TX_1~ABS:AT/ADD:TX_2~ABS:AT


64 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 64-68

ReseaRch aRticle

Solution of Multi-order Fractional Differential Equation Based 
on Conformable Derivative by Shifted Legendre Polynomial
Salim S. Mahmood1, Kamaran J. Hamad1, Milad A. kareem1, Asrin F. Shekh2

1Department of Mathematic, Faculty of Science, Soran University, Kurdistan Region, Iraq, 2Department Information Technology, Erbil 
Polytechnic University, Kurdistan Region, Iraq

ABSTRACT

The aim of this article is the way for finding approximation solution of multi-order fractional differential equation with conformable 
sense with use approximated function by shifted Legendre polynomial, the method is easy and powerful for get our results of linear and 
non-linear equation, the background idea behind this method is finding system of algebra after achieve messing variable is that mean 
obtains approximate solution, a few example are illustrate for presented how much our method is capable.

Keywords: Conformable derivative, fractional derivative, shifted Legendre polynomial, fractional deferential equation, Python program

INTRODUCTION

There are numerous phenomena in various disciplines 
of research can work quite well represented by models 
using fractional calculus mathematic techniques. In many 

circumstances, there are no analytic or precise solutions to 
fractional differential equations. As a result, novel approaches 
for deriving analytical and approximation fractional differential 
equation solutions have been devised.[1,2]

Little number techniques and methods have gotten a lot 
of attention in recent years and have proven to be a useful tool 
for fractional calculus.

According to the studies, the Caputo fractional derivative 
is commonly incorporated in fractional calculus computed using 
spectral methods. To compute fractional differential equations with 
Caputo derivatives, Legendre polynomial has been employed.[3]

In addition, Chebyshev wavelet is used for solving non-
linear fractional differential equation,[4,5] operation matrix is 
used for solving fractional multi-order differential equation,[3] 
and spectral collocation method is used for compute this kind 
of differential equation by Laguerre polynomial (LP).[6]

The aim of this article is finding a solution to our fractional 
equations both kinds linear and non-linear using shifted LP. In 
this case, if number of N is large means our approximate, you 
can’t use hand writing for find solution, so we prefer write a 
program for such case by Python.

Hence, in our method, try to find approximating solution 
of multi-order fractional differential equation based on 
conformable derivative using shifted Legendre polynomial 
with program of Python.

PRELIMINARIES

Definition 1: Given function h: [0, ∞)→R then the 
conformable fractional derivative of h of order α is determined 
by

D h x
h x x h x

x�

�

�( ) lim
( ) ( )

, , ( , )�
� �

� � �
�

�




0

1

0 0 1  

Some time, write ha (x) for Da h (x) to indicate the 
conformable derivative of h of order α

Definition 2: In this paper, we need to use Legendre 
polynomials in distance x=[0,1] so using change variable 
x=2x−1, these polynomials can be obtained as follows

E x
n
k

n k
k

xn
n

k

n
k( ) ( ) ( )� �

�
�

�

�
�
�

�
�
�

�
�

�

�
�

�
�1

0

 (2)

Where E0(x)=1, E0(x)= 2x−1 on the interval [0,1], the shifted LP is 

defined as E x
n k x
n k k

nn
n k

k

n k

( ) ( )
( )!

( )!( !)
, , , ,...� �

�
�

��

�
� 1 0 1 2

0
2  

(3)

Corresponding Author: 
Salim S. Mahmood, Department of Mathematic, Faculty 
of Science, Soran University, Kurdistan Region, Iraq. 
E-mail: salimsaeedmahmood@gmail.com

Received: August 25, 2021 
Accepted: November 11, 2021 
Published: December 20, 2021

DOI: 10.24086/cuesj.v5n2y2021.pp64-68

Copyright © 2021 Salim S. Mahmood, Kamaran J. Hamad, Milad A. kareem. This 
is an open-access article distributed under the Creative Commons Attribution 
License (CC BY-NC-ND 4.0).

