Multistage Bernstein polynomials for the solutions of the Fractional Order Stiff Systems

1 Introduction

Fractional Order Stiff Systems have been employed to describe a variety of systems such as biology, physiology, medicine, hydraulics geology, and engineering. Fractional Order Stiff Systems are considered a new topic due to their potential applications especially in control processing. Stiff problems have been studied in many areas such as chemical engineering, non linear mechanics, biochemistry, and life sciences. Hence, the need for a reliable and efficient technique for the solution of stiff systems of differential equations is of high importance. Since 30 years, respected works have focused on the development of more advanced and efficient methods for stiff problems. BDF is a formula that is based on backward differentiation, and many modifications introduced by different authors are extended backward differentiation. First modification is (EBDF) formula introduced in Cash (1980). The MEBDF (modified EBDF) (Cash, 1983) and MF-MEBDF (matrix free MEBDF) (Hosseini and Hojjati, 1999), are modification methods of (BDF) formula used to solve stiff systems of ordinary differential equations. A-EBDF is also a modification of (BDF) applied to solve stiff systems of ordinary differential equations (Hojjati et al., 2004). Haar wavelets are used for linear and nonlinear stiff system of ordinary differential equations (Hsiao, 2004; Hsiao and Wang, 2001). Adomian decomposition method is applied on stiff problems (Saad Mahmood et al., 2005). Furthermore, modification of Homotopy perturbation methods which is called Rational Homotopy perturbation method (RHPM) is used to obtain an analytic approximation of stiff systems of ordinary differential equations (Biazar et al., 2015).

One of the important analytic methods for solving linear and nonlinear equations is Bernstein polynomials (B-polynomials). Bernstein operational matrix of differentiation proposed by Bhatti and Bracken (2007) used Bernstein polynomial basis to solve differential equation. Bernstein operational matrix for solving Lane–Emden type equations (Pandey and Kumar, 2012). operational matrices of Bernstein polynomials and their applications to solve Bessel differential equation (Yousefi and Behroozifar, 2010). Bernstein polynomials to solve fractional riccati type differential equations (Yuzbasi, 2013). Bernstein series solution of linear second-order partial differential equations with mixed conditions(Isik et al., 2012). Recently, Yiming et al. (2014) used Bernstein polynomials to find Numerical solution for the variable order linear cable equation. Approximate solutions of singular differential equations with estimation error by using Bernstein polynomials (Alshbool et al., 2015).

The interpolation polynomial used in (B-polynomials) method is a good approximation to the function $y(x)$ , and for large n. Bernstein polynomials (B-polynomials) have many useful properties. The procedure takes advantage of the continuity and unity partition properties of the basis set of B-polynomials over an interval [0, R]. This provides greater flexibility to impose boundary conditions at the end points of the interval. It also ensures that the sum at any point x of all B-polynomials is unity. For this reason we choose (B-polynomials) method.

In this paper, we present a new modification of Bernstein polynomials called Multistage Bernstein polynomials (MB-polynomials). The proposed method minimizes the error of the result of Fractional and ordinary Order Stiff Systems which is solved by standard Bernstein polynomials method. Moreover, the solution provided by MB-polynomials is valid in larger x than standard B-polynomials. The results of MB-polynomials are compared with B-polynomials, A-EBDFs methods, HPM and RHPM, and we show that MB-polynomials obtains more accurate result.

The rest of this paper is organized as follows: In Section 2, we present some definitions and properties of fractional calculus. In Section 3, some basic definitions of Fractional Order Stiff Systems are provided. In Section 4, we describe the standard B-polynomials and the MB-polynomials. Section 5, presents numerical comparisons with several methods which indicate that the MB-polynomials method is a simple, yet powerful method to give the approximate solutions for Fractional Order Stiff Systems. Finally we summarize our work in Section 6 and suggest some recommendations for future work.

2 Preliminaries and notations

In this section, we give some definitions and properties of fractional calculus according to Diethelm et al. (2005).

The following are two basic properties of the Caputo fractional derivative (Diethelm et al., 2005):

1. Let $f\in C_{-1}^n\text{,}n\in N$ . Than $D^{\alpha }f\text{,}0⩽\alpha ⩽n$ is well defined and $D^{\alpha }f\in C_{-1}$ .

