝜌)𝑒 −(1−𝜌)2 4𝑇 √𝜋 + (1−𝜌)2 2 + 𝑇] + 𝑚√𝑇 𝜌√𝜋 − 𝑚 ( 1 𝜌2 ( 𝑇 4 + √𝑇 √𝜋 )). (A24) According to the analysis using HPM, we find that. U(𝜌,T) = 1 − 1 2𝜌 (1 − 𝑒𝑟𝑓 ( 𝜌−1 2√𝑇 )), (A25) 𝑉(𝜌, 𝑇) = lim 𝑝→1 𝑉(𝜌) ≈ 𝑉0 + 𝑉1 ≈ 1 2𝜌 (1 − 𝑒𝑟𝑓 ( 𝜌−1 2√𝑇 )) + 𝑚 4𝜌 [(𝑇 + (1−𝜌)2 2 ) (𝑒𝑟𝑓 ( (1−𝜌) 2√𝑇 )) + √𝑇(1−𝜌)𝑒 −(1−𝜌)2 4𝑇 √𝜋 + (1−𝜌)2 2 + 𝑇] + 𝑚√𝑇 𝜌√𝜋 − 𝑚( 1 𝜌2 ( 𝑇 4 + √𝑇 √𝜋 ). (A26) The dimensionless current is I=[ 𝜕𝑉(𝜌,𝑇) 𝜕𝜌 ] 𝜌=1 = 𝑚𝑇 4 + 𝑚𝑇−1 2√𝑇√𝜋 − 1 2 . (A27)