Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation error

In the present paper, we apply the fractional-order Bessel wavelets (FBWs) for solving optimal control problems with variable-order (VO) fractional dynamical system. The VO fractional derivative operator is proposed in the sense of Caputo type. To solve the considered problem, the collocation method based on FBWFs, pseudo-operational matrix of VO fractional derivative and the dual operational matrix is proposed. In fact, we convert the problem with unknown coefficients in the constraint equations, performance index and conditions to an optimisation problem, by utilising the proposed method. Also, the convergence of the method with details is discussed. At last, to demonstrate the high precision of the numerical approach, we examine several examples.

[1]  Delfim F. M. Torres,et al.  A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems , 2016, J. Optim. Theory Appl..

[2]  Manuel Duarte Ortigueira,et al.  Fractional Calculus for Scientists and Engineers , 2011, Lecture Notes in Electrical Engineering.

[3]  Hossein Jafari,et al.  Application of Legendre wavelets for solving fractional differential equations , 2011, Comput. Math. Appl..

[4]  Mohammad Hossein Heydari,et al.  A new direct method based on the Chebyshev cardinal functions for variable-order fractional optimal control problems , 2018, J. Frankl. Inst..

[5]  Carlos F.M. Coimbra,et al.  The variable viscoelasticity oscillator , 2005 .

[6]  Carlos F.M. Coimbra,et al.  On the Selection and Meaning of Variable Order Operators for Dynamic Modeling , 2010 .

[7]  Boying Wu,et al.  A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations , 2017, J. Comput. Appl. Math..

[8]  Delfim F. M. Torres,et al.  Minimal modified energy control for fractional linear control systems with the Caputo derivative , 2010, 1004.3113.

[9]  Yuanlu Li,et al.  Solving a nonlinear fractional differential equation using Chebyshev wavelets , 2010 .

[10]  Sohrab Ali Yousefi,et al.  An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix , 2016 .

[11]  Carlos F.M. Coimbra,et al.  A variable order constitutive relation for viscoelasticity , 2007 .

[12]  Saeed Kazem,et al.  Fractional-order Legendre functions for solving fractional-order differential equations , 2013 .

[13]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

[14]  Carlos F.M. Coimbra,et al.  Mechanics with variable‐order differential operators , 2003 .

[15]  Alireza Nazemi,et al.  Solving fractional optimal control problems with fixed or free final states by Haar wavelet collocation method , 2016, IMA J. Math. Control. Inf..

[16]  Nasser H. Sweilam,et al.  Legendre spectral-collocation method for solving some types of fractional optimal control problems , 2014, Journal of advanced research.

[17]  Yadollah Ordokhani,et al.  Numerical Solution of 1D and 2D Fractional Optimal Control of System via Bernoulli Polynomials , 2018 .

[18]  Carlos F.M. Coimbra,et al.  On the variable order dynamics of the nonlinear wake caused by a sedimenting particle , 2011 .

[19]  Dumitru Baleanu,et al.  Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution , 2017, Commun. Nonlinear Sci. Numer. Simul..

[20]  George M. Phillips,et al.  Theory and applications of numerical analysis , 1976, The Mathematical Gazette.

[21]  Behrouz Parsa Moghaddam,et al.  An efficient cubic spline approximation for variable-order fractional differential equations with time delay , 2017 .

[22]  Carlos F.M. Coimbra,et al.  Variable Order Modeling of Diffusive-convective Effects on the Oscillatory Flow Past a Sphere , 2008 .

[23]  Sohrab Effati,et al.  Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation , 2018 .

[24]  Carlo Cattani,et al.  Wavelets method for solving fractional optimal control problems , 2016, Appl. Math. Comput..

[25]  W. W. Bell,et al.  Special Functions for Scientists and Engineers , 1968 .

[26]  Mehdi Dehghan,et al.  A numerical technique for solving fractional optimal control problems , 2011, Comput. Math. Appl..

[27]  Zakieh Avazzadeh,et al.  A new Wavelet Method for Variable‐Order Fractional Optimal Control Problems , 2018 .

[28]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[29]  M. Dehghan,et al.  The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems , 2011 .

[30]  O. Agrawal.  A Quadratic Numerical Scheme for Fractional Optimal Control Problems , 2008 .

[31]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[32]  Gennady N. Chuev,et al.  A CAS Wavelet Method for Solving Nonlinear Fredholm Integro- Differential Equations of Fractional Order , 2011 .

[33]  Delfim F. M. Torres,et al.  A survey on fuzzy fractional differential and optimal control nonlocal evolution equations , 2017, J. Comput. Appl. Math..

[34]  H. S. Nik,et al.  A Bessel collocation method for solving fractional optimal control problems , 2015 .

[35]  K. Parand,et al.  Application of Bessel functions for solving differential and integro-differential equations of the fractional order ☆ , 2014 .

[36]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

[37]  Hongguang Sun,et al.  Random-order fractional differential equation models , 2011, Signal Process..

[38]  Yadollah Ordokhani,et al.  On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay , 2019, Numer. Linear Algebra Appl..

[39]  Robert J. Marks,et al.  Differintegral interpolation from a bandlimited signal's samples , 1981 .

[40]  C. Chui.  Wavelets: A Mathematical Tool for Signal Analysis , 1997 .

[41]  Yadollah Ordokhani,et al.  Fractional-order Bernoulli wavelets and their applications , 2016 .

[42]  Clara-Mihaela Ionescu,et al.  The role of fractional calculus in modeling biological phenomena: A review , 2017, Commun. Nonlinear Sci. Numer. Simul..

[43]  Zakieh Avazzadeh,et al.  Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion , 2018, Commun. Nonlinear Sci. Numer. Simul..

[44]  B. West.  Fractional Calculus in Bioengineering , 2007 .

[45]  Yadollah Ordokhani,et al.  Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations , 2018, Appl. Math. Comput..

[46]  Fawang Liu,et al.  A characteristic difference method for the variable-order fractional advection-diffusion equation , 2013 .

[47]  W. Zahra,et al.  Non standard finite difference method for solving variable order fractional optimal control problems , 2017 .