A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials

Abstract

The aim of this paper is to investigate, from the numerical point of view, the Jacobi polynomials to solve fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). A direct numerical method for solving a general class of FVPs and FOCPs is presented. The fractional derivative in FVPs is in the Caputo sense and in FOCPs is in the Riemann–Liouville sense. The Rayleigh–Ritz method is introduced for the numerical solution of FVPs containing left or right Caputo fractional derivatives. Rayleigh–Ritz method is one of the well-known direct methods used for the solution of variational problems. In this technique, at first, we expand the unknown function in terms of the modified Jacobi polynomials and then we derive a compact form of fractional derivative of the unknown function in terms of the Jacobi polynomials. Examples indicate that the new technique has high accuracy and is very efficient to implement.

Keywords

Caputo derivative Riemann–Liouville derivative fractional variational problems fractional optimal control problems Jacobi polynomials

Get full access to this article

View all access options for this article.

References

Agrawal

(1989) General formulation for the numerical solution of optimal control problems. International Journal of Control 50: 627–638.

Agrawal

(2002) Formulation of Euler–Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications 272: 368–379.

Agrawal

(2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics 38: 323–337.

Agrawal

(2006) Fractional variational calculus and the transversality conditions. Journal of Physics A 39: 10375–10384.

Agrawal

(2007a) Fractional variational calculus in terms of Riesz fractional derivatives. Journal of Physics A 40: 6287–6303.

Agrawal

(2007b) Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative. Journal of Vibration and Control 13: 1217–1237.

Agrawal

(2008) A general finite element formulation for fractional variational problems. Journal of Mathematical Analysis and Applications 337: 1–12.

Almeida

Khosravian-Arab

Shamsi

(2012) A generalized fractional variational problem depending on indefinite integrals: Euler–Lagrange equation and numerical solution. Journal of Vibration and Control 19: 2177–2186.

Almeida

Torres

DFM

(2009) Calculus of variations with fractional derivatives and fractional integrals. Applied Mathematics Letters 22: 1816–1820.

10.

Atanacković

Konjik

Pilipović

(2008) Variational problems with fractional derivatives: Euler–Lagrange equations. Journal of Physics A 41: 095201.

11.

Atanacković

Oparnica

Pilipović

(2007) On a nonlinear distributed order fractional differential equation. Journal of Mathematical Analysis and Applications 328: 590–608.

12.

Baleanu

(2008) New applications of fractional variational principles. Reports on Mathematical Physics 61: 199–206.

13.

Bastos

Ferreira

Torres

(2011) Discrete-time fractional variational problems. Signal Processing 91: 513–524.

14.

Betts

(2001) Practical Methods for Optimal Control Using Nonlinear Programming, Philadelphia, PA: SIAM.

15.

Cresson

(2007) Fractional embedding of differential operators and Lagrangian systems. Journal of Mathematical Physics 48: 033504.

16.

Datta

Mohan

(1995) Orthogonal Functions in Systems and Control, Singapore: World Scientific.

17.

Dehghan

Manafian

Saadatmandi

(2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations 26: 448–479.

18.

Diethelm

(2010) The Analysis of Fractional Differential Equations, Berlin: Springer-Verlag.

19.

Doha

Bhrawy

Baleanu

Ezz-Eldien

(2013) On shifted Jacobi spectral approximations for solving fractional differential equations. Applied Mathematics and Computation 219: 8042–8056.

20.

Dreisigmeyer

Young

(2003) Nonconservative Lagrangian mechanics: a generalized function approach. Journal of Physics A 36: 8297–8310.

21.

Dreisigmeyer

Young

(2004) Extending Bauer’s corollary to fractional derivatives. Journal of Physics A 37: L117–L121.

22.

Esmaeili

Shamsi

(2011) A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Communications in Nonlinear Science and Numerical Simulation 16: 3646–3654.

23.

Eslahchi

Dehghan

Parvizi

(2014) Application of the collocation method for solving nonlinear fractional integro-differential equations. Journal of Computational and Applied Mathematics 257: 105–128.

24.

Frederico

GSF

Torres

DFM

(2007) A formulation of Noether’s theorem for fractional problems of the calculus of variations. Journal of Mathematical Analysis and Applications 334: 834–846.

25.

Gautschi

(2004) Orthogonal Polynomials: Computation and Approximation, New York: Oxford University Press.

26.

