June  2022, 12(2): 395-426. doi: 10.3934/naco.2021013

Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

Department of Mechanical Engineering, Buali Sina University, Hamedan, Iran

Received  August 2020 Revised  March 2021 Published  June 2022 Early access  April 2021

Caputo derivative operational matrices of the arbitrary scaled Legendre and Chebyshev wavelets are introduced by deriving directly from these wavelets. The Caputo derivative operational matrices are used in quadratic optimization of systems having fractional or integer orders differential equations. Using these operational matrices, a new quadratic programming wavelet-based method without doing any integration operation for finding solutions of quadratic optimal control of traditional linear/nonlinear fractional time-delay constrained/unconstrained systems is introduced. General strategies for handling different types of the optimal control problems are proposed.

Citation: Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 395-426. doi: 10.3934/naco.2021013
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications: New York, NY, USA, 1972.

[2]

H. T. Banks and J. A. Burns, Hereditary control problem: numerical methods based on averaging approximations, SIAM Journal on Control and Optimization, 16 (1978), 169-208.  doi: 10.1137/0316013.

[3]

M. E. Benattia and B. Kacem, Numerical solution for solving fractional differential equations using shifted Chebyshev wavelet, General Letters in Mathematics, 3 (2017), 101-110. 

[4]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Applied Geophysics, 91 (1971), 134-147.  doi: 10.1007/BF00879562.

[5]

P. Chang and A. Isah, Legendre wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications in solving fractional order Brusselator system, Journal of Physics: Conference Series, 693 (2016), IOP Publishing. doi: 10.1088/1742-6596/693/1/012001.

[6]

S.-B. Chen, S. Soradi-Zeid, H. Jahanshahi, R. Alcaraz, J. F. Gómez-Aguilar, S. Bekiros and Y.-M. Chu, Optimal Control of time-delay fractional equations via a joint application of radial basis functions and collocation method, Entropy, 22 (2020), 1213. doi: 10.3390/e22111213.

[7]

K. B. Datta and B. M. Mohan, Orthogonal functions in systems and control, Advanced Series in Electrical and Computer Engineering, World Scientific Publishing Co., (1995). doi: 10.1142/2476.

[8]

K. Diethelm and A. D. Freed, On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, in Scientific Computing in Chemical Engineering II(eds. F. Keil, W. Mackens, H. Voß, J. Werther), Springer, Heidelberg, (1999), 217–224. doi: 10.1007/978-3-642-60185-9_24.

[9]

L. GaulP. Klein and S. Kempfle, Damping description involving fractional operators, Mechanical Systems and Signal Processing, 5 (1991), 81-88.  doi: 10.1016/0888-3270(91)90016-X.

[10]

W. G. Glöckle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein, Biophysical Journal, 68 (1995), 46-53.  doi: 10.1016/S0006-3495(95)80157-8.

[11]

A. Graps, An introduction to wavelets, IEEE Computational Science and Engineering, 2 (1995), 50-61.  doi: 10.1109/99.388960.

[12]

R. W. Hamming, Numerical Methods for Scientists and Engineers, McGraw Hill Book Company, Inc., ISBN 0-486-65241-6, USA, 1962.

[13]

S. HosseinpourA. Nazemi and E. Tohidi, Müntz-Legendre spectral collocation method for solving delay fractional optimal control problems, Journal of Computational and Applied Mathematics, 351 (2019), 344-363.  doi: 10.1016/j.cam.2018.10.058.

[14]

C. HuaP. X. Liu and X. Guan, Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems, IEEE Transactions on Industrial Electronics, 56 (2009), 3723-3732.  doi: 10.1109/TIE.2009.2025713.

[15]

A. Isah and P. Chang, Chebyshev Wavelet Operational Matrix of Fractional Derivative Through Wavelet-Polynomial Transformation and Its Applications on Fractional Order Differential Equations, Proceedings of the International Conference on Computing, Mathematics and Statistics, Springer, Singapore, 2017. doi: 10.1007/978-981-10-2772-7_22.

