Absolute error at

## Abstract

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.

### Keywords

- FODEs
- numerical solution
- Laguerre polynomials
- operational matrices

## 1. 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

for

## 2. 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

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

** Definition 2.** Caputo fractional derivative is defined as

where

** Theorem 1.** The FODE given by

has a unique solution, such that

** Lemma 1.** Therefore in view of this result, if

is written as

where

The above lemma is also stated as

** Definition 3.** The famous Laguerre polynomials are represented by

They are orthogonal on

where

is the weight function and

Now let

We set the above two vectors into their inner product and represent the column matrix by

Again as

which is written as

We call

Hence the coefficient

In vector form we can write Eq. (5) as

where

### 2.1 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

### 2.2 Error analysis

The proof of the following results can be found with details in [20].

** Lemma 2.** Let

** Theorem 2.** For error analysis, we state the theorem such that,

where

Now let

with the following inner product and norm

## 3. 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

where

where

* Proof*. We apply the fractional order integral of order

Since from (7), we have

Therefore Eq. (7) implies that

which is equal to

We approximate

By using the relation of orthogonality, we can find coefficients

So Eq. (8) implies

which is the desired result.

** Lemma 4.** Let

where

where

* Proof*. Leaving the proof as it is very similar to the proof of the above lemma.

** Lemma 5.** We consider a function

where

where

with

* Proof*. By considering the general term of

Using the famous Laplace transform, we have from (10)

Now using Laguerre polynomials, we have

where

To get the desired result, we evaluate the above (11) relation for

## 4. 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.

### 4.1 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

Using the initial condition to get

Finally the Sylvester-type algebraic equation is obtained as

Solving the Sylvester matrix for

Since the exact solution is given by

where

Approximating the solution through the proposed method and plotting the exact as well as numerical solution by using scale

** Case 2**.

We take

Applying Lemma 1 to Eq. (14), we get

Using the conditions by putting

Equation (15) implies

where

and

Hence

So Eq. (13) implies

which is further solved for

For Case 2, we give the following example.

At

Upon using the suggested method, we see from the subplot at the left of Figure 2 that exact and numerical solutions are very close to each other for very low scale level. Also, the absolute error is given in subplot at the right of Figure 2.

### 4.2 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

Let

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

while source functions are approximated as

and

Therefore the consider system on using (19)–(22), (18) becomes

On further rearrangement we have

which further can be written as

In matrix form we write as

We solve this system of matrix equation for

where

Upon computation of matrices

** Example 3.** We now provide its example by considering the system of FODEs:

By taking

where the external source functions are given by

CPU time (s) | Absolute error | Absolute error | CPU time (s) | |
---|---|---|---|---|

0 | 30.5 | 0.00003 | 0.000006 | 32.5 |

0.15 | 32.7 | 0.000016 | 0.000034 | 33.3 |

o.35 | 35.8 | 0.000013 | 0.00003 | 33.9 |

0.65 | 33.6 | 0.000012 | 0.00003 | 35.6 |

0.87 | 34.8 | 0.000018 | 0.000036 | 36.5 |

1 | 35.9 | 0.00003 | 0.000006 | 36.8 |

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

** Case 2.** Similarly for the coupled system of FODEs with boundary conditions, we consider

Let us assume

Applying Lemma 1 to Eq. (24), we get

where

Similarly

Equation (25) implies that

Let

Hence Eq. (26) implies

approximating

On using (24)–(29), system (23) can be written as

On rearrangement of terms, the above equations give

In matrix form, we can write

We convert the system to algebraic equation by considering

so that the system is of the form

and solving the given equation for the unknown matrix

** Example 4.** As an example, we consider the Caputo fractional differential equation for the coupled system with the boundary conditions as

At

where the source functions are given by

We approximate the solution at the considered method by taking scale level

Absolute error | CPU time (s) | Absolute error | CPU time (s) | |
---|---|---|---|---|

0 | 0.011 | 49.4 | 0.010 | 50.0 |

0.15 | 0.0062 | 50.3 | 0.0052 | 52.5 |

0.35 | 0.0058 | 51.2 | 0.0047 | 54.6 |

0.65 | 0.006 | 51.5 | 0.005 | 55.5 |

0.85 | 0.0075 | 52.6 | 0.007 | 56.4 |

1 | 0.011 | 53.8 | 0.010 | 56.2 |

## 5. 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.

## Competing interests

We declare that no competing interests exist regarding this manuscript.

