Exact Solution of Linear System of Fractional D

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International Journal of Mechanical and Industrial Technology ISSN 2348-7593 (Online) Vol. 10, Issue 2, pp: (1-7), Month: October 2022 - March 2023, Available at: www.researchpublish.com

Exact Solution of Linear System of Fractional Differential Equations with Constant Coefficients

School of Mathematics and Statistics, Zhaoqing University, Guangdong, China

DOI: https://doi.org/10.5281/zenodo.7391108

Published Date: 02-December-2022

Abstract: In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional derivative, the exact solution of linear system of fractional differential equations with constant coefficients is obtained. A new multiplication of fractional analytic functions plays an important role in this paper. In addition, we also provide some examples to illustrate the application of our results. In fact, our results are generalizations of these results in ordinary differential equations.

Keywords: Jumarie type of R-L fractional derivative, exact solution, linear systemof fractional differential equations with constant coefficients, new multiplication, fractional analytic functions.

I. INTRODUCTION

The history of fractional calculus is almost as long as the development of traditional calculus. In 1695, the concept of fractional derivative first appeared in a famous letter between L’Hospital and Leibniz. Many great mathematicians have further developed this field, such as Euler, Lagrange, Laplace, Fourier, Abel, Liouville, Riemann, and Weyl. In the past few decades, fractional calculus has played a very important role in physics, electrical engineering, economics, biology, control theory, and other fields [1-7].

However, the definition of fractional derivative is not unique. Common definitions include Riemann-Liouville (R-L) fractional derivative, Caputo fractional derivative, Grunwald-Letnikov (G-L) fractional derivative, Jumarie type of R-L fractional derivative [8-12]. Since the Jumarie type of R-L fractional derivative makes the derivative of constant function equal to zero, it is easier to use this definition to connect fractional calculus with classical calculus.

In this paper, based on Jumarie’s modified R-L fractional derivative, we obtain the exact solution of linear system of fractional differential equations with constant coefficients. A new multiplication of fractional analytic functions plays an important role in this article. Moreover, two examples are provided to illustrate the application of our results. And our results are generalizations of these results in ordinary differential equations.

II. PRELIMINARIES

Firstly, the fractional calculus used in this paper and some important properties are introduced below Definition 2.1 ([13]): Assume that 0<�� ≤1, and ��0 is a real number. The Jumarie’s modified R-L ��-fractional derivative is defined by ( �� ��0 �� ��)[��(��)]= 1 Γ(1 ��) �� ����∫ ��(��) ��(��0) (�� ��)�� ���� �� ��0 , (1) where Γ( ) is the gamma function.

Proposition 2.2 ([14]): If ��,��,��0,�� are real numbers and �� ≥�� >0, then ( �� ��0 �� ��)[(�� ��0)��]= Γ(��+1) Γ(�� ��+1)(�� ��0)�� �� , (2)

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International Journal of Mechanical and Industrial Technology ISSN 2348-7593 (Online) Vol. 10, Issue 2, pp: (1-7), Month: October 2022 - March 2023, Available at: www.researchpublish.com

( �� ��0 �� ��)[��]=0 (3)

In the following, we introduce the definition of fractional analytic function.

Definition 2.3 ([15]): Assume that ��,��0, and ���� are real numbers for all ��, ��0 ∈(��,��), and 0<�� ≤1. If the function ����:[��,��]→�� can be expressed as an ��-fractional power series, that is, ����(����)=∑ ���� Γ(����+1)(�� ��0)���� ∞ ��=0 on some open interval containing ��0, then we say that ����(����) is ��-fractional analytic at ��0. In addition, if ����:[��,��]→�� is continuous on closed interval [��,��] and it is ��-fractional analytic at every point in open interval (��,��), then ���� is called an ��-fractional analytic function on [��,��].

