7.2
CiteScore
3.7
Impact Factor
Generic selectors
Exact matches only
Search in title
Search in content
Post Type Selectors
Search in posts
Search in pages
Filter by Categories
ABUNDANCE ESTIMATION IN AN ARID ENVIRONMENT
Case Study
Editorial
Invited review
Letter to the Editor
Original Article
REVIEW
Review Article
SHORT COMMUNICATION
7.2
CiteScore
3.7
Impact Factor
Generic selectors
Exact matches only
Search in title
Search in content
Post Type Selectors
Search in posts
Search in pages
Filter by Categories
ABUNDANCE ESTIMATION IN AN ARID ENVIRONMENT
Case Study
Editorial
Invited review
Letter to the Editor
Original Article
REVIEW
Review Article
SHORT COMMUNICATION
View/Download PDF

Translate this page into:

Original article
28 (
1
); 37-40
doi:
10.1016/j.jksus.2015.05.006

Generalized variational formulations for extended exponentially fractional integral

Key Laboratory of Measurement Technology and Instrumentation of Hebei Province, Yanshan University, Qinhuangdao 066004, PR China
Stainless Steel Wire and Rod Rolling Plant, Taiyuan Iron and Steel Group Co, Ltd, Taiyuan 030003, PR China

⁎Corresponding author. Tel.: +86 13383602033. czheng@ysu.edu.cn (Cheng-Bo Zheng)

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

Recently, the fractional variational principles as well as their applications yield a special attention. For a fractional variational problem based on different types of fractional integral and derivatives operators, corresponding fractional Lagrangian and Hamiltonian formulation and relevant Euler–Lagrange type equations are already presented by scholars. The formulations of fractional variational principles still can be developed more. We make an attempt to generalize the formulations for fractional variational principles. As a result we obtain generalized and complementary fractional variational formulations for extended exponentially fractional integral for example and corresponding Euler–Lagrange equations. Two illustrative examples are presented. It is observed that the formulations are in exact agreement with the Euler–Lagrange equations.

Keywords

Fractional calculus
Generalized variational formulation
Euler–Lagrange equation
Extended exponentially fractional integral
1

1 Introduction

Fractional calculus represents a generalization of ordinary differentiation and integration to arbitrary order. It is an area of current strong research with many different and important applications in different fields of sciences ranging from geophysical fluid dynamics to quantum field theory (Malinowska and Torres, 2012; Yang, 2012).

During the last few years a special attention was devoted to the fractional variational principles as well as their applications (Baleanu, 2008). The formulation of the fractional variational principles has an important role for elaboration of a consistent fractional quantization method for both discrete and continuous systems. The first attempt to find the fractional Lagrangian and Hamiltonian is due to Riewe (1996, 1997), who first applied fractional calculus to a non-conservative mechanics modeling, and formed the fractional Euler–Lagrange equations and the fractional Hamilton equations. The research made by Riewe opened the booming of the fractional variational principle. Since then, the fractional variational principles have been becoming one of the most popular researching areas. Important contributions were obtained by many scholars, for example, Klimek (2001, 2002), Agrawal (2002, 2006, 2007, 2010), Agrawal and Baleanu (2007), Baleanu and Muslih (2005a, 2005b), Muslih and Baleanu (2005), Baleanu (2006), Baleanu et al. (2013), Rabei et al. (2007), Atanackovi’c (2008), Atanackovi’c and Pilipovi’c (2011), Atanackovic et al. (2012), He (2011, 2014), He et al. (2012), Malinowska and Torres (2010), Almeida and Torres (2011), Almeida (2012), Almeida and Malinowska (2014), El-Nabulsi (2011a, 2011b, 2014), Odzijewicz et al. (2012), Yang et al. (2013), Yang and Baleanu (2013), Bourdin et al. (2014) and Bahrami et al. (2015) and their collaborators and so on. These scholars from different angles put forward different kinds of fractional models and methods, and established corresponding fractional Lagrangian and Hamiltonian formulation and relevant Euler–Lagrange type equations. The formulations of fractional variational principles should still be more developed, continually.

In this paper, we will make an attempt to generalize the formulations for some fractional variational principles. The present paper is organized as follows: In Section 2, the extended exponentially fractional integral is reviewed briefly. In Section 3, the generalized variational formulations for the fractional variational principle based on extended exponentially fractional integral are proposed. In Section 4, two illustrative examples are given.