[16]

M. Ishteva, Properties and Applications of the Caputo Fractional Operator, Department of Mathematics, University of Karlsruhe, Karlsruhe 5, 2005.

[17]

M. A. Johnson and F.C. Moon, Experimental characterization of quasiperiodicity and chaos in a mechanical system with delay, International Journal of Bifurcation and Chaos, 9 (1999), 49-65.  doi: 10.1142/S0218127499000031.

[18]

D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004.

[19]

K. C. Kiwiel, A dual method for certain positive semidefinite quadratic programming problems, SIAM Journal on Scientific and Statistical Computing, 10 (1989), 175-186.  doi: 10.1137/0910013.

[20]

D. Kraft, A Software Package for Sequential Quadratic Programming, Wiss. Berichtswesen d. DFVLR, 1988.

[21]

L. Li and J.-G. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM Journal on Mathematical Analysis, 50 (2018), 2867-2900.  doi: 10.1137/17M1160318.

[22]

W. LiS. Wang and V. Rehbock, A 2nd-order one-step numerical integration scheme for a fractional differential equation, Numerical Algebra, Control and Optimization, 7 (2017), 273-287.  doi: 10.3934/naco.2017018.

[23]

W. LiS. Wang and V. Rehbock, Numerical solution of fractional optimal control, Journal of Optimization Theory and Applications, 180 (2019), 556-573.  doi: 10.1007/s10957-018-1418-y.

[24]

M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., North-Holland, 1978.

[25]

I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics, Optimization & Information Computing, 5 (2017), 302-324.  doi: 10.19139/soic.v5i4.341.

[26]

I. Malmir, A novel wavelet-based optimal linear quadratic tracker for time-varying systems with multiple delays, preprint, arXiv: 1802.05618.

[27]

I. Malmir, Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models, Applied Mathematical Modelling, 69 (2019), 621-647.  doi: 10.1016/j.apm.2018.12.009.

[28]

I. Malmir, Legendre wavelets with scaling in time-delay systems, Statistics, Optimization & Information Computing, 7 (2019), 235-253.  doi: 10.19139/soic.v7i1.460.

[29]

I. Malmir, A new fractional integration operational matrix of chebyshev wavelets in fractional delay systems, Fractal and Fractional, 3 (2019), 46. doi: 10.3390/fractalfract3030046.

[30]

I. Malmir and S. H. Sadati, Transforming linear time-varying optimal control problems with quadratic criteria into quadratic programming ones via wavelets, Journal of Applied Analysis, 26 (2020), 131-151.  doi: 10.1515/jaa-2020-2011.

[31]

I. Malmir, A general framework for optimal control of fractional nonlinear delay systems by wavelets, Statistics, Optimization & Information Computing, 8 (2020), 858-875.  doi: 10.19139/soic-2310-5070-939.

[32]

H. R. Marzban and H. Pirmoradian, A novel approach for the numerical investigation of optimal control problems containing multiple delays, Optimal Control Applications and Method, (2017), 1–24. doi: 10.1002/oca.2349.

[33]

H. R. Marzban, Solution of a specific class of nonlinear fractional optimal control problems including multiple delays, Optimal Control Applications and Method, (2020), 1–28. doi: 10.1002/oca.2661.

[34]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, 2003.

[35]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Hoboken, NJ, USA, 1993.

[36]

D. S. Naidu, Optimal Control Systems, Idaho State University, Pocatello, Idaho, USA, CRC PRESS, 2003.

[37]

J. Nocedal and S. J. Wright, Sequential Quadratic Programming, Numerical optimization, (2006), 529–562. doi: 10.1007/978-0-387-40065-5_18.

[38]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.

[39]

E. SafaieM. Farahi and M. Farmani-Ardehaie, An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials, Computational and Applied Mathematics, 34 (2015), 831-846.  doi: 10.1007/s40314-014-0142-y.

[40]

A. Secer and S. Altun, A new operational matrix of fractional derivatives to solve systems of fractional differential equations via Legendre wavelets, Mathematics, 6 (2018), 238. doi: 10.3390/math6110238.