Definition 2.4 ([16]): If 0<�� ≤1, ��0 is a real number, and ����(����) and ����(����) are two ��-fractional analytic functions defined on an interval containing ��0 , ����(����)=∑ ���� Γ(����+1)(�� ��0)���� =∑ ���� ��! ( 1 Γ(��+1)(�� ��0)��)⨂�� ∞ ��=0 ∞ ��=0 , (4) ����(����)=∑ ���� Γ(����+1)(�� ��0)���� =∑ ���� ��! ( 1 Γ(��+1)(�� ��0)��)⨂�� . ∞ ��=0 ∞ ��=0 (5)

Then ����(����)⊗����(����) =∑ ���� Γ(����+1)(�� ��0)���� ∞ ��=0 ⊗∑ ���� Γ(����+1)(�� ��0)���� ∞ ��=0 =∑ 1 Γ(����+1)(∑ (����)���� ������ �� ��=0 ) ∞ ��=0 (�� ��0)���� (6) In other words, ����(����)⊗����(����) =∑ ���� ��! ( 1 Γ(��+1)(�� ��0)��)⨂�� ∞ ��=0 ⊗∑ ���� ��! ( 1 Γ(��+1)(�� ��0)��)⨂�� ∞ ��=0 =∑ 1 ��!(∑ (����)���� ������ �� ��=0 ) ∞ ��=0 ( 1 Γ(��+1)(�� ��0)��)⨂�� . (7)

Definition 2.5 ([17]): If 0<α≤1, and �� is a real number. The ��-fractional exponential function is defined by ����(����)=∑ ������ Γ(����+1) =∑ 1 ��!( 1 Γ(��+1) ����)⨂�� ∞ ��=0 ∞ ��=0 (8)

In addition, the ��-fractional cosine and sine function are defined as follows: ��������(����)=∑ ( 1)����2���� Γ(2����+1) =∑ ( 1)�� (2��)! ( 1 Γ(��+1) ����)⨂2�� ∞ ��=0 ∞ ��=0 , (9) and ��������(����)=∑ ( 1)����(2��+1)�� Γ((2��+1)��+1) =∑ ( 1)�� (2��+1)!( 1 Γ(��+1) ����)⨂(2��+1) ∞ ��=0 ∞ ��=0 (10)

Theorem 2.6 ([17]): If 0<�� ≤1 and ��,�� are two real numbers. Then ����((��+����)����)=����(������)⊗(��������(������)+����������(������)). (11)

III. RESULTS AND EXAMPLES

Inthis section, the mainresults includingthe exact solutionof linear systemoffractional differential equations withconstant coefficients are obtained. On the other hand, two examples are provided to illustrate the application of our results.

Definition 3.1: Let 0<�� ≤1, �� be a positive integer. The matrix form of linear system of fractional differential equations with constant coefficients is ( ��0 �� ��)[����(����)]=������(����), (12)

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where ����(����)= [

���� 1(����) ���� 2(����) ⋮ ���� ��(����)] and �� is an ��×�� constant matrix.

Definition 3.2: If 0<�� ≤1, and �� is an ��×�� constant matrix. Then we define ����(������)=∑ 1 Γ(����+1) ���������� ∞ ��=0 =∑ 1 ��! ���� ( 1 Γ(��+1) ����)⨂�� ∞ ��=0 . (13)

Theorem 3.3: If 0<�� ≤1, and �� is a ��×�� matrix. Then the linear system of ��-fractional differential equations with constant coefficients ( ��0 �� ��)[����(����)]=������(����) (14) has the solution ����(����)=����(������)����(0) (15) Proof If ����(����)=����(������)����(0), then ( ��0 �� ��)[����(����)] =( ��0 �� ��)[����(������)����(0)] =( ��0 �� ��)[∑ 1 Γ(����+1) ���������� ∞ ��=0 ����(0)] =∑ ( ��0 �� ��)[ 1 Γ(����+1) ����������] ����(0) ∞ ��=0 =��[∑ 1 Γ(����+1) ���������� ∞ ��=0 ] ����(0) =������(������)����(0) =������(����)

Therefore, the desired result holds. Q.e.d. Definition 3.4: Suppose that 0<�� ≤1, and �� is a positive integer. If ���� 1(����), ���� 2(����), ⋯ , ���� ��(����) are linearly independent solutions of ( ��0 �� ��)[����(����)]= ������(����), then the matrix Φ��(����)=[���� 1(����),���� 2(����), ,���� ��(����)] (16) is called a fundamental matrix solution of ( ��0 �� ��)[����(����)]=������(����).