2

2 Extended fractional integral

Definition 1

Let f be a continuous function in the interval [ a , b ] . For t [ a , b ] , the left and right extended fractional integral of order α > 0 are defined by:

(1)
K ( z ) ( - α ) f ( t ) = 1 Γ ( α ) 0 z f ( ζ ) ( cosh z - cosh ζ ) α - 1 d ζ
where the multiplicity of ( cosh z - cosh ζ ) α - 1 is removed by requiring log ( cosh z - cosh ζ ) to be real when cosh z - cosh ζ > 0 .

Eq. (1) is called an extended exponentially fractional integral (El-Nabulsi, 2011a).

3

3 Generalized variational formulation

Problem 1

Given the smooth generalized Lagrangian function

L ( q , v , t ) : R n × R n × [ a , b ] R assumed to be a C 2 –function with respect to all its arguments. Find the stationary points of the extended exponentially fractional integral
(2)
S = 1 Γ ( α ) a t [ L ( q , v , τ ) + p ( q ̇ - v ) ] · ( cosh t - cosh τ ) α - 1 d τ ,
under the initial condition
(3)
q ( a ) = q a ,
where q is the generalized coordinate, q ̇ can only be used as the derivative of q , v is the generalized velocity which is defined as
(4)
v = d q d τ ,
p is the generalized momentum, τ is the intrinsic time, t is the observer time.
Theorem 1

If q , v , and p are solutions to the previous problem, i.e., q , v , and p are critical points of the functional S , then q , v , and p satisfy the following EulerLagrange equations:

(5)
v = q ̇ ,
(6)
p = L v ,
(7)
p ̇ - L q = ( α - 1 ) sinh τ cosh t - cosh τ p .
Proof

The variation of the functional S reads

(8)
δ S = 1 Γ ( α ) a t L q δ q + L v δ v + ( q ̇ - v ) δ p + p ( δ q ̇ - δ v ) ( cosh t - cosh τ ) α - 1 d τ ,
where all of q , v , and p are the independent variables.

Using the following formula of integration by part,

(9)
a t pg δ q ̇ d τ = - a t d ( pg ) d τ δ q d τ , where
(10)
g = ( cosh t - cosh τ ) α - 1 ,
(11)
g ̇ = ( 1 - α ) ( cosh t - cosh τ ) α - 2 sinh τ ,
we obtain the variation of the functional S , which takes the form
(12)
δ S = 1 Γ ( α ) a t L q - p ̇ - g ̇ g p g δ q + L v - p g δ v + ( q ̇ - v ) g δ p d τ ,

and we obtain the required result (5)–(7). □

4

4 Complementary variational formulation

Problem 2

Find the stationary points of the complementary extended exponentially fractional integral

(13)
S c = 1 Γ ( α ) a t L ( q , v , τ ) - pq ( 1 - α ) sinh τ cosh t - cosh τ - p ̇ q - pv · ( cosh t - cosh τ ) α - 1 d τ .
Theorem 2

If q , v , and p are critical points of the complement functional S c , then q , v , and p satisfy the generalized EulerLagrange Eqs. (5)–(7).

Proof

The variation of the functional S c reads 

(14)
δ S c = 1 Γ ( α ) a t L q - p ̇ - p g ̇ g δ q + L v - p δ v - q δ p ̇ - q g ̇ g + v δ p g d τ ,
where all of q , v , and p are the independent variables.

Using the following formula of integration by part,

(15)
- a t qg δ p ̇ d τ = a t d ( qg ) d τ δ p d τ , we obtain the variation of the functional, which takes the form
(16)
δ S = 1 Γ ( α ) a t L q - p ̇ - g ̇ g p g δ q + L v - p g δ v + ( q ̇ - v ) g δ p d τ ,

and we obtain the required result (5)–(7). □

5

5 Examples

Example 1

We discuss the case of generalized Caldirola-Kanai Lagrangian

(17)
L ( v , q , t ) = m ( τ ) v 2 2 - ω 2 q 2 2 .

