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.
AgrawalOP (1989) General formulation for the numerical solution of optimal control problems. International Journal of Control50: 627–638.
2.
AgrawalOP (2002) Formulation of Euler–Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications272: 368–379.
3.
AgrawalOP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics38: 323–337.
4.
AgrawalOP (2006) Fractional variational calculus and the transversality conditions. Journal of Physics A39: 10375–10384.
5.
AgrawalOP (2007a) Fractional variational calculus in terms of Riesz fractional derivatives. Journal of Physics A40: 6287–6303.
6.
AgrawalOP (2007b) Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative. Journal of Vibration and Control13: 1217–1237.
7.
AgrawalOP (2008) A general finite element formulation for fractional variational problems. Journal of Mathematical Analysis and Applications337: 1–12.
8.
AlmeidaRKhosravian-ArabHShamsiM (2012) A generalized fractional variational problem depending on indefinite integrals: Euler–Lagrange equation and numerical solution. Journal of Vibration and Control19: 2177–2186.
9.
AlmeidaRTorresDFM (2009) Calculus of variations with fractional derivatives and fractional integrals. Applied Mathematics Letters22: 1816–1820.
10.
AtanackovićTMKonjikSPilipovićS (2008) Variational problems with fractional derivatives: Euler–Lagrange equations. Journal of Physics A41: 095201.
11.
AtanackovićTMOparnicaLPilipovićS (2007) On a nonlinear distributed order fractional differential equation. Journal of Mathematical Analysis and Applications328: 590–608.
12.
BaleanuD (2008) New applications of fractional variational principles. Reports on Mathematical Physics61: 199–206.
13.
BastosNFerreiraRTorresD (2011) Discrete-time fractional variational problems. Signal Processing91: 513–524.
14.
BettsJT (2001) Practical Methods for Optimal Control Using Nonlinear Programming, Philadelphia, PA: SIAM.
15.
CressonJ (2007) Fractional embedding of differential operators and Lagrangian systems. Journal of Mathematical Physics48: 033504.
16.
DattaKBMohanBM (1995) Orthogonal Functions in Systems and Control, Singapore: World Scientific.
17.
DehghanMManafianJSaadatmandiA (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations26: 448–479.
18.
DiethelmK (2010) The Analysis of Fractional Differential Equations, Berlin: Springer-Verlag.
19.
DohaEHBhrawyAHBaleanuDEzz-EldienSS (2013) On shifted Jacobi spectral approximations for solving fractional differential equations. Applied Mathematics and Computation219: 8042–8056.
20.
DreisigmeyerDWYoungPM (2003) Nonconservative Lagrangian mechanics: a generalized function approach. Journal of Physics A36: 8297–8310.
21.
DreisigmeyerDWYoungPM (2004) Extending Bauer’s corollary to fractional derivatives. Journal of Physics A37: L117–L121.
22.
EsmaeiliSShamsiM (2011) A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Communications in Nonlinear Science and Numerical Simulation16: 3646–3654.
23.
EslahchiMRDehghanMParviziM (2014) Application of the collocation method for solving nonlinear fractional integro-differential equations. Journal of Computational and Applied Mathematics257: 105–128.
24.
FredericoGSFTorresDFM (2007) A formulation of Noether’s theorem for fractional problems of the calculus of variations. Journal of Mathematical Analysis and Applications334: 834–846.
25.
GautschiW (2004) Orthogonal Polynomials: Computation and Approximation, New York: Oxford University Press.
26.
HilferR (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.
KilbasAASrivastavaHMTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations, New York: Elsevier Science Inc.
29.
KlimekK (2001) Fractional sequential mechanics–models with symmetric fractional derivative. Czechoslovak Journal of Physics51: 1348–1354.
30.
KlimekK (2002) Stationarity-conservation laws for fractional differential equations with variable coefficients. Journal of Physics A35: 6675–6693.
31.
LakestaniMDehghanMIrandoust-PakchinS (2012) The construction of operational matrix of fractional derivatives using B-spline functions. Communications in Nonlinear Science and Numerical Simulation17: 1149–1162.
32.
LotfiADehghanMYousefiSA (2011) A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications62: 1055–1067.
33.
LotfiAYousefiSADehghanM (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 Mathematics250: 143–160.
34.
MainardiF (2010) Fractional Calculus and Waves in Linear Viscoelasticity, London: Imperial College Press.
35.
MalinowskaAB (2012) Fractional variational calculus for non-differentiable functions. Fractional Dynamics and Control97–108.
36.
MalinowskaABTorresDFM (2010) Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Computers and Mathematics with Applications59: 3110–3116.
37.
MalinowskaABTorresDFM (2011) Fractional calculus of variations for a combined Caputo derivative. Fractional Calculus and Applied Analysis14: 523–537.
38.
MetzlerRKlafterJ (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports339: 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.
MuslihSIBaleanuD (2005) Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. Journal of Mathematical Analysis and Applications304: 599–606.
42.
PodlubnyI (1999) Fractional Differential Equations, San Diego, CA: Academic Press, Inc.
43.
PoosehSAlmeidaRTorresDFM (2013a) A discrete time method to the first variation of fractional order variational functionals. Central European Journal of Physics11: 1262–1267.
44.
PoosehSAlmeidaRTorresDFM (2013b) Discrete direct methods in the fractional calculus of variations. Computers and Mathematics with Applications66: 668–676.
RieweF (1997) Mechanics with fractional derivatives. Physical Review E55: 3581–3592.
47.
RossikhinYShitikovaM (2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent result. Applied Mechanics Reviews63: 010801.
48.
SaadatmandiADehghanM (2010) A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications59: 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.
StanislavskyAA (2006) Hamiltonian formalism of fractional systems. The European Physical Journal B - Condensed Matter and Complex Systems49: 93–101.
52.
SwarztrauberPN (2002) On computing the points and weights for Gauss–Legendre quadrature. SIAM Journal on Scientific Computing24: 945–954.
53.
TarasovVZaslavskyG (2005) Fractional Ginzburg–Landau equation for fractal media. Physica A: Statistical Mechanics and its Applications354: 249–261.
54.
TarasovVZaslavskyG (2006) Nonholonomic constraints with fractional derivatives. Journal of Physics A: Mathematical and General39: 9797–9815.
55.
YousefiSALotfiADehghanM (2011) The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control17: 2059–2065.
