Bernstein collocation method for neutral type functional differential equation

Ishtiaq Ali; Ishtiaq Ali

doi:10.3934/mbe.2021140

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 3: 2764-2774. doi: 10.3934/mbe.2021140

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Research article Special Issues

Bernstein collocation method for neutral type functional differential equation

Ishtiaq Ali ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 01 February 2021 Accepted: 09 March 2021 Published: 22 March 2021

Functional differential equations of neutral type are a class of differential equations in which the derivative of the unknown functions depends on the history of the function and its derivative as well. Due to this nature the explicit solutions of these equations are not easy to compute and sometime even not possible. Therefore, one must use some numerical technique to find an approximate solution to these equations. In this paper, we used a spectral collocation method which is based on Bernstein polynomials to find the approximate solution. The disadvantage of using Bernstein polynomials is that they are not orthogonal and therefore one cannot use the properties of orthogonal polynomials for the efficient evaluation of differential equations. In order to avoid this issue and to fully use the properties of orthogonal polynomials, a change of basis transformation from Bernstein to Legendre polynomials is used. An error analysis in infinity norm is provided, followed by several numerical examples to justify the efficiency and accuracy of the proposed scheme.

Keywords:

Citation: Ishtiaq Ali. Bernstein collocation method for neutral type functional differential equation[J]. Mathematical Biosciences and Engineering, 2021, 18(3): 2764-2774. doi: 10.3934/mbe.2021140

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Abstract

1. Introduction

Very recently the concept of fractal differentiation and fractional differentiation have been combined to produce new differentiation operators ^[1,2,3,4]. The new operators were constructed using three different kernels namely, power law, exponential decay and the generalised Mittag-Leffler function. The new operators have two parameters, the first is considered as fractional order and the second as fractal dimension.

The operators were tested to model some real world problems ^[9,10,11] surprisingly, the new differential and integral operators were found to be powerful mathematical tools able to capture even hidden complexities of nature ^[5,6,16]. Very detailed attractors could be captured when modelling with such differential and integral operators ^[8]. One of the great advantage of these new operators is that they can at the same time capture processes following power law, exponential decay law, fading memory, crossover and self-similar processes. This unique and outstanding capacity makes these new operators suitable tools to model many complex real world problems. Additionally when the fractal dimension is one, we recover all the existing fractional differential and integral operators, if the fractional order is one, we recover the so-call fractal differential and integral operators. Finally if the fractal and fractional orders tend to 1 we recover classical differential and integral operators. Therefore, it is believed that these new operators are the future for modeling.

On the other hand, modeling real-life problems have been always of interest of researchers. Mathematicians especially have introduced and developed a variety of well-known and sophisticated tools to model problems arising from other sciences such as biology, finance, epidemiology, and geology. The list is not exhaustive. Moreover, they used these tools to investigate solutions to many important and crucial problems that humans are challenging in their daily life. Among these tools are differential equations and fractal differential equations for instance. Nevertheless, the continuous improvement leads constantly their researches in order to obtain better solutions that aim at reaching the reality. Fractal-fractional differential operators can be seen as an enhanced instrument for modeling important practical problems. One of the main motivations for this article was to introduce methods for solving fractal-fractional problems that have not been considered before. By using the proposed methods, approximate solutions to these problems can be determined more easily and with higher accuracy. In addition, the methods can be applied to real-world problems. In this work, we present applications of such numerical schemes in solving chaotic models that involve the new class of differential operators. We consider some well-known chaotic models with fractal-fractional differential operators, so that we can compare our results with those in the literature. The article is organized as follows: In section 2, we give a brief overview of some basic definitions of fractal-fractional differential calculus. Two efficient and effective numerical methods for determining approximate solutions to fractal-fractional problems are presented in section 3. The first is for the Caputo fractal derivative second is for Caputo-Fabrizio-Caputo fractal-fractional derivative and the thierd is for the Atangana-Baleanu-Caputo fractal-fractional derivative. The kernels used in these two definitions are singular and non-singular, respectively. Several numerical simulations for chaotic systems are described in section 4 we find the numerical analysis for Duffing attractor. Numerical analysis for EL-Ni $\hat{n}$ o Southern Oscillation is find in section 5. In section 6–9, we analysis numerical analysis for Ikeda system, numerical analysis for Dadras attractor and numerical analysis for Aizawa attractor, numerical analysis for Thomas attractor and numerical analysis for 4 Wings attractor. In section 9 we introduce a new Atangana Sonal attractor and find the solution of this. The results obtained are accurate, interesting, and meaningful. Finally, we present our overall conclusions.