The extended exponentially fractional action takes the form

(18)
S 1 = 1 Γ ( α ) a t m ( τ ) v 2 2 - ω 2 q 2 2 + p ( q ̇ - v ) · ( cosh t - cosh τ ) α - 1 d τ , where ω is the frequency, and
(19)
m ( τ ) = m 0 e - γ ( t - τ ) = m ¯ 0 e γ τ ,
where m ¯ 0 = m 0 e - γ t is an effective parameter which depends only on t and consequently, as the derivative is performed with respect to τ , it may be considered as an effective constant.

The complementary fractional action takes the form

(20)
S 1 c = 1 Γ ( α ) a t m ( τ ) v 2 2 - ω 2 q 2 2 - p ̇ q - pv - pq ( 1 - α ) sinh τ cosh t - cosh τ · ( cosh t - cosh τ ) α - 1 d τ .

The Euler–Lagrange equations are

(21)
v = q ̇ ,
(22)
p = mv ,
(23)
p ̇ + m ω 2 q = ( α - 1 ) sinh τ cosh t - cosh τ p .

For very large time, τ + , (23) is reduced to

(24)
p ̇ + m ω 2 q = ( 1 - α ) p , here,
(25)
lim τ + sinh τ cosh t - cosh τ = - 1 ,
while for very early time, τ + 0 , it is reduced to
(26)
p ̇ + m ω 2 q = 0 .
here,
(27)
p ̇ = m v ̇ + m ̇ v .

Inserting (21) into (22), we obtain

(28)
p = m q ̇ .

Inserting (28) into (23), we obtain

(29)
q ¨ + ω 2 q = - 1 - α sinh τ cosh t - cosh τ + γ q ̇ .

Hence, the generalized variational principles can propose the extended weak dissipations.

Example 2

We discuss the following special case of generalized Caldirola-Kanai Lagrangian

(30)
L = e γ τ mv 2 2 - e - γ τ m ω 2 q 2 2 .

The extended exponentially fractional action takes the form

(31)
S 2 = 1 Γ ( α ) a t e γ τ mv 2 2 - e - γ τ m ω 2 q 2 2 + p ( q ̇ - v ) ( cosh t - cosh τ ) α - 1 d τ .

The complement fractional action takes the form

(32)
S 2 c = 1 Γ ( α ) a t e γ τ mv 2 2 - e - γ τ m ω 2 q 2 2 - p ̇ q - pv - pq ( 1 - α ) sinh τ cosh t - cosh τ · ( cosh t - cosh τ ) α - 1 d τ .

The Euler–Lagrange equations are

(33)
v = q ̇ ,
(34)
p = e γ τ mv ,
(35)
p ̇ + e - γ τ m ω 2 q = ( α - 1 ) sinh τ cosh t - cosh τ p .

For very large time, τ + , (35) is reduced to

(36)
p ̇ = ( 1 - α ) p , which gives
(37)
p = p 0 e ( 1 - α ) τ ,
while for very early time, τ + 0 , it is reduced to
(38)
p ̇ + m ω 2 q = 0 .

Inserting (33) into (34), we obtain

(39)
p = e γ τ m q ̇ .

Inserting (39) into (35), we obtain

(40)
q ¨ + e - 2 γ τ ω 2 q = - 1 - α sinh τ cosh t - cosh τ + γ + m ̇ m q ̇ .

Hence, the generalized variational principles can also propose the extended weak dissipations.

6

6 Conclusion

In this paper, we obtain the generalized and complementary fractional variational formulations and corresponding Euler–Lagrange equations based on extended exponentially fractional integral. In the new actions, the parameters, q , v , and p , are all chosen as variable functions. Therefore, the Euler–Lagrange equations are reduced from second-order to first-order, which consist of velocity-displacement relations, momentum-velocity relations and equations of motion.

The result can be further extended to the fractional variational principles based on different types of fractional integral and derivatives operators, e.g., Riemann-Liouville, Caputo, Riesz, Caputo-Riesz, Erdélyi-Kober, Grünwald-Letnikov, Weyl and Marchaud etc. In addition, we hope this work will bring new opportunities in studying the fractional variational principles as well as their applications.

Acknowledgement

This work is supported by the Natural Science Foundation of Hebei Province, China (No. E2012203192).