Theorem 3.5: Let 0<�� ≤1. Then Φ��(����) is a fundamental matrix solution of ( ��0 �� ��)[����(����)]=������(����) if and only if detΦ��(����)≠0. Moreover, if detΦ��(��0 ��)≠0 for some real number ��0, then detΦ��(����)≠ 0 for all �� Theorem 3.6: Let 0<�� ≤1. If the matrix �� has eigenvalues ��1 , ��2 , ⋯ ,���� with linearly independent eigenvectors ��1,��2, ⋯ ,���� respectively. Then Φ��(����)=[����(��1����)��1,����(��2����)��2, ⋯,����(��������)����] (17) is a fundamental matrix solution of ( ��0 �� ��)[����(����)]=������(����) Proof Since for all �� =1,2,⋯,��, ( ��0 �� ��)[����(��������)����] =��������(��������)���� =������(��������)���� , (18) and detΦ��(0) =[��1,��2, ⋯,����]≠0, it follows from Theorem 3.5 that the desired result holds. Q.e.d.

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International Journal of Mechanical and Industrial Technology ISSN 2348-7593 (Online) Vol. 10, Issue 2, pp: (1-7), Month: October 2022 - March 2023, Available at: www.researchpublish.com

Theorem 3.7: If 0<�� ≤1. Then Φ��(����)=����(������) is a fundamental matrix solution of ( ��0 �� ��)[����(����)]=������(����)

Proof Since ( ��0 �� ��)[Φ��(����)] =( ��0 �� ��)[����(������)] =������(������) =��Φ��(����) (19) It follows that the desired result holds. Q.e.d. Theorem 3.8: Let 0<�� ≤1 If Φ��(����) is a fundamental matrix solution of ( ��0 �� ��)[����(����)]=������(����), then ����(������)= Φ��(����)Φ�� 1(0). (20)

Proof Since ����(������) and Φ��(����) are fundamental matrix solutions of ( ��0 �� ��)[����(����)]=������(����), it follows that ����(������)= Φ��(����)��, (21) where �� is a nonsingular constant matrix. If �� =0, then �� = Φ��(0)��. Thus, �� =Φ�� 1(0). Hence, the desired result holds. Q.e.d. Theorem 3.9: If 0<�� ≤1, ��,��,�� are ��×�� matrices, and �� is a nonsingular matrix. Then ����((��+��)����)=����(������)⨂����(������), if ���� =���� (22) [����(������)]⨂ 1 =����( ������), (23) ����(�� 1��������)=�� 1����(������)�� (24)

Proof Since ���� =����, it follows that ����(������)⨂����(������) =∑ 1 ��! ���� ( 1 Γ(��+1) ����)⨂�� ∞ ��=0 ⨂∑ 1 ��! ���� ( 1 Γ(��+1) ����)⨂�� ∞ ��=0 =(��+�� 1 Γ(��+1) ���� + 1 2! ��2 ( 1 Γ(��+1) ����)⨂2 + )⨂(��+�� 1 Γ(��+1) ���� + 1 2! ��2 ( 1 Γ(��+1) ����)⨂2 + ) =��+(��+��) 1 Γ(��+1) ���� + 1 2!(��2 +2����+��2)( 1 Γ(��+1) ����)⨂2 +⋯ =��+(��+��) 1 Γ(��+1) ���� + 1 2!(��+��)2 ( 1 Γ(��+1) ����)⨂2 +⋯ =∑ 1 ��!(��+��)�� ( 1 Γ(��+1) ����)⨂�� ∞ ��=0 =����((��+��)����). (25) On the other hand, since �� and –�� are commutative, ����(������)⨂����( ������)=����((�� ��)����)=�� (26) Thus, ����( ������)=[����(������)]⨂ 1 . Finally, ����(�� 1��������) =∑ 1 ��!(�� 1����)�� ( 1 Γ(��+1) ����)⨂�� ∞ ��=0

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=∑ 1 ��! �� 1������( 1 Γ(��+1) ����)⨂�� ∞ ��=0 =�� 1 (∑ 1 ��! ���� ( 1 Γ(��+1) ����)⨂�� ∞ ��=0 )�� =�� 1����(������)�� (27)

