A fractional nonlinear system for release oscillation of silver ions from hollow fibers

Abstract

A fractional nonlinear differential system is studied to search for its periodic solution near the equilibrium. The release oscillation of silver ions from hollow fibers is used as an example to illustrate the importance of fractional order to the release frequency. The criterion for the periodic solution is obtained and it shows that the fractional order will greatly affect its periodic property.

Introduction

A nonlinear oscillator can be generally expressed as ¹

$$y^{″}+y+N(y,y^{\prime },y^{″})=0$$

(1)

where N is nonlinear operator. For a nonlinear oscillator the following condition should be satisfied

$$\frac{N(y,x,x^{\prime })}{y}>0$$

(2)

The nonlinear oscillator given in equation (1) is equivalent to the following nonlinear system

$$\{\begin{array}{l}{y}^{\prime }=x\\ x^{\prime }=-y-N(u,x,x^{\prime })\end{array}$$

(3)

Analytical methods for nonlinear oscillators include the homotopy perturbation method,^1–9 the variational iteration method,^1,10,11 frequency formulation^12–16 and others.

In this paper we will consider the following generalized nonlinear system

$$\{\begin{array}{l}{x}^{\prime }=ax-bxy\\ y^{\prime }=cxy-dy\end{array}$$

(4)

where a, b, c and d are constants.

Equation (4) is the well-known Lotka–Volterra predator–prey model, and it can also describe a damped oscillation of a bridge or release of silver ions from hollow fibers.^17,18 As an example, the well-known van der Pol oscillator can be expressed as

$$\{\begin{array}{l}{x}^{\prime }=\mu \left(x-\frac{1}{3}{x}^3-y\right)\\ y^{\prime }=\frac{1}{\mu }x\end{array}$$

(5)

or

$$\{\begin{array}{l}{x}^{\prime }=y\\ y^{\prime }=\mu (1-x^2)y-x\end{array}$$

(6)

Fractional modification of the model

In this paper we study the fractional modification of equation (4)

$$\{\begin{array}{l}\frac{d^{\alpha }x}{d{t}^{\alpha }}=ax-bxy\\ \frac{d^{\alpha }x}{d{t}^{\alpha }}=cxy-dy\end{array}$$

(7)

where $\frac{d^{\alpha }x}{d{t}^{\alpha }}$ is He’s fractional derivative defined as ^{3 , 6 , 19}

$$\frac{{\text{d}}^{\alpha }x}{\text{d}{t}^{\alpha }}=\frac{1}{\Gamma (n-\alpha )}\frac{{\text{d}}^n}{\text{d}{t}^n}{\displaystyle {\int }_{t_0}^t{(s-x)}^{n-\alpha -1}}\left[x_0(s)-x(s)\right]\text{d}s$$

(8)

The physical meaning of the fractional derivative was discussed in literature, ^{3 , 6 , 19} and it can be widely applied to discontinuous problems.^20–23

All predator–prey models assume that time is continuous; however, the number of either prey or predator changes not smoothly with time. The discontinuous growth rate is of their intrinsic property, and a fractional model can best model its property of periodic property.

The release of ions from hollow fibers can be also described by the fractional calculus, where the water can be considered as a porous medium.

By the fractional transform ⁶

$$s=\frac{t^a}{\Gamma \left(1+a\right)}$$

(9)

Equation (6) can be converted into the following one

$$\{\begin{array}{l}\frac{dx}{ds}=f(x,y)=ax-bxy\\ \frac{dx}{ds}=g(x,y)=cxy-dy\end{array}$$

(10)

The equilibrium point locates at

$$\{\begin{array}{l}{x}_0=\frac{d}{c}\\ y_0=\frac{a}{b}\end{array}$$

(11)

The equilibrium point for van der Pol oscillator is (0,0). In this paper we search for the periodicity near the equilibrium point, so the system, equation (10), can be linearized as

$$\{\begin{array}{l}f(x,y)=f(x_0,y_0)+(x-x_0)\frac{\partial f}{\partial x}(x_0,y_0)+(y-x_0)\frac{\partial f}{\partial y}(x_0,y_0)\\ g(x,y)=g(x_0,y_0)+(x-x_0)\frac{\partial g}{\partial x}(x_0,y_0)+(y-x_0)\frac{\partial g}{\partial y}(x_0,y_0)\end{array}$$

