• Previous Article
    Green cross-dock based supply chain network design under demand uncertainty using new metaheuristic algorithms
  • JIMO Home
  • This Issue
  • Next Article
    A multi-objective decision-making model for supplier selection considering transport discounts and supplier capacity constraints
doi: 10.3934/jimo.2021050

Multi-aircraft cooperative path planning for maneuvering target detection

AVIC LEIHUA Electronic Technology Institute, Wuxi 214063, China

* Corresponding author: Yongkun Wang

Received  July 2020 Revised  January 2021 Early access  March 2021

Multi-aircraft cooperative path planning is a key problem in modern and future air combat scenario. In this paper, this problem is studied in aspect of airborne radar detection to maintain a continuous tracking of a manoeuvring air target. Firstly, the objective function is established in combination with multiple constraints considered, including Doppler blind zone constraint, radar viewing aspect constraint, baseline constraint, and so on. Then, the above optimal control problem is transformed into a nonlinear programming problem with a series of algebraic constraints by hp-adaptive Gauss pseudospectral method (HPAGPM). And it is solved by GPOPS software package based on MATLAB. Simulation results show that the optimized cooperative paths can be got to achieve continuous tracking of maneuvering air target by HPAGPM.

Citation: Yongkun Wang, Fengshou He, Xiaobo Deng. Multi-aircraft cooperative path planning for maneuvering target detection. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021050
References:
[1]

D. Benson, Gauss Pseudospectral Transcription for Optimal Control, Massachusetts Institute of Technology, 2005. Google Scholar

[2]

J. T. Betts and W. P. Huffman, Application of sparse nonlinear programming to trajectory optimization, Journal of Guidance, Control, and Dynamics, 15 (1992), 198-206.  doi: 10.2514/3.20819.  Google Scholar

[3]

K. Bousson, Single gridpoint dynamic programming for trajectory optimization, in 2005 AIAA Atmospheric Flight Mechanics Conference and Exhibit, San Francisco, USA, (2005), 1–8. doi: 10.2514/6.2005-5902.  Google Scholar

[4]

X. T. Chen and J. Z. Wang, Sliding-mode guidance for simultaneous control of impact time and angle, Journal of Guidance, Control, and Dynamics, 42 (2019), 394-401.  doi: 10.2514/1.G003893.  Google Scholar

[5]

X. M. ChengH. F. Li and R. Zhang, Efficient ascent trajectory optimization using convex models based on the Newton-Kantorovich/Pseudospectral approach, Aerospace Science and Technology, 66 (2017), 140-151.  doi: 10.1016/j.ast.2017.02.023.  Google Scholar

[6]

Y. Cherfaoui and M. Moulai, Biobjective optimization over the efficient set of multiobjective integer programming problem, Journal of Industrial and Management Optimization, 17 (2021), 117-131.  doi: 10.3934/jimo.2019102.  Google Scholar

[7]

J. M. C. Clark, P. A. Kountouriotis and R. B. Vinter, A methodology for incorporating the Doppler blind zone in target tracking algorithms, in 2008 11th International Conference on Information Fusion, Cologne, Germany, (2008), 1–8. Google Scholar

[8]

H. B. DanX. X. Wei and Z. M. Dong, Multiple UCAVs cooperative air combat simulation platform based on PSO, ACO, and game theory, IEEE Transactions on Aerospace and Electronic System Magazine, 28 (2013), 12-19.  doi: 10.1109/MAES.2013.6678487.  Google Scholar

[9]

C. L. Darby, W. W. Hager and A. V. Rao, An improved adaptive hp algorithm using pseudospectral methods for optimal control, in 2010 AIAA Guidance, Navigation, and Control Conference, Reston, USA, 2012. doi: 10.2514/6.2010-8272.  Google Scholar

[10]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.  Google Scholar

[11]

M. Gandhi and E. Theodorou, A comparison between trajectory optimization methods: Differential dynamic programming and pseudospectral optimal control, in 2016 AIAA Guidance, Navigation, and Control Conference, San Diego, California, USA, (2016), 1–16. Google Scholar

[12]

C. GoerzenZ. Kong and B. Mettler, A survey of motion planning algorithms from the perspective of autonomous UAV guidance, Journal of Intelligent and Robotic Systems, 57 (2010), 65-100.  doi: 10.1007/978-90-481-8764-5_5.  Google Scholar

[13]

