This study introduces an innovative numerical approach employing the hybrid Lerch functions to solve distributed-order fractional optimal control problems. The proposed approach combines the block-pulse functions with the Lerch polynomials to create an efficient approximation scheme. The Laplace transform method is employed to formulate the Riemann–Liouville fractional integral operator corresponding to the hybrid functions. The unknown components are represented through the hybrid Lerch function expansion, and the application of Gauss–Legendre quadrature reformulates the original model into algebraic expressions. These expressions are subsequently resolved employing the Newton–Raphson iterative procedure. A comprehensive error analysis is conducted in Sobolev norms. The effectiveness of the proposed strategy is assessed through four representative computational experiments, with results showing superior precision and computational efficiency compared to existing methods. Furthermore, the method exhibits significantly lower computational time than conventional techniques, making it particularly suitable for complex distributed-order fractional systems.
AghchiSFazliHSunH (2024) A numerical approach for solving optimal control problem of fractional order vibration equation of large membranes. Computers & Mathematics with Applications165: 19–27.
2.
AgrawalOP (2008a) A formulation and numerical scheme for fractional optimal control problems. Journal of Vibration and Control14(9-10): 1291–1299.
3.
AgrawalOP (2008b) A formulation and numerical scheme for fractional optimal control problems. Journal of Vibration and Control14(9-10): 1291–1299.
4.
AgrawalOP (2008c) A quadratic numerical scheme for fractional optimal control problems. Journal of Dynamic Systems Measurement and Control130: 011010.
5.
AgrawalOPDefterliOBaleanuD (2010) Fractional optimal control problems with several state and control variables. Journal of Vibration and Control16(13): 1967–1976.
6.
AlizadehAEffatiS (2018) An iterative approach for solving fractional optimal control problems. Journal of Vibration and Control24(1): 18–36.
7.
BassRF (2013) Real analysis for graduate students. Createspace Ind Pub.
8.
BransonD (1996) An extension of stirling numbers. Fibonacci Quarterly34(3): 213–223.
9.
BrycW (1995) Normal Distributions. Springer.
10.
CanutoCHussainiMYQuarteroniA, et al. (2006a) Spectral Methods: Fundamentals in Single Domains. Springer.
11.
CanutoCHussainiMYQuarteroniA, et al. (2006b) Spectral Methods: Fundamentals in Single Domains. Springer.
DiethelmKFordNJ (2009) Numerical analysis for distributed-order differential equations. Journal of Computational and Applied Mathematics225(1): 96–104.
14.
DuongPKwokELeeM (2016) Deterministic analysis of distributed order systems using operational matrix. Applied Mathematical Modelling40(3): 1929–1940.
15.
FredericoGSTorresDF (2007) Fractional optimal control in the sense of caputo and the fractional noether’s theorem. arXiv preprint arXiv:0712.1844.
16.
HoseiniTOrdokhaniYRahimkhaniP (2023) A numerical method based on the fractional vieta-fibonacci functions for a class of fractional optimal control problems. Iranian Journal of Science and Technology, Transactions of Electrical Engineering47(3): 1117–1128.
17.
IlieSJeffreyDJCorlessRM, et al. (2015) Computation of stirling numbers and generalizations. In 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE, pp. 57–60.
18.
JibenjaNYuttananBRazzaghiM (2018) An efficient method for numerical solutions of distributed-order fractional differential equations. Journal of Computational and Nonlinear Dynamics13(11): 111003.
19.
KeshavarzEOrdokhaniYRazzaghiM (2016) A numerical solution for fractional optimal control problems via bernoulli polynomials. Journal of Vibration and Control22(18): 3889–3903.
20.
KruchininDVKruchininVV (2013) Explicit formulas for some generalized polynomials. Applied Mathematics & Information Sciences7(5): 2083–2088.
21.
KumarNMehraM (2024) Generalized fractional-order legendre wavelet method for two dimensional distributed order fractional optimal control problem. Journal of Vibration and Control30(7-8): 1690–1705.
22.
KumarYSinghSSrivastavaN, et al. (2020) Wavelet approximation scheme for distributed order fractional differential equations. Computers & Mathematics with Applications80(8): 1985–2017.
23.
LorenzoCFHartleyTT (2002) Variable order and distributed order fractional operators. Nonlinear Dynamics29(1): 57–98.
24.
LotfiAYousefiSA (2013) A numerical technique for solving a class of fractional variational problems. Journal of Computational and Applied Mathematics237(1): 633–643.
25.
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.
26.
MainardiFMuraAPagniniG, et al. (2008) Time-fractional diffusion of distributed order. Journal of Vibration and Control14(9-10): 1267–1290.
27.
MarzbanHRazzaghiM (2005) Analysis of time-delay systems via hybrid of block-pulse functions and Taylor series. Journal of Vibration and Control11(12): 1455–1468.
28.
MashayekhiSRazzaghiM (2016) Numerical solution of distributed order fractional differential equations by hybrid functions. Journal of Computational Physics315: 169–181.
