Approximate Analytical Solution of Nonlinear Reaction’s Diffusion Equation at Conducting Polymer Ultramicroelectrodes

Table 1

Approximate expressions of concentration of substrate and current under the boundary conditions (2a) and (2b).

Conditions	Concentrations and currents		Figures
For very small values of 𝛼	𝑢=sinh(√𝛾𝜌)𝜌sinh(√𝛾)	(5)	Figures 2(a)–2(c)
	𝜓=√𝛾coth(√𝛾)−1	(6)	Figure 4(a)
For large values of 𝛼 and all value of 𝛾	𝑢=1+𝛾(12𝛼^2−6𝛼−𝛾)36𝛼^3(𝜌^2−1)+𝛾^2120𝛼^3(𝜌^4−1)	(8)	Figure 3
	𝜓=𝛾(30𝛼^2−15𝛼−𝛾)45𝛼^3	(9)	Figures 4(b) and 4(c)
For small and medium values of𝛼 and all values of 𝛾	𝑢=sinh(√𝛾𝜌)𝜌sinh(√𝛾)−𝛼𝛾sinh(√𝛾𝜌)𝜌sinh^3(√𝛾)+sinh𝛼^2𝛾(√𝛾)	(10)	Figure 2
	𝜓=√𝛾coth(√𝛾)−1+𝛼𝛾cosech^2(√𝛾)(√𝛾coth(√𝛾)−1)	(11)	Figures 4(a) and 4(c)