Y. F. Guo, D. Z. Feng and X. Wang, The earth-mars transfer trajectory optimization of solar sail based on hp-adaptive pseudospectral method, Discrete Dynamics in Nature and Society, 2018 (2018), Art. ID 6916848, 14 pp. doi: 10.1155/2018/6916848.  Google Scholar

[14]

R. P. HuangS. J. QuX. G. Yang and Z. M. Liu, Multi-stage distributionally robust optimization with risk aversion, Journal of Industrial and Management Optimization, 17 (2021), 233-259.  doi: 10.3934/jimo.2019109.  Google Scholar

[15]

G. Q. HuangY. P. Lu and Y. Nan, A survey of numerical algorithms for trajectory optimization of flight vehicles, Science China Technological Sciences, 55 (2012), 2538-2560.  doi: 10.1007/s11431-012-4946-y.  Google Scholar

[16]

T. H. KimC. H. LeeI. S. Jeon and M. J. Tahk, Augmented polynomial guidance with impact time and angle constraints, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 2806-2817.   Google Scholar

[17]

S. KangR. Tekin and F. Holzapfel, Generalized impact time and angle control via look-angle shaping, Journal of Guidance, Control, and Dynamics, 42 (2019), 695-702.  doi: 10.2514/1.G003765.  Google Scholar

[18]

A. KhatamiS. Mirghasemi and A. Khosravi, A new PSO-based approach to fire flame detection using K-Medoids clustering, Expert Systems with Applications, 68 (2017), 69-80.  doi: 10.1016/j.eswa.2016.09.021.  Google Scholar

[19]

M. MertensW. Koch and T. Kirubarajan, Exploiting Doppler blind zone information for ground moving target tracking with bistatic airborne radar, IEEE Transactions on Aerospace and Electronic Systems, 50 (2014), 130-148.  doi: 10.1109/TAES.2013.120718.  Google Scholar

[20]

F. W. Moore, Radar cross-section reduction via route planning and intelligent control, IEEE Transactions on Control Systems Technology, 10 (2016), 696-700.  doi: 10.1109/TCST.2002.801879.  Google Scholar

[21]

L. H. Nam, L. Huang, X. J. Li and J. F. Xu, An approach for coverage path planning for UAVs, in 2016 IEEE 14th International Workshop on Advanced Motion Control, Auckland, New Zealand, (2016), 411–416. doi: 10.1109/AMC.2016.7496385.  Google Scholar

[22]

N. Ozalo and O. K. Sahingoz, Optimal UAV path planning in a 3D threat environment by using parallel evolutionary algorithms, in 2013 International Conference on Unmanned Aircraft Systems, Grand Hyatt Atlanta, Atlanta, (2013), 308–317. Google Scholar

[23]

N. OzakiS. CampagnolaR. Funase and C. H. Yam, Stochastic differential dynamic programming with unscented transform for low-thrust trajectory design, Journal of Guidance, Control, and Dynamics, 41 (2018), 377-381.  doi: 10.2514/1.G002367.  Google Scholar

[24]

M. Patterson and A. Rao, Gpops-Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Transactions on Mathematical Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904.  Google Scholar

[25]

Y. H. Qu, Y. T. Zhang and Y. M. Zhang, Optimal flight path planning for UAVs in a 3-D threat environment, in 2014 International Conference on Unmanned Aircraft systems, Orlando, FL, USA, (2014), 149–155. doi: 10.1109/ICUAS.2014.6842250.  Google Scholar

[26]

A. V. Rao, D. A. Benson and C. Darby, Algorithm 902: GPOPS A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method, ACM Transactions on Mathematical Software, 37 (2010), Article 22. doi: 10.1145/1731022.1731032.  Google Scholar

[27]

J. R. RiehlG. E. Collins and J. P. Hespanha, Cooperative search by UAV teams: A model predictive approach using dynamic graphs, IEEE Transactions on Aerospace and Electronic systems, 47 (2011), 2637-2656.  doi: 10.1109/TAES.2011.6034656.  Google Scholar

[28]

V. RobergeM. Tarbouchi and G. Labonte, Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning, IEEE Transactions on Industrial Information, 9 (2013), 132-141.  doi: 10.1109/TII.2012.2198665.  Google Scholar

[29]

B. M. SathyarajL. C. JainA. Finn and S. Drake, Multiple UAVs path planning algorithms: A comparative study, Fuzzy Optimization and Decision Making, 7 (2008), 257-267.  doi: 10.1007/s10700-008-9035-0.  Google Scholar

[30]

