Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre Polynomials

This article is devoted to compute numerical solutions of some classes and families of fractional order differential equations (FODEs). For the required numerical analysis, we utilize Laguerre polynomials and establish some operational matrices regarding to fractional order derivatives and integrals without discretizing the data. Further corresponding to boundary value problems (BVPs), we establish a new operational matrix which is used to compute numerical solutions of boundary value problems (BVPs) of FODEs. Based on these operational matrices (OMs), we convert the proposed (FODEs) or their system to corresponding algebraic equation of Sylvester type or system of Sylvester type. The resulting algebraic equations are solved by MATLAB® using Gauss elimination method for the unknown coefficient matrix. To demonstrate the suggested scheme for numerical solution, many suitable examples are provided.


Introduction
The theory of integrals as well as derivatives of arbitrary order is known by the special name "fractional calculus." It has an old history just like classical calculus. The chronicle of fractional calculus and encyclopedic book can be studied in [1,2]. Researchers have now necessitated the use of fractional calculus due to its diverse applications in different fields, specially in electrical networks, signal and image processing and optics, etc. For conspicuous work on FODEs in the fields of dynamical systems, electrochemistry, advanced techniques of microorganisms culturing, weather forecasting, as well as statistics, we refer to peruse [3,4]. Fractional derivatives show valid results in most cases where ordinary derivatives do not. Also annotating that fractional order derivatives as well as fractional integrals are global operators, while ordinary derivatives are local operators. Fractional order derivative provides greater degree of freedom. Therefore from different aspects, the aforesaid areas were investigated. For instance, many researchers have provide understanding to existence and uniqueness results about FODEs, for few results, we refer [5][6][7], and many others have actualized the instinctive framework of fractional differential equations in various problems [8][9][10][11][12][13][14][15][16][17][18][19] with many references included in them.
Often it is very difficult to obtain the exact solution due to global nature of fractional derivatives in differential equations. Contrarily approximate solutions are obtained by numerical methods assorted in [20][21][22]. Various new numerical methods have been developed, among them is one famous method called "spectral method" which is used to solve problems in various realms [23]. In this method operational matrices are obtained by using orthogonal polynomials [24]. Many authors have successfully developed operational matrices by using Legender, Jacobi, and various other polynomials [25,26]. For delay differential and various other related equations, Laguerre spectral methods have been used [27][28][29][30][31][32]. Bernstein polynomials and various classes of other polynomials were also used to obtain operational matrices corresponding to fractional integrals and derivatives [33][34][35][36][37][38][39][40]. Apart from them, operational matrices were also developed with the collocation method (see Refs. [41][42][43]). Since spectral methods are powerful tools to compute numerical solutions of both ODEs and FODEs. Therefore, we bring out numerical analysis via using Laguerre polynomials of some families and coupled systems of FODEs under initial as well as boundary conditions. In this regard we investigate the numerical solutions to the given families under initial conditions and subject to boundary conditions By similar numerical techniques, we also investigate the numerical solutions to the following systems with fractional order derivatives under initial and boundary conditions as for 1 < γ ≤ 2 where f , g : 0, 1 ½ ÂR 2 ! R and z 0 , y 0 , z 1 , y 1 ∈ R: We first obtain OMs for fractional derivatives and integrals by using Laguerre polynomials. Also corresponding to boundary conditions, we construct an operational matrix which is needed in numerical analysis of BVPs. With the help of the OMs we convert the considered problem of FODEs under initial/boundary conditions to Sylvester-type algebraic equations. Solving the mentioned matrix equations by using MATLAB®, we compute the numerical solutions of the considered problems.

Preliminaries
Here we recall some basic definition results that are needed in this work onward, keeping in mind that throughout the paper we use fractional derivative in Caputo sense. Definition 1. The fractional integral of order γ >0 of a function z : 0, ∞ ð Þ!R is defined by provided the integral converges at the right sides. Further a simple and important property of 0 I γ t is given by Definition 2. Caputo fractional derivative is defined as where n is a positive integer with the property that n À 1 < γ ≤ n: For example, if 0 < γ ≤ 1, then Caputo fractional derivative becomes Lemma 1. Therefore in view of this result, if h ∈ L n 0, T ½ , then the unique solution of nonhomogenous FODE is written as where d i for i ¼ 0, 1, 2, 3 … n À 1 are real constants. The above lemma is also stated as Definition 3. The famous Laguerre polynomials are represented by L γ i t ð Þ and defined as They are orthogonal on 0, ∞ ½ : If L γ i t ð Þ and L γ j t ð Þ are Laguerre polynomials, then the orthogonality condition is given as is the weight function and Now let Z t ð Þ be any function, defined on the interval 0, ∞ ½ : We express the function in terms of Laguerre polynomials as We set the above two vectors into their inner product and represent the column matrix by Ψ t ð Þ, so that which is written as We call h i to the general term of integration Hence the coefficient c i is In vector form we can write Eq. (5) as where M = m þ 1, c M is the M terms coefficient vector and Ψ M t ð Þ is the M terms function vector.