(12)

Equation (10) becomes

$$\{\begin{array}{l}{x}^{\prime }=ax-bxy\hspace{0.17em}\approx \hspace{0.17em}a{x}_0-b{x}_0{y}_0+(a-b{y}_0)(x-x_0)-b{x}_0(y-y_0)\\ y^{\prime }=cxy-dy\hspace{0.17em}\approx \hspace{0.17em}c{x}_0{y}_0-d{y}_0+c{y}_0(x-x_0)-(c{x}_0-d)(y-y_0)\end{array}$$

(13)

We assume that the system has the periodic solution

$$\{\begin{array}{l}x=A+B\hspace{0.17em}\sin (\omega s+\alpha )\\ y=m+n\hspace{0.17em}\sin (\omega s+\beta )\end{array}$$

(14)

where A, B, m, n, $\alpha$ and $\beta$ are constants to be determined.

In view of equation (14), we write equation (13) in the form

$$\{\begin{array}{l}B\hspace{0.17em}\omega \hspace{0.17em}\cos (\omega s+\alpha )=x_0-b{x}_0{y}_0+(a-b{y}_0)(A+B\hspace{0.17em}\sin (\omega s+\alpha )-x_0)-b{x}_0(m+n\hspace{0.17em}\sin (\omega s+\beta )-y_0)\\ n\hspace{0.17em}\omega \hspace{0.17em}\cos (\omega s+\beta )=c{x}_0{y}_0-d{y}_0+c{y}_0(A+B\hspace{0.17em}\sin (\omega s+\alpha )-x_0)-(c{x}_0-d)(m+n\hspace{0.17em}\sin (\omega s+\beta )-y_0)\end{array}$$

(15)

In order to guarantee a periodic solution, the following criterion has to be satisfied

$$\{\begin{array}{l}\cos (\omega s+\alpha )=\sin (\omega s+\beta )\\ \cos (\omega s+\beta )=\sin (\omega s+\alpha )\\ a-b{y}_0=0\\ c{x}_0-d=0\\ 1-bm=0\\ B\omega =-b{x}_{0}n\\ -d+cA=0\\ n\omega =c{y}_{0}B\end{array}$$

(16)

Equation (16) implies that

$$\{\begin{array}{l}\beta =\frac{\pi }{2}-\alpha \\ \frac{d}{c}=\frac{a}{b}\\ m=\frac{1}{b}\\ A=\frac{d}{c}\\ a{B}^2+b{n}^2=0\end{array}$$

(17)

The frequency is obtained as

$$\omega =\frac{-bdn}{cB}$$

(18)

As an example, considering the case a = d = −1, b = c = 1, $\alpha =0$ , $\beta =\pi /2$ , m = 1, A = −1, B = 1, n = 1, and $\omega =1$ , we obtain the solution

$$\{\begin{array}{l}x=-1+\sin (s)\\ y=1+\sin (s+\frac{\pi }{2})\end{array}$$

(19)

or

$$\{\begin{array}{l}x=-1+\sin \left(\frac{t^a}{\Gamma (1+a)}\right)\\ y=1+\sin (\frac{t^a}{\Gamma \left(1+a\right)}+\frac{\pi }{2})\end{array}$$

(20)

Figure 1 shows that the periodicity criterion is satisfied, the frequency is not changed for $\alpha$  = 1, and the frequency increases with time for $\alpha$  < 1.

Figure 1.

Frequency change with time.

Conclusions

This short paper elucidates the periodic property of a fractional nonlinear system; the predator–prey model is used to illustrate the solution process. As the predator–prey model can describe the release of silver ions from hollow fibers or nanofiber membrane,^17,18 Figure 1 can be explained as the release oscillation, which is greatly affected by the fractional order. Generally the frequency changes with time, and when time tends to infinity, the frequency becomes infinite large.