P. Y. Volkan, A new vibrational genetic algorithm enhanced with a Voronoi diagram for path planning of autonomous UAV, Aerospace Science and Technology, 16 (2012), 47-55.   Google Scholar

[31]

B. Z. Xu, Y. J. Wang and L. Liu, Multi-stage boost aircraft trajectory optimization strategy based on hp adaptive Gauss pseudo spectral method, in 10th International Conference on Modelling, Identification and Control, Guiyang, China, 2018, 1–7. doi: 10.1109/ICMIC.2018.8529869.  Google Scholar

[32]

P. YaoZ. X. Xie and P. Ren, Optimal UAV route planning for coverage search of stationary target in river, IEEE Transactions on Control Systems Technology, 27 (2019), 822-829.  doi: 10.1109/TCST.2017.2781655.  Google Scholar

[33]

M. Zhang, Z. Zhu, Z. Zhao and X. Li, Trajectory optimization for missile-borne SAR imaging phase via Gauss Pseudospectral Method, in 2011 IEEE CIE International Conference on Radar, Chengdu, China, (2011), 867–870. doi: 10.1109/CIE-Radar.2011.6159678.  Google Scholar

show all references

References:
[1]

D. Benson, Gauss Pseudospectral Transcription for Optimal Control, Massachusetts Institute of Technology, 2005. Google Scholar

[2]

J. T. Betts and W. P. Huffman, Application of sparse nonlinear programming to trajectory optimization, Journal of Guidance, Control, and Dynamics, 15 (1992), 198-206.  doi: 10.2514/3.20819.  Google Scholar

[3]

K. Bousson, Single gridpoint dynamic programming for trajectory optimization, in 2005 AIAA Atmospheric Flight Mechanics Conference and Exhibit, San Francisco, USA, (2005), 1–8. doi: 10.2514/6.2005-5902.  Google Scholar

[4]

X. T. Chen and J. Z. Wang, Sliding-mode guidance for simultaneous control of impact time and angle, Journal of Guidance, Control, and Dynamics, 42 (2019), 394-401.  doi: 10.2514/1.G003893.  Google Scholar

[5]

X. M. ChengH. F. Li and R. Zhang, Efficient ascent trajectory optimization using convex models based on the Newton-Kantorovich/Pseudospectral approach, Aerospace Science and Technology, 66 (2017), 140-151.  doi: 10.1016/j.ast.2017.02.023.  Google Scholar

[6]

Y. Cherfaoui and M. Moulai, Biobjective optimization over the efficient set of multiobjective integer programming problem, Journal of Industrial and Management Optimization, 17 (2021), 117-131.  doi: 10.3934/jimo.2019102.  Google Scholar

[7]

J. M. C. Clark, P. A. Kountouriotis and R. B. Vinter, A methodology for incorporating the Doppler blind zone in target tracking algorithms, in 2008 11th International Conference on Information Fusion, Cologne, Germany, (2008), 1–8. Google Scholar

[8]

H. B. DanX. X. Wei and Z. M. Dong, Multiple UCAVs cooperative air combat simulation platform based on PSO, ACO, and game theory, IEEE Transactions on Aerospace and Electronic System Magazine, 28 (2013), 12-19.  doi: 10.1109/MAES.2013.6678487.  Google Scholar

[9]

C. L. Darby, W. W. Hager and A. V. Rao, An improved adaptive hp algorithm using pseudospectral methods for optimal control, in 2010 AIAA Guidance, Navigation, and Control Conference, Reston, USA, 2012. doi: 10.2514/6.2010-8272.  Google Scholar

[10]

C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.  Google Scholar

[11]

M. Gandhi and E. Theodorou, A comparison between trajectory optimization methods: Differential dynamic programming and pseudospectral optimal control, in 2016 AIAA Guidance, Navigation, and Control Conference, San Diego, California, USA, (2016), 1–16. Google Scholar

[12]

C. GoerzenZ. Kong and B. Mettler, A survey of motion planning algorithms from the perspective of autonomous UAV guidance, Journal of Intelligent and Robotic Systems, 57 (2010), 65-100.  doi: 10.1007/978-90-481-8764-5_5.  Google Scholar

[13]

Y. F. Guo, D. Z. Feng and X. Wang, The earth-mars transfer trajectory optimization of solar sail based on hp-adaptive pseudospectral method, Discrete Dynamics in Nature and Society, 2018 (2018), Art. ID 6916848, 14 pp. doi: 10.1155/2018/6916848.  Google Scholar

[14]

