Maximum absolute error for

## Abstract

This paper is concerned about the study of shifted Jacobi polynomials. By means of these polynomials, we construct some operational matrices of fractional order integration and differentiations. Based on these matrices, we develop a numerical scheme for the boundary value problems of fractional order differential equations. The construction of the procedure is new one for the coupled systems of fractional order boundary value problems. In the proposed scheme, we obtain a simple but highly accurate systems of algebraic equations. These systems are easily soluble by means of Matlab or using Mathematica. We provide some examples to which the procedure is applied to verify the applicability of our proposed method.

### Keywords

- Jacobi polynomials
- operational matrices
- fractional order differential equations
- coupled system
- boundary conditions

## 1. Introduction

Recently, fractional order differential equations have gained much attention from the researchers of mathematics, physics, computer science and engineers. This is due to the large numbers of applications of fractional order differential equations in various fields of science and applied nature. The applications of fractional order differential equations are found in physics, mechanics, viscoelasticity, photography, biology, chemistry, fluid mechanics, image and signal processing phenomenons, etc.; for more details, see [1, 2, 3, 4]. The researchers are giving more attention to fractional order differential equations, because in most cases the fractional order models are more accurate and reliable than classical order models. On the other hand, fractional order models have more degrees of freedom than integer order models. In most cases, the hereditary phenomena and memory description process are accurately modeled by means of fractional order derivatives and integrations as compare with integer order derivative and integrations. Due to these facts, researchers had given much attention to study the existence and uniqueness of positive solutions as well as multiplicity of solutions for fractional order differential equations and their systems. This area is very well explored, and large numbers of research articles can be found in literature; for details, see [5, 6, 7, 8, 9, 10]. It is very difficult to obtain exact solutions for such type of differential equations in all cases always. However, the area involving numerical solutions is in initial stage. Therefore, various numerical schemes have been developed in last few years by different researchers. The aim of these schemes was to achieve better accuracy. These schemes have their own merits and demerits. For example, in [11, 12, 13], the authors developed Adomain decomposition methods, Homotopy analysis methods, and Variational iterational methods to obtained good approximate solutions for certain fractional order differential equations. In recent years, spectral methods have got much attention as they were applied to solve some real-word problems of various fields of science and engineering. High accuracy was obtained for solving such problems [14]. Spectral method needs operational matrices for the numerical solutions, which have been constructed by using some polynomials, for example, in [15], the authors developed an operational matrix for shifted Legendre polynomials corresponding to fractional order derivative. In [16], the authors introduced operational matrix by using shifted Legendre polynomials corresponding to fractional order integrations. Similarly in [17, 18], authors constructed operational matrices for fractional order derivative by using Chebyshev and Jacobi polynomials. In all these cases, these matrices were applied to solve multiterms fractional order differential equations together with Tau-collocation method. In [19], Singh et al. derived an operational matrices for Berestein polynomials corresponding to fractional order integrations. Haar wavelet operational matrices were also developed, and some problems of fractional order differential equations were solved. In all these methods, collocation method was used together with these methods to obtain numerical solutions for fractional order differential equations. They only solved single problems.

Most of the physical and biological phenomena can be modeled in the form of coupled systems. For example, when two masses and two spring systems are modeled, it led to a coupled system of differential equations.

subject to the boundary conditions

Many coupled systems can be found in physics and in biological model, chemistry, fluid dynamics, etc.; see [16, 20, 21, 22, 23, 24, 25, 26, 27]. In fractional calculus, we generalize these systems by replacing the integer order by any fractional values that lie in certain range. All the above numerical methods were applied to solve singles problems, and to the best of our knowledge, very few articles are devoted to the numerical solutions of coupled systems or models. In all these cases, the authors have solved initial value problems of coupled systems of fractional order differential and partial differential equations; for details, see [15, 28] and the references therein. For example, in [15], authors have solved some initial value problems of coupled systems of fractional differential equations (FDEs) by using shifted Jacobi polynomials operational matrix method. However, the scheme is not properly applied yet for solving coupled systems of boundary value problems of fractional order differential equations. In this paper, we discuss the shifted Jacobi polynomial operational matrices methods to solve boundary value problems for a coupled systems of fractional order differential equations. In this scheme, we introduced a new matrix corresponding to boundary conditions, which is required for the approximate solutions. First, we solve single boundary value problems given by

subject the boundary conditions

where

subject to the boundary conditions given by

where

Our article is organized as follows: Some fundamental concepts and results of fractional calculus and Jacobi polynomials needed are provided in sections 2. Section 3 contains some operational matrices for shifted Jacobi polynomials corresponding to fractional order derivative and integrations. Also in this section, we provide new operational matrix corresponding to the boundary conditions. In Section 4, we derive main procedure for the general class of boundary value problems of fractional order differential equations as well as for coupled system of FDEs. In Section 5, we apply the proposed method to some examples. Also we plot the images of these examples. In Section 6, we provide a short conclusion of the papers.