Cihan University-Erbil Scientific Journal (CUESJ)



Mahmood, et al.: Multi-order FDE Based on CD by Shifted LP

65 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 64-68

And explicit way for finding shifted Legendre polynomial 
is “Rodrigues formula”

E x
n

d
dx

x xn

n

n
n( )

!
( )� �

1 2  (4)

Each continuous function h (x) in the interval [0,1] can 
be written using the transferred Legendre polynomials as 
follows:

h x E xai
i

N

i( ) ( )�
�
�

0

 (5)

Where, ai is obtained using the following relation:

a hi x E x dxi i� � �( ) ( ) ( )2 1
0

1

 (6)

Definition 3: The orthogonal property of these 
polynomials is as follows

E x E x dx i
i j

i j
i j

0

1 2
2 1

0
� � �

�

�

�
�
�

��
( ) ( )  (7)

The orthogonal property of the expansion of any function 
makes it possible in terms of Legendre polynomials.

Theorem 1: Let hN (x) function can be approximate by 
shifted LP like that

h ax E xN i

i

N

i( ) ( )�
�
�

0

 (8)

After that, for any derivative we use[7]

D ah x W xN i

k

i

i

N

i k
k�

��

� �( ( )) ,�
��

���  (9)

Where, α  is define as least integer greater than or equal 

to α and

W x
i k

i k k k ni k
k

i k

,
( ) ( ) ( )

( ) ( ) ( )
� ��

�

�
� � �

� � � � �
1 1
1 1 1

�
� � �

 (10)

Proof: According to the property of the conformable 
derivative and using the estimated function definition hN (x), 
we get the following results:

D ah x D E xN i

i

N

i
� �( ( )) ( ( ))�

�
�

0

 (11)

D E x
i k

i k k k n
xi

i k

k

i
k�

�

�( ( ))
( ) ( )

( ) ( ) ( )
�

� � �
� � � � �

�

�

�� 1 1
1 1 1

�
� � �

 (12)

By substitution, we get

D h x
i k

i k k k n
aN i

k

i

i

N i k
�

��

( ( ))
( ) ( )

( ) ( ) (
�

� � �
� � � � �

��

�

�� 1 1
1 1 1

�
� � � ))

xk��  (13)

NUMERICAL SCHEME

In here presented how solve the multi-order fraction 
differential equation by Python program for get analytic and 
numerical results.

Assume the initial condition for this equation which is a multi-
order fractional differential equation[8]

D h x x h x B x h x f x xBk
k

r

r
�( ( )) ( ) ( ) ( ) ( ) ( ), [ , ]� � � �

�

�

�
1

1

0 1  

(14)

h q ii
i( ) , , ,...,0 0 1 1� � ��  (15)

Where, n−1<α≤n, Bk (x), k=1,2,….r and f (x) are 
functions that are well-known to be continuous on [0,1] and 
q ii , , ,...,� �0 1 1� , constants are provided. Assume the 
fractional differential Equation (14) has an answer that may 
be represented as

0

( ) ( )
N

N i i
i

h ax E x
=

= ∑  (16)

By the initial equation, we have α  equation and is 
depend on our approximate about N so all equation number is 
N+1, except α  equation other equation get by the finding 

root of shifted Legendre polynomial E x
N � �1 �

( )  and put on 

Equation (14). So solving this system of equation is our 
approximation solution for multi-order fractional differential 
Equation (14).

Example 1: (Bagley-Torvik equation). The 
inhomogeneous Bagley-Torvik initial value problem is

D h x D h x h x x x2
3
2 1 0 1( ( )) ( ( )) ( ) , [ , ]� � � � �  (17)

With initial assumption h(0)=1, h’(0)=1 and N=2

Solution:

h a

h a a E E E

h

x E

x E x a x a x

x

N i

i

N

i

i

i

i

( )

( ) ( ) ( ) ( )

(

�

� � ��

�

�

�

�
0

2

0

2

0 0 1 1 2 2

2 )) � � � � � �a a x a a x a x a0 1 1 2
2

2 22 6 6

So use initial condition for finding the α equation and for 
any derivative, we use

D ah x W xN i

k

i

i

N

i k
k�

��

� �( ( )) ,�
��

���

h

D

a a a

h a a
2 0 1 2

1
2 1 2

0 1 1

0 2 6 1 2

( ) ( )

(( ( )) ( )

� � � � �

� � � �

For finding N+1 equation, we need use the root of the polynomial
E x E x x x
N � �

� � � � � �
1 1 2 1 0 0 5

�
( ) ( ) .

So put the root in the Equation (17) after, we calculate all 
structure instead each x variable.



