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(2)

Substituting (2) into (1) and equating terms having identical powers of ε , we obtain a number of differential equations that can be integrated, recursively, to determine the unknown functions: x 0 ( t ), x 1 ( t ), x 2 ( t )…

(3)

where k and α denote positive reaction diffusion and saturation parameters, respectively, for the mentioned process; y is the mediator concentration and x the distance (Thiagarajan et al. Nike Wmns Free Run Flyknit 307 Hyper Turq/Blackvoltrio Teal discount factory outlet find great sale online really online Manchester cheap online INWCH
).

(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)

and f ( x ) is the right hand side of (13).

(17)
(18)
(19)
(20)

We consider, as a case study, the following values for parameters: α =0.1, α =1, and α =1.5 for k =0.1,1,5,10,20,50, and 100.

Nonlinear phenomena appear in such broad scientific fields like applied mathematics, physics, and engineering. Scientists in those disciplines face, constantly, with the task of finding solutions for nonlinear ordinary differential equations. As a matter of fact, the possibility of finding analytical solutions for those cases is very difficult and cumbersome.

Figure 1

Fourth order Runge Kutta numerical solution for ( 3 ) (symbols) and proposed solution ( 20 ) (solid line) for α=0.1 .

Figure 2

Fourth order Runge Kutta numerical solution for ( 3 ) (symbols) and proposed solution ( 20 ) (solid line) for α=1.0 .

Figure 3

Fourth order Runge Kutta numerical solution for ( 3 ) (symbols) and proposed solution ( 20 ) (solid line) for α=1.5 .

The PM method provides in general, better results for small perturbation parameters ε <<1 (see (1)) and when are included the most number of terms from (2). To be precise, ε is a parameter of smallness; measures how much larger is the contribution of linear term L ( x ) than N ( x ) in (1). Although Figure adidas Originals Womens Originals Superstar 80S Trainers US7 White latest cheap price outlet sale sale in China mICfto5L
for α =0.1 satisfies that condition, Figure 2 and Figure 3 show that (20) provides a good approximation as solution to (3); despite of the fact that perturbation parameters α =1 and α =1.5, cannot be considered small. Since that the transport and kinetics are quantified in terms of k and α , it is important that our solutions have good accuracy for a wide range of values for both parameters.

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