2. Let $n-1⩽\alpha ⩽n\text{,}n\in N$ and $f\in C_{\mu }^n\text{,}\mu ⩾-1$ . Then

   (3)
   $$(J^{\alpha }{D}^{\alpha })f(x)=f(x)-{\displaystyle \sum _{k=0}^{n-1}}{f}^{(k)}(0^{+})\frac{x^k}{k!}\text{.}$$

For the Caputo derivative we have

3 Fractional Order Stiff Systems

We consider a stiff system of FDEs:

First, we write system (6) in the form

4 Solution by Bernstein polynomials (B-polynomials)

The Bernstein polynomials of degree m are defined by $$B_{i\text{,}m}(x)=\left(\begin{array}{c}\hfill m\hfill \\ \hfill i\hfill \end{array}\right)x^i{(1-x)}^{m-i}\text{,}i=0\text{,}1\text{,}\dots \text{,}m\text{,}$$ where the binomial coefficient is $$\left(\begin{array}{c}\hfill m\hfill \\ \hfill i\hfill \end{array}\right)=\frac{m!}{i!(m-i)!}\text{.}$$ There are $m+1$ nth-degree Bernstein polynomials. For mathematical convenience, we usually set $B_{i\text{,}m}=0$ , if $i<0$ or $i>m$ .

In general, we approximate any function $y_j(x)$ with the first $(m+1)$ Bernstein polynomials as

To solve the system (16), we have to find collocation points $x_0\text{,}{x}_1\text{,}\dots \text{,}{x}_m$ which will be substituted in (16), then we will have $(m-1)$ equations, with initial condition in (17). Now we have (m) equations where the unknowns are $c_i$ , which can be solved by using Newton’s iterative method. To find the collocation points $x_0\text{,}{x}_1\text{,}\dots \text{,}{x}_m$ , follow as

Therefore, according to B-polynomials, the m-term approximations for the solutions of (6) can be expressed as

5 Numerical experiments

In this section, some numerical examples are given to illustrate the properties and effectiveness of the method. We also compare the approximate solution with some other numerical solutions.

We can approximate solution of the system as

Hence, the solution to the stiff system (23) is:

• Stage 1. On the intervals $[0\text{,}{x}_1)\text{,}[x_1\text{,}{x}_2)\text{,}[x_2\text{,}{x}_3)\dots [x_{k-1}\text{,}1)$ , the collocation points $(x_0=1\text{,}{x}_1=0.1\dots x_8=0.8)$ which were obtained from (18).

• Stage 2. On the intervals $[1\text{,}{x}_1)\text{,}[x_1\text{,}{x}_2)\text{,}[x_2\text{,}{x}_3)\dots [x_{k-1}\text{,}2)$ , but the collocation points which will be used are $(x_0=2\text{,}{x}_1=1.1\dots x_{13}=1.8)$ .

  $⋮$

• Stage 5. On the intervals $[5\text{,}{x}_1)\text{,}[x_1\text{,}{x}_2)\text{,}[x_2\text{,}{x}_3)\dots [x_{k-1}\text{,}6)$ . The collocation points which will be used are $(x_0=5\text{,}{x}_1=5.1\dots x_{13}=5.8)$ .

The theoretical solution when $\alpha =1$ is $$\begin{array}{cc}\hfill & y_1=\frac{1}{2}\left({\text{e}}^{-0.5}+{\text{e}}^{-20x}(\cos (20x)+\sin (20x))\right)\text{,}\hfill \\ \hfill & y_2=\frac{1}{2}\left({\text{e}}^{-0.5}-{\text{e}}^{-20x}(\cos (20x)-\sin (20x))\right)\text{,}\hfill \\ \hfill & y_3=-\frac{1}{2}\left({\text{e}}^{-0.5}+{\text{e}}^{-20x}(\cos (20x)-\sin (20x))\right)\text{.}\hfill \end{array}$$ In Table 3 we present the numerical solutions which are applied by the 9-term B-polynomials and the 9-term MB-polynomials, with $\alpha =0.90$ and the step-length is $h=0.2$ . Evidently, the classical B-polynomials solutions are not valid for a long time. MB-polynomials method solves this problem and solutions are valid for large x. In Table 4 we listed the error of the computed solution obtained by the MB-polynomials and compared it with B-polynomials and A-EBDF (Hojjati et al., 2004). To evaluate the approximation values of the solution at a given x, the step-length is $h=0.2$ , with $m=14\text{,}j=1\text{,}2\text{,}3\text{,}k=6$ and $r=10$ .

6 Conclusions

In this paper, the MB-polynomials is considered a simple modification of the standard B-polynomials. We applied MB-polynomials to solve Fractional Order Stiff Systems. Comparison between MB-polynomials and other several methods as B-polynomials, A-EBDF, HPM and RHBM indicates that MB-polynomials can solve stiff problems more accurately with less iterations, also MB-polynomials is considered valid in large x than standard B-polynomials. The subjects of our future works can be exemplified by applying MB-polynomials for solving different systems, like Chaotic Fractional Order Systems and Lorenz system.