International Scholarly Research Notices

Research Article

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

Table 2

Approximate expressions of concentration of substrate, and current under the boundary conditions (12a)–(12c).
ConditionsConcentrations and currentsFigures
For very small values of 𝛼 √ 𝑢 = s i n h ( 𝛾 𝜌 ) 𝜌 √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] ) (17)Figures 5 and 6
√ 𝜓 = √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] ) (18)Figure 8
For large values of 𝛼 ( 𝛼 ≥ 1 ) and all values of 𝛾 𝑢 = 1 + 𝛾 ( 1 2 𝛼 2 𝑣 − 6 𝛼 𝑣 − 𝛾 𝑣 − 2 𝛾 ) 3 6 𝛼 3 𝑣 𝜌 2 2 − 1 − 𝑣 + 𝛾 2 1 2 0 𝛼 3 𝜌 4 4 − 1 − 𝑣 (20)Figure 7
𝜓 = 𝛾 ( 3 0 𝛼 2 𝑣 − 1 5 𝛼 𝑣 − 𝛾 𝑣 − 5 𝛾 ) 4 5 𝛼 3 𝑣 (21)Figure 9
For small and medium values of 𝛼 and all values of 𝛾 √ 𝑢 = s i n h ( 𝛾 𝜌 ) 𝜌 √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ − √ 𝛾 ] ) 𝛼 𝛾 s i n h ( 𝛾 𝜌 ) 𝜌 √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] ) 3 + 𝛼 𝛾 √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] ) 2 (22)Figures 5 and 6
√ 𝜓 = √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ − √ 𝛾 ] ) 𝛼 𝛾 ( √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] ) √ s i n h [ 1 𝛾 ] + 𝑣 ( √ √ 𝛾 c o s h [ √ 𝛾 ] − s i n h [ 𝛾 ] ) 3 (23)Figure 8