2. Preliminaries on fractal-fractional calculus

Definition 2.1. ^[3] The fractal fractional derivative of f(t) with order $\varpi-\kappa$ in the Riemann-Liouville sense is defined as follows:

$\begin{equation} ^{FF-RL}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{1}{\Gamma(m-\kappa)}\frac{d}{dt^\varpi}\int_{0}^{t}(t-s)^{m-\kappa-1}f(s)ds, \end{equation}$

(2.1)

where $m-1 < \kappa, \varpi\leq m\in\mathbb{N}$ and $\frac{df(s)}{ds^\varpi} = \lim\limits_{t\rightarrow s} \frac{f(t)-f(s)}{t^\varpi-s^\varpi}.$

Definition 2.2. ^[3] The Caputo-Fabrizio fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Riemann-Liouville sense is defined as follows:

$\begin{equation} ^{FF-CFR}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{M(\kappa)}{(1-\kappa)}\frac{d}{dt^\varpi}\int_{0}^{t}\exp\left( -\frac{\kappa}{1-\kappa} (t-s)\right)f(s)ds, \end{equation}$

(2.2)

where $\kappa > 0, \; \varpi\geq m\in\mathbb{N}$ and $M(0) = M(1) = 1.$

Definition 2.3. ^[2] The Atangan-Baleanu fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Riemann-Liouville sense is defined as follows:

$\begin{equation} ^{FF-ABR}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{AB(\kappa)}{(1-\kappa)}\frac{d}{dt^\varpi}\int_{0}^{t}E_\kappa\left( -\frac{\kappa}{1-\kappa} (t-s)^\kappa\right)f(s)ds, \end{equation}$

(2.3)

where $0 < \kappa, \; \varpi\leq 1$ and $AB(\kappa) = 1-\kappa+\frac{\kappa}{\Gamma(\kappa)}.$

Definition 2.4. ^[3] The fractal-fractional derivative of with order $\varpi-\kappa$ in the Liouville-Caputo sense is defined as follows:

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{1}{\Gamma{(m-\kappa)}}\frac{d}{dt^\varpi}\int_{0}^{t}(t-s)^{m-\kappa-1}\left(\frac{d}{ds^\tau}f(s)\right)ds, \end{equation}$

(2.4)

where $m-1 < \kappa, \; \varpi\leq m\in\mathbb{N}$ and $\frac{df(s)}{ds^\varpi} = \lim\limits_{t\rightarrow s} \frac{f(t)-f(s)}{t^\varpi-s^\varpi}.$

Definition 2.5. ^[2] The Caputo-Fabrizio fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Liouville-Caputo sense is defined as follows:

$\begin{equation} ^{FF-CFC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{M(\kappa)}{(1-\kappa)}\int_{0}^{t}\exp\left( -\frac{\kappa}{1-\kappa} (t-s)\right) \left( \frac{d}{ds^\varpi}f(s)\right)ds, \end{equation}$

(2.5)

where $\kappa > 0, \; \varpi\geq m\in\mathbb{N}$ and $M(0) = M(1) = 1.$

Definition 2.6. ^[4] The Atangana-Baleanu fractal-fractional derivative of $f(t)$ with order $\varpi-\kappa$ in the Liouville-Caputo sense is defined as follows:

$\begin{equation} ^{FF-ABC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{AB(\kappa)}{(1-\kappa)}\int_{0}^{t}E_\kappa\left( -\frac{\kappa}{1-\kappa} (t-s)^\kappa\right)\left( \frac{d}{ds^\varpi}f(s)\right) ds, \end{equation}$

(2.6)

where $0 < \kappa, \; \varpi\leq 1$ and $AB(\kappa) = 1-\kappa+\frac{\kappa}{\Gamma(\kappa)}.$

Definition 2.7. ^[2] The Liouville-Caputo fractal-fractional integral of $f(t)$ with order $\kappa$ is defined as follows:

$\begin{equation} ^{FF-C}\mathfrak{I}_{0, t}^{\kappa}\left\lbrace f(t)\right\rbrace = \frac{\varpi}{\Gamma(\kappa)}\int_{0}^{t}(t-s)^{\kappa-1}s^{\varpi-1}f(s)ds. \end{equation}$

(2.7)

Definition 2.8. ^[3] The Caputo-Fabrizio fractal-fractional integral of $f(t)$ with order $\kappa$ is defined as follows:

$\begin{equation} ^{FF-CF}\mathfrak{I}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{\kappa\varpi}{M(\kappa)}\int_{0}^{t}s^{\kappa-1}f(s)ds+\frac{\varpi(1-\kappa)t^{\varpi-1}}{M(\kappa)}f(t). \end{equation}$

(2.8)

Definition 2.9. ^[3] The Atangana-Baleanu fractal-fractional integral of $f(t)$ with order $\kappa$ is defined as follows:

$\begin{equation} ^{FF-AB}\mathfrak{I}_{0, t}^{\kappa, \varpi}\left\lbrace f(t)\right\rbrace = \frac{\kappa\varpi}{AB(\kappa)}\int_{0}^{t}s^{\kappa-1}(t-s)^{\kappa-1}f(s)ds+\frac{\varpi(1-\kappa)t^{\varpi-1}}{AB(\kappa)}f(t). \end{equation}$

(2.9)

3. Numerical approximation of fractal-fractional calculus

In this section, we have given three numerical schemes for Caputo-fractal-fractional, Caputo-Fabrizio-fractal fractional and the Atangana-Baleanu fractal-fractional derivative operators ^[7].

3.1. Caputo-fractal-fractional derivative

Consider the following differential equations in the fractal-fractional Liouville-Caputo sense

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u(t)\right\rbrace = f(t, u(t), v(t)). \end{equation}$

(3.1)

Equation (3.1) can be converted to the Volterra case and the numerical scheme of this system using a Caputo-fractal-fractional approach at $t_{n+1}$ is given by

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi}{\Gamma(\kappa)}\int_0^{t_{n+1}}s^{\varpi-1}(t_{n+1-s})^{\kappa-1}f(s, u, v)ds. \end{equation}$

(3.2)

We can approximate the above integral to

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi}{\Gamma(\kappa)}\sum\limits_{j = 0}^{n}\int_{t_j}^{t_{j+1}}s^{\varpi-1}(t_{n+1-s})^{\kappa-1}f(s, u, v)ds. \end{equation}$

(3.3)

With in the finite interval $[t_j, t_{j+1}]$ , we approximate the function $s^{\varpi-1}, f(t, u, v)$ using the Lagrangian piecewise interpolation such that

$\begin{equation} \mathbb{E}(s) = \frac{s-t_{j-1}}{t_j-t_{j-1}}t_j^{\varpi-1}f(t_j, u^j, v^j)- \frac{s-t_{j}}{t_j-t_{j-1}}t_{j-1}^{\varpi-1}f(t_{j-1}, u^{j-1}, v^{j-1}). \end{equation}$

(3.4)

So we obtain

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi}{\Gamma(\kappa)}\sum\limits_{j = 0}^{n}\int_{t_j}^{t_{j+1}}s^{\varpi-1}(t_{n+1-s})^{\kappa-1}\mathbb{E}(s)ds. \end{equation}$

(3.5)

Solving the integral of the right hand side, we obtain the following numerical scheme

$\begin{equation} \begin{split} u(t_{n+1}) & = u^0+\frac{\varpi(\Delta t)^\kappa}{\Gamma(\kappa+2)}\sum\limits_{j = 0}^{n}\left[t_{j}^{\varpi-1}f(t_{j}, u_{j}, v^{j})\left( (n+1-j)^\kappa(n-j+2+\kappa)\right.\right.\\ &\left.\left.-(n-j)^\kappa(n-j+2+2\kappa)\right)-t_{j-1}^{\varpi-1}f(t_{j-1}, u_{j-1}, v^{j-1}) \left( (n+1-j)^{\kappa+1}\right.\right.\\ &\left.\left.-(n-j)^\kappa(n-j+1+\kappa)\right)\right]. \end{split} \end{equation}$

