We use a hybrid local meshless method to solve the distributed optimal control problem of a system governed by parabolic partial differential equations with Caputo fractional time derivatives of order α ∈ (0, 1]. The presented meshless method is based on the linear combination of moving least squares and radial basis functions in the same compact support, this method will change between interpolation and approximation. The aim of this paper is to solve the system of coupled fractional partial differential equations, with necessary and sufficient conditions, for fractional distributed optimal control problems using a combination of moving least squares and radial basis functions. To keep matters simple, the problem has been considered in the one-dimensional case, however the techniques can be employed for both the two- and three-dimensional cases. Several test problems are employed and results of numerical experiments are presented. The obtained results confirm the acceptable accuracy of the proposed method.
AgrawalOP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics38(1–4): 323–337.
2.
AgrawalOPDefterliOBaleanuD (2010) Fractional optimal control problems with several state and control variables. Journal of Vibration and Control16: 1967–1976.
3.
AlipourMRostamyDBaleanuD (2013) Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. Journal of Vibration and Control19(16): 2523–2540.
4.
Antil H, Otárola E and Salgado AJ (2016) A space-time fractional optimal control problem: analysis and discretization. SIAM Journal on Control and Optimization 54(3): 1295–1328.
5.
ArmentanoMG (2001) Error estimates in Sobolev spaces for moving least square approximations. SIAM Journal Numerical Analysis39: 38–51.
6.
ArmentanoMGDuranRG (2001) Error estimates for moving least square approximations. Applied Numerical Mathamatics37: 397–416.
7.
BhrawyAHEzz-EldienSS (2015) A new Legendre operational technique for delay fractional optimal control problems. Calcolo. 1–23.
8.
BhrawyAHDohaEHTenreiro MachadoJAet al. (2015) An efficient numerical scheme for solving multidimensional fractional optimal control problems with a quadratic performance index. Asian Journal of Control17(6): 2389–2402.
9.
BolligEFFlyerNErlebacherG (2012) Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs. Journal of Computational Physics231(21): 7133–7151.
10.
BuhmannMD (2003) Radial Basis Functions: Theory and Implementations, Cambridge: Cambridge University Press.
11.
ChangRYYangSY (1986) Solution of two-point boundary value problems by generalized orthogonal polynomials and application to optimal control of lumped and distributed parameter systems. International Journal of Control43: 1785–1802.
12.
DefterliO (2010) A numerical scheme for two-dimensional optimal control problems with memory effect. Computers and Mathematics with Applications59(5): 1630–1636.
13.
DehghanMMohammadiV (2014) The numerical solution of Fokker–Planck equation with radial basis functions (RBFs) based on the meshless technique of Kansa’s approach and Galerkin method. Engeeniring Analysis and Boundary Element47: 38–63.
14.
DehghanMNikpourA (2013a) Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Applied Mathematics and Modeling37: 8578–8599.
15.
DehghanMNikpourA (2013b) The solitary wave solution of coupled Klein–Gordon–Zakharov equations via two different numerical methods. Computer Physics Communications184(9): 2145–2158.
16.
DehghanMShokriA (2007) A numerical method for two-dimensional Schrdinger equation using collocation and radial basis functions. Computational Mathematics and Application54: 136–146.
17.
DehghanMShokriA (2008) A numerical method for solution of the two- dimensional sine-Gordon equation using the radial basis functions. Mathematics and Computers in Simulation79: 700–715.
18.
DohaEHBhrawyAHBaleanuDet al. (2015) An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Advances in Difference Equations (1). 1–17.
19.
Ezz-EldienSS (2016) New quadrature approach based on operational matrix for solving a class of fractional variational problems. Journal of Computational Physics317: 362–381.
20.
Ezz-EldienSSDohaEHBaleanuDet al. (2015) A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. Journal of Vibration and Control. 1077546315573916.
21.
Ezz-EldienSSHafezRMBhrawyAHet al. (2016) New numerical approach for fractional variational problems using shifted Legendre orthonormal polynomials. Journal of Optimization Theory and Applications. 1–26.
22.
HartEECoxSJDjidjeliK (2011) Compact RBF meshless methods for photonic crystal modelling. Journal of Computational Physics230(12): 4910–4921.
23.