[41]

R. Serban and L. R. Petzold, COOPT-a software package for optimal control of large-scale differential–algebraic equation systems, Mathematics and Computers in Simulation, 56 (2001), 187-203.  doi: 10.1016/S0378-4754(01)00289-0.

[42]

Y. Wang and Y. Chen, Shifted Legendre polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model, Applied Mathematical Modelling, 81 (2020), 159-176.  doi: 10.1016/j.apm.2019.12.011.

[43]

Y. Zhou and Z. Wang, Optimal feedback control for linear systems with input delays revisited, Journal of Optimization Theory and Applications, (2014), 989–1017. doi: 10.1007/s10957-014-0532-8.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications: New York, NY, USA, 1972.

[2]

H. T. Banks and J. A. Burns, Hereditary control problem: numerical methods based on averaging approximations, SIAM Journal on Control and Optimization, 16 (1978), 169-208.  doi: 10.1137/0316013.

[3]

M. E. Benattia and B. Kacem, Numerical solution for solving fractional differential equations using shifted Chebyshev wavelet, General Letters in Mathematics, 3 (2017), 101-110. 

[4]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and Applied Geophysics, 91 (1971), 134-147.  doi: 10.1007/BF00879562.

[5]

P. Chang and A. Isah, Legendre wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications in solving fractional order Brusselator system, Journal of Physics: Conference Series, 693 (2016), IOP Publishing. doi: 10.1088/1742-6596/693/1/012001.

[6]

S.-B. Chen, S. Soradi-Zeid, H. Jahanshahi, R. Alcaraz, J. F. Gómez-Aguilar, S. Bekiros and Y.-M. Chu, Optimal Control of time-delay fractional equations via a joint application of radial basis functions and collocation method, Entropy, 22 (2020), 1213. doi: 10.3390/e22111213.

[7]

K. B. Datta and B. M. Mohan, Orthogonal functions in systems and control, Advanced Series in Electrical and Computer Engineering, World Scientific Publishing Co., (1995). doi: 10.1142/2476.

[8]

K. Diethelm and A. D. Freed, On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, in Scientific Computing in Chemical Engineering II(eds. F. Keil, W. Mackens, H. Voß, J. Werther), Springer, Heidelberg, (1999), 217–224. doi: 10.1007/978-3-642-60185-9_24.

[9]

L. GaulP. Klein and S. Kempfle, Damping description involving fractional operators, Mechanical Systems and Signal Processing, 5 (1991), 81-88.  doi: 10.1016/0888-3270(91)90016-X.

[10]

W. G. Glöckle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein, Biophysical Journal, 68 (1995), 46-53.  doi: 10.1016/S0006-3495(95)80157-8.

[11]

A. Graps, An introduction to wavelets, IEEE Computational Science and Engineering, 2 (1995), 50-61.  doi: 10.1109/99.388960.

[12]

R. W. Hamming, Numerical Methods for Scientists and Engineers, McGraw Hill Book Company, Inc., ISBN 0-486-65241-6, USA, 1962.

[13]

S. HosseinpourA. Nazemi and E. Tohidi, Müntz-Legendre spectral collocation method for solving delay fractional optimal control problems, Journal of Computational and Applied Mathematics, 351 (2019), 344-363.  doi: 10.1016/j.cam.2018.10.058.

[14]

C. HuaP. X. Liu and X. Guan, Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems, IEEE Transactions on Industrial Electronics, 56 (2009), 3723-3732.  doi: 10.1109/TIE.2009.2025713.

[15]

A. Isah and P. Chang, Chebyshev Wavelet Operational Matrix of Fractional Derivative Through Wavelet-Polynomial Transformation and Its Applications on Fractional Order Differential Equations, Proceedings of the International Conference on Computing, Mathematics and Statistics, Springer, Singapore, 2017. doi: 10.1007/978-981-10-2772-7_22.