(3.6)

3.2. Caputo-Fabrizio-Caputo fractal-fractional derivative

Consider the following differential equations in the Caputo-Fabrizio-fractal-fractional derivative

$\begin{equation} ^{FF-CFC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u(t)\right\rbrace = f(t, u(t), v(t)). \end{equation}$

(3.7)

Applying the Caputo-Fabrizio integral, we obtain

$\begin{equation} u(t) = u^0+\frac{\varpi t^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t, u, v)+\frac{\kappa \varpi}{M(\kappa)}\int_0^{t}s^{\varpi-1}f(s, u, v)ds. \end{equation}$

(3.8)

Here we present the detailed derivation of the numerical scheme. Thus, at $t_{n+1}$ we have

$\begin{equation} u(t_{n+1}) = u^0+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_n, u^n, v^n)+\frac{\kappa \varpi}{M(\kappa)}\int_0^{t_{n+1}}s^{\varpi-1}f(s, u, v)ds. \end{equation}$

(3.9)

Taking the difference between the consecutive terms, we obtain

$\begin{equation} \begin{split} u(t_{n+1}) & = u^n+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_n, u^n, v^n)-\frac{\varpi t_{n-1}^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_{n-1}, u^{n-1}, v^{n-1})\\ &+\frac{\kappa \varpi}{M(\kappa)}\int_{t_{n}}^{t_{n+1}}s^{\varpi-1}f(s, u, v)ds. \end{split} \end{equation}$

(3.10)

Now using the Lagrange polynomial piece-wise interpolation and integrating, we obtain

$\begin{equation} \begin{split} u(t_{n+1}) & = u^n+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_n, u^n, v^n)-\frac{\varpi t_{n-1}^{\varpi-1}(1-\kappa) }{M(\kappa)}f(t_{n-1}, u^{n-1}, v^{n-1})\\ &+\frac{\kappa \varpi}{M(\kappa)}\times\left[\frac{3}{2}(\Delta t)t_n^{\varpi-1}f(t_n, u^n, v^n) -\frac{\Delta t}{2}t_{n-1}^{\varpi-1}f(t_{n-1}, u^{n-1}, v^{n-1})\right]. \end{split} \end{equation}$

(3.11)

3.3. Atangana-Baleanu-Caputo fractal-fractional derivative

Consider the following differential equations in the Atangana-Baleanu fractal-fractional derivative in the Liouville-Caputo sense

$\begin{equation} ^{FF-ABC}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u(t)\right\rbrace = f(t, u(t), v(t)). \end{equation}$

(3.12)

Applying the Atangana-Baleanu integral, we have

$\begin{equation} u(t) = u(0)+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t, u, v)+\frac{\kappa \varpi}{AB (\kappa)\Gamma(\kappa)}\int_{0}^{t}s^{\varpi-1}(t-s)^{\kappa-1}f(s, u, v)ds. \end{equation}$

(3.13)

At $t_{n+1}$ , we have the following

$\begin{equation} \begin{split} u(t_{n+1}) & = u(0)+\frac{\kappa \varpi}{AB (\kappa)\Gamma(\kappa)}\int_{0}^{t_{n+1}}s^{\varpi-1}(t_{n+1}-s)^{\kappa-1}f(s, u, v)ds\\ & +\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t_n, u^n, v^n), \end{split} \end{equation}$

(3.14)

the above system can be expressed as

$\begin{equation} \begin{split} u(t_{n+1}) & = u(0)+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t_n, u^n, v^n)\\ &+\frac{\kappa \varpi}{AB (\kappa)\Gamma(\kappa)}\sum\limits_{j = 0}^{n}\int_{t_{j}}^{t_{j+1}}s^{\varpi-1}(t_{n+1}-s)^{\kappa-1}f(s, u, v)ds. \end{split} \end{equation}$

(3.15)

Approximating $s^{\varpi-1f(s, u, v)}$ in $[t_j, t_{j+1}]$ , we have the following numerical scheme