R. P. HuangS. J. QuX. G. Yang and Z. M. Liu, Multi-stage distributionally robust optimization with risk aversion, Journal of Industrial and Management Optimization, 17 (2021), 233-259.  doi: 10.3934/jimo.2019109.  Google Scholar

[15]

G. Q. HuangY. P. Lu and Y. Nan, A survey of numerical algorithms for trajectory optimization of flight vehicles, Science China Technological Sciences, 55 (2012), 2538-2560.  doi: 10.1007/s11431-012-4946-y.  Google Scholar

[16]

T. H. KimC. H. LeeI. S. Jeon and M. J. Tahk, Augmented polynomial guidance with impact time and angle constraints, IEEE Transactions on Aerospace and Electronic Systems, 49 (2013), 2806-2817.   Google Scholar

[17]

S. KangR. Tekin and F. Holzapfel, Generalized impact time and angle control via look-angle shaping, Journal of Guidance, Control, and Dynamics, 42 (2019), 695-702.  doi: 10.2514/1.G003765.  Google Scholar

[18]

A. KhatamiS. Mirghasemi and A. Khosravi, A new PSO-based approach to fire flame detection using K-Medoids clustering, Expert Systems with Applications, 68 (2017), 69-80.  doi: 10.1016/j.eswa.2016.09.021.  Google Scholar

[19]

M. MertensW. Koch and T. Kirubarajan, Exploiting Doppler blind zone information for ground moving target tracking with bistatic airborne radar, IEEE Transactions on Aerospace and Electronic Systems, 50 (2014), 130-148.  doi: 10.1109/TAES.2013.120718.  Google Scholar

[20]

F. W. Moore, Radar cross-section reduction via route planning and intelligent control, IEEE Transactions on Control Systems Technology, 10 (2016), 696-700.  doi: 10.1109/TCST.2002.801879.  Google Scholar

[21]

L. H. Nam, L. Huang, X. J. Li and J. F. Xu, An approach for coverage path planning for UAVs, in 2016 IEEE 14th International Workshop on Advanced Motion Control, Auckland, New Zealand, (2016), 411–416. doi: 10.1109/AMC.2016.7496385.  Google Scholar

[22]

N. Ozalo and O. K. Sahingoz, Optimal UAV path planning in a 3D threat environment by using parallel evolutionary algorithms, in 2013 International Conference on Unmanned Aircraft Systems, Grand Hyatt Atlanta, Atlanta, (2013), 308–317. Google Scholar

[23]

N. OzakiS. CampagnolaR. Funase and C. H. Yam, Stochastic differential dynamic programming with unscented transform for low-thrust trajectory design, Journal of Guidance, Control, and Dynamics, 41 (2018), 377-381.  doi: 10.2514/1.G002367.  Google Scholar

[24]

M. Patterson and A. Rao, Gpops-Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Transactions on Mathematical Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904.  Google Scholar

[25]

Y. H. Qu, Y. T. Zhang and Y. M. Zhang, Optimal flight path planning for UAVs in a 3-D threat environment, in 2014 International Conference on Unmanned Aircraft systems, Orlando, FL, USA, (2014), 149–155. doi: 10.1109/ICUAS.2014.6842250.  Google Scholar

[26]

A. V. Rao, D. A. Benson and C. Darby, Algorithm 902: GPOPS A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method, ACM Transactions on Mathematical Software, 37 (2010), Article 22. doi: 10.1145/1731022.1731032.  Google Scholar

[27]

J. R. RiehlG. E. Collins and J. P. Hespanha, Cooperative search by UAV teams: A model predictive approach using dynamic graphs, IEEE Transactions on Aerospace and Electronic systems, 47 (2011), 2637-2656.  doi: 10.1109/TAES.2011.6034656.  Google Scholar

[28]

V. RobergeM. Tarbouchi and G. Labonte, Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning, IEEE Transactions on Industrial Information, 9 (2013), 132-141.  doi: 10.1109/TII.2012.2198665.  Google Scholar

[29]

B. M. SathyarajL. C. JainA. Finn and S. Drake, Multiple UAVs path planning algorithms: A comparative study, Fuzzy Optimization and Decision Making, 7 (2008), 257-267.  doi: 10.1007/s10700-008-9035-0.  Google Scholar

[30]

P. Y. Volkan, A new vibrational genetic algorithm enhanced with a Voronoi diagram for path planning of autonomous UAV, Aerospace Science and Technology, 16 (2012), 47-55.   Google Scholar

[31]