[16]

M. Ishteva, Properties and Applications of the Caputo Fractional Operator, Department of Mathematics, University of Karlsruhe, Karlsruhe 5, 2005.

[17]

M. A. Johnson and F.C. Moon, Experimental characterization of quasiperiodicity and chaos in a mechanical system with delay, International Journal of Bifurcation and Chaos, 9 (1999), 49-65.  doi: 10.1142/S0218127499000031.

[18]

D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004.

[19]

K. C. Kiwiel, A dual method for certain positive semidefinite quadratic programming problems, SIAM Journal on Scientific and Statistical Computing, 10 (1989), 175-186.  doi: 10.1137/0910013.

[20]

D. Kraft, A Software Package for Sequential Quadratic Programming, Wiss. Berichtswesen d. DFVLR, 1988.

[21]

L. Li and J.-G. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM Journal on Mathematical Analysis, 50 (2018), 2867-2900.  doi: 10.1137/17M1160318.

[22]

W. LiS. Wang and V. Rehbock, A 2nd-order one-step numerical integration scheme for a fractional differential equation, Numerical Algebra, Control and Optimization, 7 (2017), 273-287.  doi: 10.3934/naco.2017018.

[23]

W. LiS. Wang and V. Rehbock, Numerical solution of fractional optimal control, Journal of Optimization Theory and Applications, 180 (2019), 556-573.  doi: 10.1007/s10957-018-1418-y.

[24]

M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., North-Holland, 1978.

[25]

I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics, Optimization & Information Computing, 5 (2017), 302-324.  doi: 10.19139/soic.v5i4.341.

[26]

I. Malmir, A novel wavelet-based optimal linear quadratic tracker for time-varying systems with multiple delays, preprint, arXiv: 1802.05618.

[27]

I. Malmir, Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models, Applied Mathematical Modelling, 69 (2019), 621-647.  doi: 10.1016/j.apm.2018.12.009.

[28]

I. Malmir, Legendre wavelets with scaling in time-delay systems, Statistics, Optimization & Information Computing, 7 (2019), 235-253.  doi: 10.19139/soic.v7i1.460.

[29]

I. Malmir, A new fractional integration operational matrix of chebyshev wavelets in fractional delay systems, Fractal and Fractional, 3 (2019), 46. doi: 10.3390/fractalfract3030046.

[30]

I. Malmir and S. H. Sadati, Transforming linear time-varying optimal control problems with quadratic criteria into quadratic programming ones via wavelets, Journal of Applied Analysis, 26 (2020), 131-151.  doi: 10.1515/jaa-2020-2011.

[31]

I. Malmir, A general framework for optimal control of fractional nonlinear delay systems by wavelets, Statistics, Optimization & Information Computing, 8 (2020), 858-875.  doi: 10.19139/soic-2310-5070-939.

[32]

H. R. Marzban and H. Pirmoradian, A novel approach for the numerical investigation of optimal control problems containing multiple delays, Optimal Control Applications and Method, (2017), 1–24. doi: 10.1002/oca.2349.

[33]

H. R. Marzban, Solution of a specific class of nonlinear fractional optimal control problems including multiple delays, Optimal Control Applications and Method, (2020), 1–28. doi: 10.1002/oca.2661.

[34]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, 2003.

[35]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Hoboken, NJ, USA, 1993.

[36]

D. S. Naidu, Optimal Control Systems, Idaho State University, Pocatello, Idaho, USA, CRC PRESS, 2003.

[37]

J. Nocedal and S. J. Wright, Sequential Quadratic Programming, Numerical optimization, (2006), 529–562. doi: 10.1007/978-0-387-40065-5_18.

[38]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.

[39]

E. SafaieM. Farahi and M. Farmani-Ardehaie, An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials, Computational and Applied Mathematics, 34 (2015), 831-846.  doi: 10.1007/s40314-014-0142-y.

[40]

A. Secer and S. Altun, A new operational matrix of fractional derivatives to solve systems of fractional differential equations via Legendre wavelets, Mathematics, 6 (2018), 238. doi: 10.3390/math6110238.