HilferR (2000) Applications of Fractional Calculus in Physics, Singapore: World Scientific.
24.
HonYCCheungKFMaoXZet al. (1999) Multiquadric solution for shallow water equations. ASCE Journal Hydraulic Engineering125(5): 524–533.
25.
LevinD (1998) The approximation power of moving least-squares. Mathematics of Computation67: 1517–1531.
26.
LiJChengAHDChenCS (2003) A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Engineering Analysis with Boundary Elements27(3): 251–257.
27.
KansaEJ (1990a) Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-I. Computers and Mathematics with Application19: 127–145.
28.
KansaEJGeiserJ (2013) Numerical solution to time-dependent 4D inviscid Burgers’ equations. Engineering Analysis with Boundary Elements37(3): 637–645.
29.
KansaEJHonYC (2000) Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations. Computers and Mathematics with applications39(7): 123–137.
30.
KosecG (2012) Simulation of multiphase thermo-fluid phenomena by a local meshless numerical approach. Informatica36(2): 221–222.
31.
KulishVVLageJL (2002) Application of fractional calculus to fluid mechanics. Journal of Fluids Engineering124(3): 803–806.
32.
LancasterPSalkauskasK (1981) Surfaces generated by moving least squares methods. Mathematics of Computation155: 141–158.
33.
LevinD (1998) The approximation power of moving least-squares. Mathematics of Computation67: 1517–1531.
34.
LiJChengAHDChenCS (2003) A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Engineering Analysis with Boundary Elements27(3): 251–257.
35.
LotfiADehghanMYousefiSA (2011) A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications62(3): 1055–1067.
36.
MittalRCJiwariR (2012) A differential quadrature method for numerical solutions of Burgers’-type equations. International Journal of Numerical Method22(7): 880–895.
37.
MohebbiAAbbaszadehMDehghanM (2013) The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics. Engineering Analysis with Boundary Elements37(2): 475–485.
38.
MonaghanJJ (1982) Why particle methods work. SIAM Journal on Scientific and Statistical Computing3: 422–433.
39.
NazemiAKheyrinatajF (2015) Parabolic optimal control problems with a quintic B-spline dynamic model. Nonlinear Dynamics80(1–2): 653–667.
40.
Pepper D and Sarler B (2005) Application of meshless methods for thermal analysis. Journal of Mechanical Engineering 51(7): 476–483.
41.
RakhshanSAKamyadAVEffatiS (2016) An efficient method to solve a fractional differential equation by using linear programming and its application to an optimal control problem. Journal of Vibration and Control22(8): 2120–2134.
42.
SabatierJAgrawalOPMachadoJT (2007) Advances in Fractional Calculus, Dordrecht: Springer.
43.
SchabackR (1995) Error estimates and condition numbers for radial basis function interpolation. Advances in Computational Mathematics3: 251–264.
44.
Schaback R (1997) Reconstruction of multivariate functions from scattered data. Doctoral dissertation, Inst. f. numerische und angewandte Mathematik, Univ.
45.
VermaAJiwariRKumarS (2014) A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein–Gordon equation. International Journal of Numerical Methods for Heat and Fluid Flow24(7): 1390–1404.
46.
WangB (2015) A local meshless method based on moving least squares and local radial basis functions. Engineering Analysis with Boundary Elements50: 395–401.
47.
WangHQinQH (2006) A meshless method for generalized linear or nonlinear Poisson-type problems. Engineering Analysis with Boundary Elements30(6): 515–521.
48.
WendlandH (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics4: 389–396.
49.
WendlandH (1999) Meshless Galerkin methods using radial basis functions. Mathematics of Computation68: 1521–1531.
50.
WendlandH (2005) Scattered data approximation, Cambridge: Cambridge University Press.
51.
WuZMSchabackR (1993) Local error estimates for radial basis function interpolation of scattered data. IMA Journal of Numerical Analysis13: 13–27.
52.
YaoGSarlerBChenCS (2011) A comparison of three explicit local meshless methods using radial basis functions. Engineering Analysis with Boundary Elements35(3): 600–609.
53.
YousefiSALotfiADehghanM (2011) The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control17(13): 2059–2065.
54.
ZuppaC (2003) Error estimates for moving least square approximations. Bulletin of the Brazilian Mathematical Society34: 231–249.