Hilfer

(2000) Applications of Fractional Calculus in Physics, Singapore: World Scientific.

27.

Kayedi-Bardeh A, Eslahchi MR and Dehghan M (2014) A method for obtaining the operational matrix of the fractional Jacobi functions and applications. Journal of Vibration and Control 20: 736–748.

28.

Kilbas

Srivastava

Trujillo

(2006) Theory and Applications of Fractional Differential Equations, New York: Elsevier Science Inc.

29.

Klimek

(2001) Fractional sequential mechanics–models with symmetric fractional derivative. Czechoslovak Journal of Physics 51: 1348–1354.

30.

Klimek

(2002) Stationarity-conservation laws for fractional differential equations with variable coefficients. Journal of Physics A 35: 6675–6693.

31.

Lakestani

Dehghan

Irandoust-Pakchin

(2012) The construction of operational matrix of fractional derivatives using B-spline functions. Communications in Nonlinear Science and Numerical Simulation 17: 1149–1162.

32.

Lotfi

Dehghan

Yousefi

(2011) A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications 62: 1055–1067.

33.

Lotfi

Yousefi

Dehghan

(2013) Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. Journal of Computational and Applied Mathematics 250: 143–160.

34.

Mainardi

(2010) Fractional Calculus and Waves in Linear Viscoelasticity, London: Imperial College Press.

35.

Malinowska

(2012) Fractional variational calculus for non-differentiable functions. Fractional Dynamics and Control 97–108.

36.

Malinowska

Torres

DFM

(2010) Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Computers and Mathematics with Applications 59: 3110–3116.

37.

Malinowska

Torres

DFM

(2011) Fractional calculus of variations for a combined Caputo derivative. Fractional Calculus and Applied Analysis 14: 523–537.

38.

Metzler

Klafter

(2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339: 1–77.

39.

Mohebbi A, Abbaszadeh M and Dehghan M (2013) Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Computer Methods in Applied Mechanics and Engineering 264: 163–177.

40.

Mohebbi A, Abbaszadeh M and Dehghan M (2014) Solution of two-dimensional modifed anomalous fractional subdiffusion equation via radial basis functions (RBF) meshless method. Engineering Analysis with Boundary Elements 38: 72–82.

41.

Muslih

Baleanu

(2005) Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. Journal of Mathematical Analysis and Applications 304: 599–606.

42.

Podlubny

(1999) Fractional Differential Equations, San Diego, CA: Academic Press, Inc.

43.

Pooseh

Almeida

Torres

DFM

(2013a) A discrete time method to the first variation of fractional order variational functionals. Central European Journal of Physics 11: 1262–1267.

44.

Pooseh

Almeida

Torres

DFM

(2013b) Discrete direct methods in the fractional calculus of variations. Computers and Mathematics with Applications 66: 668–676.

45.

Riewe

(1996) Nonconservative Lagrangian and Hamiltonian mechanics. Physical Review E 53: 1890–1899.

46.

Riewe

(1997) Mechanics with fractional derivatives. Physical Review E 55: 3581–3592.

47.

Rossikhin

Shitikova

(2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent result. Applied Mechanics Reviews 63: 010801.

48.

Saadatmandi

Dehghan

(2010) A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications 59: 1326–1336.

49.

Saadatmandi A and Dehghan M (2011) A tau approach for solution of the space fractional diffusion equation. Computers and Mathematics with Applications 62: 1135–1142.

50.

Saadatmandi A, Dehghan M and Azizi MR (2012) The Sinc-Legendre collocation method for a class of fractional convection-diffusion equation with variable coefficients. Communication in Nonlinear Science and Numerical Simulation 17: 4125–4136.

51.

Stanislavsky

(2006) Hamiltonian formalism of fractional systems. The European Physical Journal B - Condensed Matter and Complex Systems 49: 93–101.

52.

Swarztrauber

(2002) On computing the points and weights for Gauss–Legendre quadrature. SIAM Journal on Scientific Computing 24: 945–954.

53.

Tarasov

Zaslavsky

(2005) Fractional Ginzburg–Landau equation for fractal media. Physica A: Statistical Mechanics and its Applications 354: 249–261.

54.

Tarasov

Zaslavsky

(2006) Nonholonomic constraints with fractional derivatives. Journal of Physics A: Mathematical and General 39: 9797–9815.

55.

Yousefi

Lotfi

Dehghan

(2011) The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control 17: 2059–2065.