Mahmood, et al.: Multi-order FDE Based on CD by Shifted LP

66 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 64-68

D

D

h

h a

h a

a a

2
2 2

3
2

2 2

2 0 2

0 5 12

0 5 12 0 5

0 5
1
2

( ( . ))

( ( . )) .

( . )

�

�

� �

 

Substitute each one of D h D h2
2

3
2

20 5 0 5( ( . )), ( ( . ))  and 

h2 (0.5) in the Equation (17) and instead of each x put root of 
shifted Legendre polynomial

12 12 0 5
1
2

1
1
2

23
2

6 2
3
2

3

2 2 0 2

0 2

a a a a

a a

� � � � �

� � � � �

.

( ) ( )
 

So we have a linear system after solve by matrix operation, 
we get

1 1 1

0 2 6

1 0
23
2

6 2

1

1

3
2

0

1

2

�
�

�

�

�

�
�
�
�
�

�

�

�
�
�
�
�

�

�

�
�
�

�

�

�
�
�
�

�

�

�
�
�
�
�

a
a
a

��

�

�
�
�
�
�

� � �

� �

a

h

a a

x x

0 1 2

2

3
2

1
2

0

1

, ,

( )

Therefore, the obtained solution is the exact solution of 
this problem, code of Python for this example is given.[9]

Example 2: Suppose the following fractional linear 
differential equation with variable coefficients

D h x D h x xD h x x

h x x x x x x

3
2

1
2

2 3

2 3 1

2 4 7 0

( ( )) ( ( )) ( ( )) ( )

( ) , (

� � � �

� � � � � ,, ]1  

With the primary conditions h(0)=0, h’(0)=0 and N=2

Solution:

h a

h E E E

h

x E

x a x a x a x

x a a x

N i

i

N

i( )

( ) ( ) ( ) ( )

( )

�

� � �

� � �

�
�

0

2 0 0 1 1 2 2

2 0 12 aa a x a x a1 2
2

2 26 6� � �

For finding N+1 equation, we need use the root of the polynomial
E x E x x x
N � �

� � � � � �
1 1 2 1 0 0 5

�
( ) ( ) .

So we put the root in the Equation (18) after, we calculate 
all structure instead each x variable

D h a

h a

h a

D

D

3
2

2 2

1
2

1
2 1

1
2

2 1

0 5 12
1
2

0 5 2

0 5 2 0 5

( ( . )) ( )

( ( . ))

( ( . )) .

�

�

� �66 5 12 0 5

0 5
1
2

2 2

2 0 2

a o a

a ah

. .

( . )

�

� �

We substitute each one of D h D h D h
3
2

2
1

2

1
2

20 5 0 5 0 5( ( . )), ( ( . )), ( ( . ))  

and h2 (0.5) in the Equation (18) and instead of each x we put root (0.5)

1
2

7 8 2352813742238570 5 03921356237095 10 1 2a a a� � � �. . ( )

h a a a

D h a a
2 0 1 2

1
2 1 2

0 0 2

0 2 6 0 3

( ) ( )

(( ( )) ( )

� � � � �

� � � �

So we have a linear system, we solved by matrix operation

1
2

7 8 2352813742238570

1 1 1

0 2 6

0

1

2

.

�
�

�

�

�
�
�
�
��

�

�

�
�
�
�
��

�

�

�
�
�

�

�

a
a
a

��
�
�
�
�

�

�
�
�

�

�

�
�
�

5 03921356237095

0

0

.

a a a

h x x

0 1 2

2
2

1
3

1
2

1
6

= = =

=

, ,

( )

Therefore, the obtained solution is the precise solution to 
this problem.

Example 3: Consider the following problem with an 
initial value.

D h x D h x h x x x4
7
2 3 9 0 1( ( )) ( ( )) ( ) , ( , ]� � � �  (18)

With initial condition h(0)=0, h’(0)= h”(0)=0 and 
h”’(0)=6, N=4

h x a E a E x a E x a E x a E x a E x

h

N i

i

N

i( ) ( ) ( ) ( ) ( ) ( )

(

� � � � � �
�
�

0

0 0 1 1 2 2 3 3 4 4

4 xx a x a a x a a a x

a a a

) � � �� � � � �� � �

� � �

70 20 140 6 30 90

2 6 12 2
4

4
3 4

3
2 3 4

2

1 2 3 00 4 0 1 2 3 4a x a a a a a� � � � � � �

For finding N+1 equation we need use the root of the polynomial 
E x E x x x
N � �

� � � � � �
1 1 2 1 0 0 5

�
( ) ( ) .