## 2. Preliminaries

In this section, we recall some basic definitions and results of fractional calculus, also we will give fundamental properties, definitions related to Jacobi polynomials as in [1, 4, 6, 7, 8, 14, 17, 18, 28, 29, 30, 31].

** Definition 2.1.**The fractional integral of order

provided the integral converges at the right sides. Further, a simple and important property of

provided that the right side is pointwise defined on

The following results are needed in the sequel.

** Lemma 2.2.1.**[6], Let

for arbitrary

### 2.1 The shifted Jacobi polynomials and its fundamental properties

In this section, we provide basic properties of shifted Jacobi polynomials. The famous Jacobi polynomials

By means of the substitution

The orthogonality conditions of the shifted Jacobi polynomials are given by

and

Any function

where the shifted Jacobi coefficient vector is denoted by

## 3. Operational matrices

In this section, we provide some operational matrices for fractional order differentiation as well as for fractional order integration. Further, in this section, we give construction of new matrix for the boundary conditions.

** Theorem 3.1.**Let

where

where

* Proof.*The proof of this theorem is same as given in [14, 17] in lemma

_{,} then the operational matrix corresponding to the fractional order integration is given by

where

where

* Proof.*The proof of this theorem is available in [14]. Therefore, we omit the proof.

where

where

where

and

* Proof.*Consider the general term of

Now by applying Laplace transform, we get

Now putting (27) in (26), we get

Now if

Representing

By the use of orthogonality relation (11), we have

Now by means of Laplace transform, we have

Thus putting (32) in (31) and using (12), we get

Now (29) implies that

where

Evaluating the result (34) for different values of

## 4. Applications of operational matrices

In this section, we show fundamental importance of operational matrices of fractional order derivative and integration. We apply them to solve some multiterms fractional order boundary value problems of fractional differential equations. Consider the following general fractional differential equation with constant coefficient and given boundary conditions

subject the boundary conditions

where

To obtain the solutions of the (35) in terms of shifted Jacobi polynomials, we proceed as

Applying

By means of boundary conditions we get

using the approximation

Now from (39) we have

Putting (36)–(40) in (35), which yields

Which is a simple algebraic equation. Solving this equation for

### 4.1 Coupled system of boundary value problems for fractional order differential equations

In this subsection, we use operational matrices to derive procedure for the numerical solutions of coupled system. We consider the following general coupled system of FDEs as

subject to the boundary conditions given by

where

and

Applying

Now taking fractional order derivative of (46), (47), we have

and

Putting (45)–(49) together with the use of approximation

Rearranging the terms in above system (50) and using the notation for simplicity

then witting in matrix form, we have

which is an algebraic equation and can easily be solved for unknown matrices

## 5. Numerical examples

In this section, we will apply our proposed scheme to some practical problems.

where

1 | ||||||

In Table 2, we have shown the maximum absolute error for various choices of

Let the exact solution

We approximate the solutions of this problem with proposed method by setting

The absolute error for different choices of

From Table 4, we observed that as the orders of derivatives tend to their corresponding integer values, the approximate solutions tend to its exact value, which demonstrate the accuracy of numerical solutions obtained in our proposed method.

In the following examples, we solve some coupled systems under the given conditions by our proposed method.

** Example 3.**Consider the coupled system given by

subject to the boundary conditions

We solve this problem under the given parameters

Let the exact solutions at

We approximate the coupled systems at scale level

Absolute error for specific

From Table 5, we see the effect of

** Example 4.**Consider the mathematical models of fractionally damped coupled system of spring masses system whose model is given by

subject to the boundary conditions

Where the source terms are given by

Where

Let the exact solutions at

The above model is approximated for the solutions by our proposed methods (Figures 6 and 7). We observed that our method provides best approximate solutions to the problems for small-scale level

Absolute error for various

We observe from Table 7 that values of _{,} the absolute errors are given in Table 8.

From Table 8, it is obvious that when the orders of the derivatives approach to their integral values, the error is reducing and the approximate solutions are converging to the exact solutions. This phenomenon indicates that the proposed method is highly accurate.

## 6. Conclusion and discussion

In this article, we have studied shifted Jacobi polynomials. Based on these polynomials, we recalled some already existing matrices of fractional order derivative and integrations from the literature as well as we constructed a new operational matrix corresponding to boundary conditions. By means of these operational matrices, we converted the system of fractional differential equations to simple and easily soluble system of algebraic equations. There is no need of Tau-collocation method. The simple algebraic equations were easily solved, and the result were plotted. From the plot, we observe that our proposed method is highly accurate and can be applied to a variety of problems of fractional order ordinary as well as partial differential equations. We also compared our results to the exact solutions and observed that our method gave satisfactory results. The proposed method can easily and accurately can be applied to a variety of problems of applied mathematics, engineering, and physics, etc.

## Acknowledgments

The authors Aziz Khan, Kamal Shah and Thabet Abdeljawad would like to thank Prince Sultan university for the support through TAS research lab.

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