29.
MashayekhiSOrdokhaniYRazzaghiM (2012) Hybrid functions approach for nonlinear constrained optimal control problems. Communications in Nonlinear Science and Numerical Simulation17(4): 1831–1843.
30.
MashayekhiSRazzaghiMWattanataweekulM (2016) Analysis of multi-delay and piecewise constant delay systems by hybrid functions approximation. Differential Equations and Dynamical Systems24(1): 1–20.
31.
MorgadoMLRebeloMFerrasLL, et al. (2017) Numerical solution for diffusion equations with distributed order in time using a chebyshev collocation method. Applied Numerical Mathematics114: 108–123.
32.
NaberM (2004) Distributed order fractional sub-diffusion. Fractals12(01): 23–32.
33.
OdibatZMomaniSErturkVS (2008) Generalized differential transform method: application to differential equations of fractional order. Applied Mathematics and Computation197(2): 467–477.
34.
OmidiniyaPAlipourM (2020) A semi-analytic method for solving a class of non-linear optimal control problems. International Journal of Industrial Electronics, Control and Optimization3: 407–414.
35.
RabieiKOrdokhaniYBabolianE (2017) The boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dynamics88(2): 1013–1026.
36.
RahimkhaniPOrdokhaniY (2019) Generalized fractional-order bernoulli–legendre functions: an effective tool for solving two-dimensional fractional optimal control problems. IMA Journal of Mathematical Control and Information36(1): 185–212.
37.
RahimkhaniPOrdokhaniY (2021) Numerical investigation of distributed-order fractional optimal control problems via bernstein wavelets. Optimal Control Applications and Methods42(1): 355–373.
38.
RahimkhaniPOrdokhaniYLimaP (2019) An improved composite collocation method for distributed-order fractional differential equations based on fractional chelyshkov wavelets. Applied Numerical Mathematics145: 1–27.
39.
RahimkhaniPOrdokhaniYSabermahaniS (2025) An accurate wavelets-collocation technique for neutral delay distributed-order fractional optimal control problems. Optimal Control Applications and Methods46(1): 94–113.
40.
RazzaghiMVoTN (2022) Numerical solutions for distributed-order fractional optimal control problems by using müntz–legendre wavelets. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences478(2258): 20210617.
41.
SabermahaniSOrdokhaniY (2024) Solving distributed-order fractional optimal control problems via the fibonacci wavelet method. Journal of Vibration and Control30(1-2): 418–432.
42.
SabermahaniSOrdokhaniYRabieiK, et al. (2023) Solution of optimal control problems governed by volterra integral and fractional integro-differential equations. Journal of Vibration and Control29(15-16): 3796–3808.
43.
SabermahaniSOrdokhaniYRahimkhaniP (2024a) Touchard–ritz method to solve variable-order fractional optimal control problems. Iranian Journal of Science and Technology, Transactions of Electrical Engineering48(3): 1189–1198.
44.
SabermahaniSOrdokhaniYRazzaghiM (2024b) Numerical solution of different kinds of fractional-order optimal control problems using generalized lucas wavelets and the least squares method. Optimal Control Applications and Methods45(6): 2702–2721.
45.
SabermahaniSOrdokhaniYRahimkhaniP (2025) Modified wavelets technique for multitype 2d fractional optimal control problems. Journal of Vibration and Control31(13-14): 2576–2590.
46.
SahuPSaha RayS (2018) Comparison on wavelets techniques for solving fractional optimal control problems. Journal of Vibration and Control24(6): 1185–1201.
47.
ShabaniZTajadodiH (2019) A numerical scheme for constrained optimal control problems. International Journal of Industrial Electronics Control and Optimization2(3): 233–238.
48.
SinghVKPandeyRKSinghS (2010) A stable algorithm for hankel transforms using hybrid of block-pulse and legendre polynomials. Computer Physics Communications181(1): 1–10.
49.
TangXLiuZWangX (2015) Integral fractional pseudospectral methods for solving fractional optimal control problems. Automatica62: 304–311.
50.
WangXTLiYM (2009) Numerical solutions of integrodifferential systems by hybrid of general block-pulse functions and the second chebyshev polynomials. Applied Mathematics and Computation209(2): 266–272.
51.
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.
52.
YuttananBRazzaghiM (2019) Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Applied Mathematical Modelling70: 350–364.
53.
ZakyMA (2018a) A legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dynamics91(4): 2667–2681.
54.
ZakyMA (2018b) A legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dynamics91(4): 2667–2681.
55.
ZakyMAMachadoJT (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Communications in Nonlinear Science and Numerical Simulation52: 177–189.
56.
ZeidSSYousefiM (2016) Approximated solutions of linear quadratic fractional optimal control problems. Journal of Applied Mathematics Statistics and Informatics12(2): 83–94.
57.
ZhuLFanQ (2013) Numerical solution of nonlinear fractional-order volterra integro-differential equations by scw. Communications in Nonlinear Science and Numerical Simulation18(5): 1203–1213.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
