An Efficient Approach of Homotopic Asymptotic for System Differential Equations of Non Integer Order

Abstract

The momentous objective of this paper is to using a method with a free parameter which named optimum asymptotic homotopic method for finding solution of system differential equations featuring fractional derivative. To better understand the methodology and view its advantages compared to similar examples are presented.

Appendix: Illustration of Test Example 1 with Details

Consider test example 1:

$$\begin{aligned} \left\{ \begin{array}{l } D^{\mu }_{t}u-v\,u_{xx}-u\,v_{xx}=f,\\ D^{\mu }_{t}v-v\,v_{xx} +u\,u_{xx}=g,\\ \end{array} \right. \end{aligned}$$

(29)

With considering

$$\begin{aligned} u(x,t,p)=u_{0}+ \sum _{i=1}^{\infty }u_{i}p^{i},\quad v(x,t,p)=v_{0}+ \sum _{i=1}^{\infty }v_{i}p^{i}, \end{aligned}$$

(30)

and

$$\begin{aligned} H(p)=p\, c_1+ p^2 c_2+ \cdots , \quad H'(p)=p\, d_1+ p^2 d_2+ \cdots , \end{aligned}$$

(31)

and using of O.HAM, for the first equation of (29) we can gain:

$$\begin{aligned}&\left( D^{\mu }u_{0}+\right. p D^\mu u_1+ \left. p^2 D^\mu u_2+ \cdots -h(x,t)\right) \\&\quad -\left( p D^{\mu }u_{0}+p^2 D^\mu u_1+p^3 D^\mu u_2 +\cdots -p\,h(x,t)\right) \\&\quad -\left( p\, c_1+ p^2 c_2+ \cdots \right) \bigg ( \left( D^{\mu }u_{0}+ p D^\mu u_1+p^2 D^\mu u_2+\cdots \right) \\&\quad -\left( v_{0}+p\,v_1+p^2v_2+\cdots \right) \left( u_{0}+p\,u_1+p^2u_2+\cdots \right) _{xx} \\&\quad -\left( u_{0}+p\,u_1+p^2u_2+\cdots \right) \left( v_{0}+p\,v_1+p^2v_2+\cdots \right) _{xx}-f(x,t)\bigg )=0. \end{aligned}$$

Equating the coefficients of different power in “p”, we have the following system of partial differential equations

$$\begin{aligned} p^{0}&: D^{\mu } u_{0}=f(x,t), \\ p^{1}&: D^{\mu } u_{1}=-c_1\left( u_{0}(v_{0})_{xx}+v_{0}(u_{0})_{xx} \right) ,&\\ p^{2}&: D^{\mu } u_{2}=(1+c_1)D^\mu u_1-c_2\left( u_{0}(v_{0})_{xx}+v_{0}(u_{0})_{xx} \right)&\\&\qquad \quad \qquad -c_1\left( v_{0}(u_{1})_{xx}+v_{0}(u_{1})_{xx}+u_{0}(v_{1})_{xx}+u_{1}(v_{0})_{xx}\right) ,&\\&\cdots&\\ p^{n}&: D^{\mu } u_{n}=(1+c_1)D^\mu u_{n-1}&\\&\qquad \quad \qquad -\left( \sum _{m=1}^{n}c_m \sum _{i=0}^{n-m}\left( u_i\right) _{xx} v_{n-m-i}+u_i \left( v_{n-m-i}\right) _{xx}\right) +\sum _{i=2}^{n-1}c_i D^{\mu } u_{n-i}. \end{aligned}$$

Using of O.HAM, for the second equation of (29) we can gain:

$$\begin{aligned}&\left( D^{\mu }v_{0}+\right. p D^\mu v_1+ \left. p^2 D^\mu v_2+ \cdots -h(x,t)\right) \\&\quad -\left( p D^{\mu }v_{0}+p^2 D^\mu v_1 +p^3 D^\mu v_2+\cdots -p\,h(x,t)\right) \\&\quad -\left( p\, d_1+ p^2 d_2+ \cdots \right) \bigg ( \left( D^{\mu }v_{0}+ p D^\mu v_1+p^2 D^\mu v_2+\cdots \right) \\&\quad -\left( v_{0}+p\,v_1+p^2v_2+\cdots \right) \left( v_{0}+p\,v_1+p^2v_2+\cdots \right) _{xx} \\&\quad +\left( u_{0}+p\,u_1+p^2u_2+\cdots \right) \left( u_{0}+p\,u_1+p^2u_2+\cdots \right) _{xx}-h(x,t)\bigg )=0. \end{aligned}$$

Equating the coefficients of different power in “p”, we have the following system of partial differential equations

$$\begin{aligned} p^{0}&: D^{\mu } v_{0}=h(x,t), \\ p^{1}&: D^{\mu } v_{1}=-d_1\left( v_{0}(v_{0})_{xx}-u_{0}(u_{0})_{xx} \right) ,&\\ p^{2}&: D^{\mu } v_{2}=(1+d_1)D^\mu v_1-d_2\left( v_{0}(v_{0})_{xx}-u_{0}(u_{0})_{xx} \right) ,&\\&\qquad \quad \qquad -d_1\left( v_{0}(v_{1})_{xx}+v_{1}(u_{0})_{xx}\right) +d_1\left( u_{0}(u_{1})_{xx}+u_{1}(u_{0})_{xx}\right) ,&\\&\cdots&\\ p^{n}&: D^{\mu } v_{n}=(1+d_1)D^\mu v_{n-1}&\\&\qquad \qquad \quad +\left( \sum _{m=1}^{n}d_m \sum _{i=0}^{n-m}u_i \left( u_{n-m-i}\right) _{xx}-v_i \left( v_{n-m-i}\right) _{xx}\right) + \sum _{i=2}^{n-1}d_i D^{\mu } u_{n-i}. \end{aligned}$$