Example 3.10: Let 0<�� ≤1 Find the solution of the initial-value problem of linear system of ��-fractional differential equations with constant coefficients ( ��0 �� ��)[���� 1(����) ���� 2(����)]=[2 1 0 2][���� 1(����) ���� 2(����)] , [���� 1(0) ���� 2(0)]=[ 21] (28) Solution Since �� =[2 1 0 2]=[2 0 0 2]+[0 1 0 0] , and [2 0 0 2], [0 1 0 0] are commutative, it follows from Theorem 3.9 that ����(������) =���� ([2 1 0 2]����) =���� ([2 0 0 2]����)⨂���� ([0 1 0 0]����) =[����(2����) 0 0 ����(2����)]⨂(��+[0 1 0 0] 1 Γ(��+1) ���� + 1 2![0 1 0 0]2 ( 1 Γ(��+1) ����)⨂2 + ) =[����(2����) 0 0 ����(2����)]⨂[1 1 Γ(��+1) ���� 0 1 ] =[����(2����) 1 Γ(��+1) ����⨂����(2����) 0 ����(2����) ]. (29)

And hence, by Theorem 3.3, the solution is [���� 1(����) ���� 2(����)] =[����(2����) 1 Γ(��+1) ����⨂����(2����) 0 ����(2����) ][ 21] =[����(2����)⨂(2 1 Γ(��+1) ����) ����(2����) ] (30)

That is, ���� 1(����)=����(2����)⨂(2 1 Γ(��+1) ����), (31) ���� 2(����)= ����(2����) (32)

Example 3.11: If 0<�� ≤1 Find the solution of the initial-value problem of linear system of ��-fractional differential equations with constant coefficients ( ��0 �� ��)[���� 1(����) ���� 2(����)]=[ 3 5 5 3][���� 1(����) ���� 2(����)] , [���� 1(0) ���� 2(0)]=[ 1 3 ]. (33)

Solution Let �� =[ 3 5 5 3], then det(���� ��)=0 implies that the eigenvalues of �� are ��1 =3+5��, ��2 =3 5�� We can easily obtain the eigenvector of ��1 is ��1 =[1��], and the eigenvector of ��2 is ��2 =[��1]. Therefore, by Theorem 3.6 we obtain a fundamental matrix solution of this linear system of ��-fractional differential equations,

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��(����)=[����((3+5��)����) ������((3 5��)����) ������((3+5��)����) ����((3 5��)����)] (34)

Using Theorem 2.6 and Theorem 3.8 yields ����(������) = Φ��(����)Φ�� 1(0) =[����((3+5��)����) ������((3 5��)����) ������((3+5��)����) ����((3 5��)����)][1 �� �� 1] 1 =[����((3+5��)����) ������((3 5��)����) ������((3+5��)����) ����((3 5��)����)]12[1 �� �� 1] = 1 2[ ����((3+5��)����)+����((3 5��)����) ������((3+5��)����)+������((3 5��)����) ������((3+5��)����) ������((3 5��)����) ����((3+5��)����)+����((3 5��)����) ] =[ ����(3����)⨂��������(5����) ����(3����)⨂��������(5����) ����(3����)⨂��������(5����) ����(3����)⨂��������(5����)] (35)

Thus, by Theorem 3.3, the solution is [���� 1(����) ���� 2(����)] =����(������)����(0) =[ ����(3����)⨂��������(5����) ����(3����)⨂��������(5����) ����(3����)⨂��������(5����) ����(3����)⨂��������(5����)][ 1 3 ] =[����(3����)⨂( ��������(5����)+3��������(5����)) ����(3����)⨂(3��������(5����)+��������(5����)) ] . (36) That is, ���� 1(����)=����(3����)⨂( ��������(5����)+3��������(5����)), (37) ���� 2(����)=����(3����)⨂(3��������(5����)+��������(5����)) (38)

IV. CONCLUSION

In this paper, based on Jumarie’s modified R-L fractional derivative, we obtain the exact solution of linear system of fractional differential equations with constant coefficients. A new multiplication of fractional analytic functions plays an important role in this article. On the other hand, we give some examples to illustrate the application of our results. In fact, the results we obtained are generalizations of these results in ordinary differential equations. In the future, we will continue to study the problems in fractional differential equations and applied mathematics.

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