$\begin{equation} \begin{split} u(t_{n+1}) & = u(0)+\frac{\varpi t_n^{\varpi-1}(1-\kappa) }{AB(\kappa)}f(t_n, u^n, v^n)+\frac{\varpi\Delta t^\kappa}{AB(\kappa)\Gamma(\kappa+2)}\\ &\times\sum\limits_{j = 0}^{n}\left[t_j^{\varpi-1} f(t_j, u^j, v^j)[(n+1-j)^\kappa(n-j+2+\kappa)-(n-j)^\kappa(n-j+2+2\kappa)]\right. \\ &\left.-t_{j-1}^{\varpi-1}f(t_{j-1}, u^{j-1}, v^{j-1})\left( (n+1-j)^{\kappa+1}-(n-j)^\kappa(n-j+1+\kappa)\right) \right]. \end{split} \end{equation}$

(3.16)

4. Numerical analysis for Duffing attractor

The Duffing Pendulum is a kind of a forced oscillator with damping, the mathematical model represented in state variable as when we apply Caputo-fractal-fractional derivative is given by

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u_1(t)\right\rbrace = Au_2(t), \end{equation}$

(4.1)

$\begin{equation} ^{FF-C}\mathfrak{D}_{0, t}^{\kappa, \varpi}\left\lbrace u_2(t)\right\rbrace = u_1(t)-u_1^3(t)-au_2(t)+b\cos(\omega t), \end{equation}$

(4.2)

where initial conditions are $u_1(0) = 0.01$ , $u_2(0) = 0.5$ and the constant are $\kappa = 0.25,$ $b = 0.3$ , $\omega = 1$ , $h = 0.01$ , and $t = 100.$ The numerical scheme is given by

$u_1(t_{n+1}) = u_1^0 +\frac{A\varpi(\Delta t)^\kappa}{\Gamma(\kappa+2)}\sum\limits_{j = 0}^{n}\left[t_{j}^{\varpi-1}u_2(t)^j\left( (n+1-j)^\kappa(n-j+2+\kappa)\\ -(n-j)^\kappa(n-j+2+2\kappa)\right)\right.\\ \left.-t_{j-1}^{\varpi-1}Au_2(t)^{j-1} \left( (n+1-j)^{\kappa+1}-(n-j)^\kappa(n-j+1+\kappa)\right)\right], \\ u_2(t_{n+1}) = u_1^0+\frac{\varpi(\Delta t)^\kappa}{\Gamma(\kappa+2)}\sum\limits_{j = 0}^{n}\left[t_{j}^{\varpi-1}\left( u_1(t)^j-u_1^3(t)^j-au_2(t^j-\tau)+b\cos\omega t^j\right)\right. \\ \left. \times \left( (n+1-j)^\kappa(n-j+2+\kappa)-(n-j)^\kappa(n-j+2+2\kappa)\right)\right.\\ \left.-\left( t_{j-1}^{\varpi-1}u_1(t)^{j-1}-u_1^3(t)^{j-1}-au_2(t^{j-1}-\tau)+b\cos\omega t^{j-1} \right)\right. \\ \left. \left( (n+1-j)^{\kappa+1}-(n-j)^\kappa(n-j+1+\kappa)\right)\right].$

(4.3)

Numerical simulation of the Caputo-power law case are depicted in – for different values of fractional order $\kappa$ and fractal dimension $\varpi$ .

Figure 1. Numerical simulation for

$\kappa = \varpi = 1$ and A = 1 with classical differentiation for Duffing attractor.

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Mathematical Biosciences and Engineering

Bernstein collocation method for neutral type functional differential equation

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Abstract

1. Introduction

2. Preliminaries on fractal-fractional calculus

3. Numerical approximation of fractal-fractional calculus

3.1. Caputo-fractal-fractional derivative

3.2. Caputo-Fabrizio-Caputo fractal-fractional derivative

3.3. Atangana-Baleanu-Caputo fractal-fractional derivative

4. Numerical analysis for Duffing attractor

5. Numerical analysis for EL-Niˆn \hat{n} o Southern Oscillation

6. Numerical analysis for Ikeda system

7. Numerical analysis for Dadras attractor

8. Numerical analysis for Aizawa attractor

9. Numerical analysis for Thomas attractor

10. Numerical analysis for 4 Wings attractor

11. AS attractor

12. Conclusions

Acknowledgment

Conflict of interest

References

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5. Numerical analysis for EL-Ni $\hat{n}$ o Southern Oscillation