References

  1. , . Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl.. 2002;272:368-379.
    [Google Scholar]
  2. , . Fractional variational calculus and the transversality conditions. J. Phys. A: Math. Gen.. 2006;39:10375-10384.
    [Google Scholar]
  3. , . Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor.. 2007;40:6287-6303.
    [Google Scholar]
  4. , . Generalized variational problems and Euler–Lagrange equations. Comput. Math. Appl.. 2010;59:1852-1864.
    [Google Scholar]
  5. , , . Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control. 2007;13(9–10):1269-1281.
    [Google Scholar]
  6. , . Fractional variational problems with the Riesz-Caputo derivative. Appl. Math. Lett.. 2012;25(2):142-148.
    [Google Scholar]
  7. , , . Fractional variational principle of Herglotz. Discrete Cont. Dyn. Syst. B. 2014;19(8):2367-2381.
    [Google Scholar]
  8. , , . Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul.. 2011;16:1490-1500.
    [Google Scholar]
  9. , . Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A: Math. Theor.. 2008;41:095201.
    [Google Scholar]
  10. , , . Hamilton’s principle with variable order fractional derivatives. Fract. Calc. Appl. Anal.. 2011;14(1):94-109.
    [Google Scholar]
  11. , . Complementary variational principles with fractional derivatives. Acta Mech.. 2012;223(4):685-704.
    [Google Scholar]
  12. , . A new approach on fractional variational problems and Euler–Lagrange equations. Commun. Nonlinear Sci. Numer. Simul.. 2015;23(1–3):39-50.
    [Google Scholar]
  13. , . Fractional Hamiltonian analysis of irregular systems. Signal Proc.. 2006;86(10):2632-2636.
    [Google Scholar]
  14. , . New applications of fractional variational principles. Rep. Math. Phys.. 2008;61
    [Google Scholar]
  15. , , . Formulation of Hamiltonian equations for fractional variational problems. Czech J. Phys.. 2005;55(6):633-642.
    [Google Scholar]
  16. , , . Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr.. 2005;72(2–3):119-121.
    [Google Scholar]
  17. , , , . A fractional variational approach to the fractional basset-type equation. Rep. Math. Phys.. 2013;72(1):57-64.
    [Google Scholar]
  18. , . Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition – application to fractional variational problems. Differ. Integral Equ.. 2014;27(7–8):743-766.
    [Google Scholar]
  19. , . Fractional variational problems from extended exponentially fractional integral. Appl. Math. Comput.. 2011;217:9492-9496.
    [Google Scholar]
  20. , . Universal fractional Euler–Lagrange equation from a generalized fractional derivate operator. Cent. Eur. J. Phys.. 2011;9(1):250-256.
    [Google Scholar]
  21. , . Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent. Comput. Appl. Math.. 2014;33(1):163-179.
    [Google Scholar]
  22. , . A new fractal derivation. Therm. Sci.. 2011;15:S145-S147.
    [Google Scholar]
  23. , . A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys.. 2014;53(11):3698-3718.
    [Google Scholar]
  24. , . Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A. 2012;376(4):257-259.
    [Google Scholar]
  25. , . Fractional sequential mechanics–models with symmetric fractional derivative. Czech J. Phys.. 2001;51:1348-1354.
    [Google Scholar]
  26. , . Lagrangean and Hamiltonian fractional sequential mechanics. Czech J. Phys.. 2002;52:1247-1253.
    [Google Scholar]
  27. , , . Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl.. 2010;59:3110-3116.
    [Google Scholar]
  28. , , . Introduction to the Fractional Calculus of Variations. London: Imp. Coll. Press; .
  29. , , . Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl.. 2005;304:599-606.
    [Google Scholar]
  30. , . Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal.. 2012;75(3):1507-1515.
    [Google Scholar]
  31. , . The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl.. 2007;327:891-897.
    [Google Scholar]
  32. , . Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E. 1996;53(2):1890-1899.
    [Google Scholar]
  33. , . Mechanics with fractional derivatives. Phys. Rev. E. 1997;55(3):3581-3592.
    [Google Scholar]
  34. , . Advanced Local Fractional Calculus and Its Applications. New York: World Science Publisher; .
  35. , , . Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci.. 2013;17(2):625-628.
    [Google Scholar]
  36. , . Cantortype cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A. 2013;377(28–30):1696-1700.
    [Google Scholar]
Show Sections