## Author contribution

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

## References

- 1.
Butzer PL, Westphal U. An Introduction to Fractional Calculus. Singapore: World Scientific; 2000 - 2.
Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives. Switzerland: Gordon and Breach; 1993 - 3.
Scalas E, Raberto M, Mainardi F. Fractional calculus and continous time finance. Physica A: Statistical Mechanics and its Applications. 2000; 284 :376-384 - 4.
Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000 - 5.
Amairi M, Aoun M, Najar S, Abdelkrim MN. A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation. Applied Mathematics and Computation. 2010; 217 (5):2162-2168 - 6.
Deng J, Ma L. Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Applied Mathematics Letters. 2000; 23 :676-680 - 7.
Girejko E, Mozyrska D, Wyrwas M. A sufficient condition of viability for fractional differential equations with the Caputo derivative. Journal of Mathematical Analysis and Applications. 2011; 38 :146-154 - 8.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional Calculus Models and Numerical Methods. Singapore: World Scientific; 2009 - 9.
Guy J. Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Applied Mathematical Modelling. 2008; 32 :836-859 - 10.
Sabatier JATMJ, Agrawal OP, Machado JT. Advances in Fractional Calculus. Dordrecht: Springer; 2007 - 11.
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006 - 12.
Lakshmikantham I, Leela S. Theory of Fractional Dynamical Systems. Cambridge, UK: Cambridge Scientific Publishing; 2009 - 13.
Li CP, Deng WH. Remarks on fractional derivatives. Applied Mathematics and Computation. 2007; 187 :777-784 - 14.
Li CP, Dao XH, Guo P. Fractional derivatives in complex planes. Nonlinear Analysis: Theory Methods & Applications. 2009; 71 :5-6 - 15.
Li C, Gong Z, Qian D, Chen Y. On the bound of the Lyapunov exponents for the fractional differential systems. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2010; 20 (1):013127 - 16.
Oldham KB, Spanier J. The Fractional Calculus. New York: Acad. Press; 1974 - 17.
Ortigueira MD. Comments on modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions. Applied Mathematical Modelling. 2009; 33 :2534-2537 - 18.
Qian DL, Li CP, Agarwal RP, Wong PJY. Stability analysis of fractional differential system with Riemann-Liouville derivative. Mathematical and Computer Modelling. 2010; 52 :862-874 - 19.
West BJ, Bologna M, Grigolini P. Physics of Fractional Operators. New York: Springer; 2003 - 20.
Yang S, Xiao A, Su H. Convergence of the variational iteration method for solving multi-order fractional differential equations. Computers & Mathematics with Applications. 2010; 60 :2871-2879 - 21.
Ray SS, Bera RK. Solution of an extraordinary differential equation by adomian decomposition method. Journal of Applied Mathematics. 2004; 4 :331338 - 22.
Hashim I, Abdulaziz O, Momani S. Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation. 2009; 14 :674-684 - 23.
Bengochea G. Operational solution of fractional differential equations. Applied Mathematics Letters. 2014; 32 :48-52 - 24.
Khalil H, Khan RA. The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations. International Journal of Computer Mathematics. 2015; 92 (7):1452-1472 - 25.
Doha EH, Bhrawy AH, Ezz-Eldien SS. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Applied Mathematical Modelling. 2011; 35 :5662-5672 - 26.
Esmaeili S, Shamsi M, Luchko Y. Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials. Computers & Mathematics with Applications. 2011; 62 :918-929 - 27.
Odibat Z, Momani S, Erturk VS. Generalized differential transform method an application to differential equations of fractional order. Applied Mathematics and Computation. 2008; 197 :467-477 - 28.
Baleanu D, Mustafa OG, Agarwal RP. An existence result for a superlinear fractional differential equation. Applied Mathematics Letters. 2010; 23 :1129-1132 - 29.
Baleanu D, Mustafa OG, Agarwal RP. On the solution set for a class of sequential fractional differential equations. Journal of Physics A. 2010; 43 :385-209 - 30.
Doha EH, Abd-Elhameed WM. Efficient solutions of multidimensional sixth-order boundary F value problems using symmetric generalized Jacobi-Galerkin method. Abstract and Applied Analysis. 2012; 2012 :12 - 31.
Bhrawy AH, Al-Shomrani MM. A Jacobi dual-Petrov Galerkin-Jacobi collocation method for solving Korteweg-de Vries equations. Abstract and Applied Analysis. 2012; 2012 :14 - 32.
Singh AK, Singh VK, Singh VK. The Bernstein operational matrix of integration. Applied Mathematical Sciences. 2009; 3 :2427-2436 - 33.
Bhrawy AH, Alofi AS, Ezz-Eldien SS. A quadrature tau method for fractional differential equations with variable coefficients. Applied Mathematics Letters. 2011; 24 :2146-2152 - 34.
Bhrawy AH, Mohammed MA. A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Advances in Difference Equations. 2012; 2012 :8 - 35.
Khalil H, Khan RA. New operational matrix of integration and coupled system of Fredholm integral equations. Chinese Journal of Mathematics. 2014; 16 :12 - 36.
Khan RA, Khalil H. A new method based on Legendre polynomials for solution of system of fractional order partial differential equations. International Journal of Computer Mathematics. 2014; 91 (12):2554-2567 - 37.
Khalil H, Khan RA. A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation. Computers & Mathematics with Applications. 2014; 67 :1938-1953 - 38.
Guo BY, Wang LL. Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation. Journal of Computational and Applied Mathematics. 2006; 190 :304-324 - 39.
Gulsu M, Gurbuz B, Ozturk Y, Sezer M. Laguerre polynomial approach for solving linear delay difference equations. Applied Mathematics and Computation. 2011; 217 :6765-6776 - 40.
Bhrawy AH, Taha TM, Machado JAT. A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dynamics. 2015; 81 (3):1023-1052 - 41.
Diethelm K, Ford NJ. Numerical solution of the Bagley Torvik equation. BIT Numerical Mathematics. 2002; 42 (1):490-500 - 42.
Akyuz-Dascioglu A, Isler N. Bernstein collocation method for solving nonlinear differential equations. Mathematical and Computational Applications. 2013; 18 :293-300 - 43.
Shah K. Using a numerical method by omitting discretization of data to study numerical solutions for boundary value problems of fractional order differential equations. Mathematical Methods in the Applied Sciences. 2019; 42 :6944-6959. DOI: 10.1002/mma.5800