B. Z. Xu, Y. J. Wang and L. Liu, Multi-stage boost aircraft trajectory optimization strategy based on hp adaptive Gauss pseudo spectral method, in 10th International Conference on Modelling, Identification and Control, Guiyang, China, 2018, 1–7. doi: 10.1109/ICMIC.2018.8529869.  Google Scholar

[32]

P. YaoZ. X. Xie and P. Ren, Optimal UAV route planning for coverage search of stationary target in river, IEEE Transactions on Control Systems Technology, 27 (2019), 822-829.  doi: 10.1109/TCST.2017.2781655.  Google Scholar

[33]

M. Zhang, Z. Zhu, Z. Zhao and X. Li, Trajectory optimization for missile-borne SAR imaging phase via Gauss Pseudospectral Method, in 2011 IEEE CIE International Conference on Radar, Chengdu, China, (2011), 867–870. doi: 10.1109/CIE-Radar.2011.6159678.  Google Scholar

Figure 1.  An illustration for multi-aircraft air combat
Figure 2.  Target radial velocity results without path planning in two-to-one scenario
Figure 3.  Flight trajcetories in two-to-one scenario
Figure 4.  Target radial velocity in two-to-one scenario
Figure 5.  The airborne radar blind zone in two-to-one scenario with HPAGPM
Figure 6.  The airborne radar blind zone in two-to-one scenario with GPM
Figure 7.  Azimuth angle of the target relative to the aircraft in two-to-one scenario
Figure 8.  The normal accelerations in two-to-one scenario
Figure 9.  Flight trajcetories in four-to-one scenario
Figure 10.  Target radial velocity in four-to-one scenario
Figure 11.  Azimuth angle of the target relative to the aircraft in four-to-one scenario
Figure 12.  The normal accelerations in four-to-one scenario
Table 1.  Compraison results for theree methods
Method Aboved 2v1 scenario 50 scenarios
run time blind zone time solution probability
PSO 195s 58s 80%
GPM 176s 51s 92%
HPAGPM 109s 40s 96%
Method Aboved 2v1 scenario 50 scenarios
run time blind zone time solution probability
PSO 195s 58s 80%
GPM 176s 51s 92%
HPAGPM 109s 40s 96%
[1]

Jinlong Guo, Bin Li, Yuandong Ji. A control parametrization based path planning method for the quad-rotor uavs. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021009

[2]

Matthias Gerdts, René Henrion, Dietmar Hömberg, Chantal Landry. Path planning and collision avoidance for robots. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 437-463. doi: 10.3934/naco.2012.2.437

[3]

Yi Gao, Rui Li, Yingjing Shi, Li Xiao. Design of path planning and tracking control of quadrotor. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021063

[4]

Sheng-I Chen, Yen-Che Tseng. A partitioning column approach for solving LED sorter manipulator path planning problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021055

[5]

Zhong-Qing Wang, Ben-Yu Guo, Yan-Na Wu. Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1019-1038. doi: 10.3934/dcdsb.2009.11.1019

[6]

Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209

[7]

Xin Yang, Nan Wang, Lingling Xu. A parallel Gauss-Seidel method for convex problems with separable structure. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 557-570. doi: 10.3934/naco.2020051

[8]

Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685

[9]

Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021007

[10]

Yuanchang Sun, Lisa M. Wingen, Barbara J. Finlayson-Pitts, Jack Xin. A semi-blind source separation method for differential optical absorption spectroscopy of atmospheric gas mixtures. Inverse Problems & Imaging, 2014, 8 (2) : 587-610. doi: 10.3934/ipi.2014.8.587

[11]

Zheng Chang, Haoxun Chen, Farouk Yalaoui, Bo Dai. Adaptive large neighborhood search Algorithm for route planning of freight buses with pickup and delivery. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1771-1793. doi: 10.3934/jimo.2020045

[12]

Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056

[13]

Hamid Reza Marzban, Hamid Reza Tabrizidooz. Solution of nonlinear delay optimal control problems using a composite pseudospectral collocation method. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1379-1389. doi: 10.3934/cpaa.2010.9.1379

[14]

Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021149

[15]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[16]

Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141

[17]

Derek H. Justice, H. Joel Trussell, Mette S. Olufsen. Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method. Mathematical Biosciences & Engineering, 2006, 3 (2) : 419-440. doi: 10.3934/mbe.2006.3.419

[18]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189

[19]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327

[20]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

2020 Impact Factor: 1.801

Article outline

Figures and Tables

[Back to Top]