Representation of Laguerre polynomial with Caputo fractional order derivative
If the Caputo fractional order derivative is applied to Laguerre polynomial, by considering whole function constant except t k : We use the definition of Caputo fractional order derivative for t k to obtain (6) as

Error analysis
The proof of the following results can be found with details in [20].
Theorem 2. For error analysis, we state the theorem such that, a be any integer and 0 ≤ s ≤ a, and then with the following inner product and norm

Operational matrices corresponding to fractional derivatives and integrals
Here in this section, we provide the required OMs via Laguerre polynomials of fractional derivatives and integrals. Lemma 3. Let Ψ M t ð Þ be a function vector; the fractional integral of order γ for the function Ψ M t ð Þ can be generalized as where G γ NÂN is the OM of integration of fractional order γ and given by Proof. We apply the fractional order integral of order γ to the Laguerre polynomials Since from (7), we have Therefore Eq. (7) implies that which is equal to We approximate t kþ γ in (8) with Laguerre polynomials, i.e.
By using the relation of orthogonality, we can find coefficients which is the desired result. Lemma 4. Let Ψ M t ð Þ be a function vector; then the fractional derivative of order γ for Ψ M t ð Þ is generalized as where W γ MÂM is the OM of derivative of order γ , defined as in (9) where Proof. Leaving the proof as it is very similar to the proof of the above lemma. Lemma 5. We consider a function Z t ð Þ defined on 0, ∞ ½ and y t where Q γ MÂM is the operational matrix, given by , where Proof. By considering the general term of Ψ M t ð Þ Using the famous Laplace transform, we have from (10) £ð Now using Laguerre polynomials, we have where C i,j is calculated by using orthogonality as To get the desired result, we evaluate the above (11) relation for i ¼ 0, 1, … , m and j ¼ 0, 1, … , m.

Main result
In this section, we discuss some cases of FODEs with initial condition as well as boundary conditions. The approximate solution obtained through desired method is compared with the exact solution. Similarly we investigate numerical solutions to various coupled systems under some initial conditions as well as boundary conditions.

Treatment of FODEs under initial and boundary conditions
Here we discuss different cases. Case 1. In the first case, we consider the fractional order differential equation we see that and applying 0 I γ t by the Lemma 1, on (12) we write Using the initial condition to get e 0 ¼ z 0 and approximate z 0 as Finally the Sylvester-type algebraic equation is obtained as Solving the Sylvester matrix for Ł M , we get the numerical value for z t ð Þ.
Since the exact solution is given by where E γ is the Mittag-Leffler representation, and at γ ¼ 1, z t ð Þ ¼ e Àt :. Approximating the solution through the proposed method and plotting the exact as well as numerical solution by using scale M ¼ 8 corresponding to γ ¼ 1 in Figure 1, we see that the proposed method works very well.
We take Applying Lemma 1 to Eq. (14), we get Using the conditions by putting t ¼ 0 and t ¼ 1 to get e 0 ¼ z 0 and Þt is the smooth function of t and constants; we approximate it as which is further solved for K M to get the required numerical solution.
For Case 2, we give the following example.

Coupled systems of linear FODEs under initial and boundary conditions
In this subsection, we consider different forms of coupled systems of FODEs with the initials as well as boundary conditions. Case 1. First we take the coupled system of FODEs as with the conditions Applying Lemma 1 to Eq. (20), we get Using the initial conditions given in Eq. (19), from Eq. (21), we get We take approximation as and Figure 2.
The plot of exact and approximate solution for Example 2 for Case 2.
By taking γ ¼ 1, the exact solution is obtained as where the external source functions are given by f t ð Þ ¼ cos t ð Þ þ e Àt þ 2e t and g t ð Þ ¼ e Àt þ sin t ð Þ þ 2 cos t ð Þ: The exact solution z ex , y ex can be computed by any method of ODEs. Approximating the problem by the considered method, we see that the computed numerical and exact solutions have close agreement at very small-scale level. The corresponding accuracy has been recorded in Table 1. Further the comparison between exact and numerical solution and the results about absolute error have been demonstrated in Figures 3 and 4, respectively. In Figure 3 we are given the comparison between exact solution and approximate solutions by using proposed method. Similarly the absolute errors have been described in Figure 4.
By comparing the exact and numerical solution through the proposed method, we observe that our numerical solution does not show any disagreement with the exact solution as can be seen in Figure 3. The absolute errors ∥z app À z ex ∥ and ∥y app À y ex ∥ plotted at the scale M ¼ 5 are very low as given in Figure 4, which describes the efficiency of the proposed method.
Case 2. Similarly for the coupled system of FODEs with boundary conditions, we consider where the source functions are given by We approximate the solution at the considered method by taking scale level M ¼ 5: One can see that numerical plot and exact solution plot coincide very well as shown in Figure 5. Similarly the absolute error has been plotted at the given scale M ¼ 5 in Figure 6, which is very low. The lowest value of absolute error ∥z app À z ex ∥ and ∥y app À y ex ∥ indicates efficiency of the proposed method. The table shows the   Table 2 which provides the information about efficiency of the proposed method.

Conclusion
We have successfully used the class of orthogonal polynomials of Laguerre polynomials to establish a numerical method to compute the numerical solution of FODEs and their coupled systems under some initial and boundary conditions. By using these polynomials, we have obtained some operational matrices corresponding to fractional order derivatives and integration. Also we have computed a new matrix corresponding to boundary conditions for boundary value problems of FODEs. Using the aforementioned matrices, we have converted the considered problem of FODEs to Sylvester-type algebraic equations. To obtain the numerical solution, we easily solved the desired algebraic equations by taking help from MATLAB®. Corresponding to the established procedure, we have provided numbers of examples to demonstrate our results. Also some error analyses have been provided along with graphical representations. By increasing the scale level, the accuracy is increased and vice versa. On the other hand, when the fractional order is approaching to integer value, the solutions tend to the exact solutions of the considered FODE. Therefore in each example, we have compared the exact and approximate solution and found that both the solutions were in closure contact with each other. Hence the established method can be very helpful in solving many classes and systems of FODEs under both initial and boundary conditions. In future the shifted Laguerre polynomials can be used to compute numerical solutions of partial differential equations of fractional order.

Author contribution
All authors equally contributed this paper and approved the final version.

Competing interests
We declare that no competing interests exist regarding this manuscript.