So we put the root in the Equation (19) after, we calculate 
all structure instead each x variable so our equation is

a
a a

a0
2 4

3

42
3
8

1680 840 2
1

512
0 1� ��

�
�

�

�
� � �� � � � � ( )

So by initial condition, we have α equation

h a a a a a

h a a a aD
4 0 1 2 3 4

1
4 1 2 3 4

0 0 2

0 2 6 12 20 0 3

( ) ( )

( ( )) (

� � � � � � �

� � � � � � ))

( ( )) ( )

( ( ))

D

D

h a a a

h a a

2
4 2 3 4

3
4 3 4

0 12 60 180 0 4

0 120 840 6 0

� � � � �

� � � � �� ( )5

Now use Newton iteration method for finding the messing 
variable and our result will be

a a a a and a0 1 2 3 4

1
4

9
20

1
4

1
20

0= = = = =, , ,    



Mahmood, et al.: Multi-order FDE Based on CD by Shifted LP

67 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 64-68

If we put those number in our h4 (x) our result will be h4 

(x)=x3 which is the precise solution to this issue.

CONCLUSION

In this paper, using the method and technique for get 
approximate solution of fractional differential equations with 
initial conditions and conformable sense are used to characterize 
the fractional derivatives, the solution can be easily analyzed 
for any number of different values of x since it is written as 
a truncated Legendre series. Furthermore, the suggested 
strategy’s validity and applicability are proved by numerical 
findings. The numerical results show that as the number of N 
terms is raised, the method converges. With tiny N, we get the 
precise answer for several examples, all examples are achieved 
by the use of Python programming to create quick algorithms. 
The method presented here can be used to solve the high order 
of N for conformable fractional differential equations as well as 
conformable fractional partial differential equations.

REFERENCES
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Shawagfeh. New operational matrix for shifted legendre 
polynomials. Punjab University Journal of Mathematics, vol. 47, 
no. 1, pp. 1-23, 2020.

2. R. A. Khalil. A new definition of fractional derivative. Journal 
of Computational and Applied Mathematics, vol. 264, pp. 65-70, 

2014.
3. A. B. Kisabo, U. C. Uchenna, and F. A. Adebimpe. Newton’s 

method for solving non-linear system of algebraic equations 
(NLSAEs) with MATLAB/Simulink® and MAPLE®. American 
Journal of Mathematical and Computer Modelling, vol. 2, no. 4, 
pp. 117-131, 2017.

4. S. S. Ezz-Eldien, J. A. Machado, Y. Wang and A. A. Aldraiweesh. 
An algorithm for the approximate solution of the fractional 
Riccati differential equation. International Journal of Nonlinear 
Sciences and Numerical Simulation, vol. 20, no. 6, pp. 661-674, 
2019.

5. L. Yuanlu. Solving a nonlinear fractional differential equation 
using Chebyshev wavelets. Communications in Nonlinear Science 
and Numerical Simulation, vol. 15, no. 9, pp. 2284-2292, 2010.

6. E. Hesameddini and E. Asadollahifar. Numerical solution of 
multi-order fractional differential equation. IJNAO, vol. 5, no. 1, 
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7. M. Abul-Ez, M. Zayed, A. Youssef and M. De la Sen. On 
conformable fractional Legendre polynomials and their 
convergence properties with applications. Alexandria Engineering 
Journal, vol. 59, no. 6, pp. 5231-5245, 2020.

8. H. Çerdik Yaslan and F. Mutlu. Numerical solution of the 
conformable differential equations via shifted Legendre 
polynomials. International Journal of Computer Mathematics, 
vol. 97, no.5, pp. 1016-1028, 2020.

9. M. M. Khader, T. S. El Danaf and A. S. Hendy. Efficient spectral 
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APPENDIX

Result of Python code for example 1



Mahmood, et al.: Multi-order FDE Based on CD by Shifted LP

68 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2021, 5 (2): 64-68

Result of Python code for example 2

Result of Python code for example 3