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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. MERUMOTE Womens Wedges Sandals High Platform Open Toe Ankle Strap Shoes Natural cheap sale tumblr buy online authentic cheap hot sale hnc104
).

(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 1 for α =0.1 satisfies that condition, Figure FSJ Women Summer Casual Low Stacked Block Heels Sandals Open Toe Buckled Strap Shoes Size 415 US Black Matte high quality for sale H5WaJpWK
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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