[41]

R. Serban and L. R. Petzold, COOPT-a software package for optimal control of large-scale differential–algebraic equation systems, Mathematics and Computers in Simulation, 56 (2001), 187-203.  doi: 10.1016/S0378-4754(01)00289-0.

[42]

Y. Wang and Y. Chen, Shifted Legendre polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model, Applied Mathematical Modelling, 81 (2020), 159-176.  doi: 10.1016/j.apm.2019.12.011.

[43]

Y. Zhou and Z. Wang, Optimal feedback control for linear systems with input delays revisited, Journal of Optimization Theory and Applications, (2014), 989–1017. doi: 10.1007/s10957-014-0532-8.

Figure 1.  Numerical solutions for Example 1
Figure 2.  $ {x}^*(t) $ and $ u^*(t) $ for Case 2 of Example 4
Figure 3.  $ x^*(t) $ and $ u^*(t) $ for Example 5, $ \alpha = 1 $
Figure 4.  Optimal control for Example 6
Table 1.  $ u^* $ for some $ t $ in Example 1, $ k = 2 $
$ t $ Exact CW, Type III, Method 1, $ \xi=2 $, $ M=7 $ CW, Type III, Method 2, $ \xi=2 $, $ M=7 $ CW, Type I, $ \xi=2 $, $ M=7 $ LW, Type I, $ \xi=4 $, $ M=7 $
0 $ -0.0870988 $ $ -0.0865881 $ $ -0.0870909 $ $ -0.0870909 $ $ -0.0870982 $
0.2 $ -0.0336738 $ $ -0.0335485 $ $ -0.0336745 $ $ -0.0336745 $ $ -0.0336737 $
0.4 $ \phantom{+}0.0218134 $ $ \phantom{+}0.0219177 $ $ \phantom{+}0.0218149 $ $ \phantom{+}0.0218149 $ $ \phantom{+}0.0218134 $
0.6 $ \phantom{+}0.0774030 $ $ \phantom{+}0.0773811 $ $ \phantom{+}0.0774012 $ $ \phantom{+}0.0774012 $ $ \phantom{+}0.0774030 $
0.8 $ \phantom{+}0.1301664 $ $ \phantom{+}0.1298957 $ $ \phantom{+}0.1301676 $ $ \phantom{+}0.1301676 $ $ \phantom{+}0.1301664 $
1 $ \phantom{+}0.1758728 $ $ \phantom{+}0.1757065 $ $ \phantom{+}0.1758696 $ $ \phantom{+}0.1758696 $ $ \phantom{+}0.1758731 $
1.2 $ \phantom{+}0.2205516 $ $ \phantom{+}0.2079875 $ $ \phantom{+}0.2205631 $ $ \phantom{+}0.2205631 $ $ \phantom{+}0.2205505 $
1.4 $ \phantom{+}0.2751223 $ $ \phantom{+}0.2785798 $ $ \phantom{+}0.2751004 $ $ \phantom{+}0.2751004 $ $ \phantom{+}0.2751218 $
1.6 $ \phantom{+}0.3417751 $ $ \phantom{+}0.3464972 $ $ \phantom{+}0.3417959 $ $ \phantom{+}0.3417959 $ $ \phantom{+}0.3417753 $
1.8 $ \phantom{+}0.4231851 $ $ \phantom{+}0.4093533 $ $ \phantom{+}0.4231753 $ $ \phantom{+}0.4231753 $ $ \phantom{+}0.4231855 $
2 $ \phantom{+}0.5226194 $ $ \phantom{+}0.5872923 $ $ \phantom{+}0.5226835 $ $ \phantom{+}0.5226835 $ $ \phantom{+}0.5226206 $
$ t $ Exact CW, Type III, Method 1, $ \xi=2 $, $ M=7 $ CW, Type III, Method 2, $ \xi=2 $, $ M=7 $ CW, Type I, $ \xi=2 $, $ M=7 $ LW, Type I, $ \xi=4 $, $ M=7 $
0 $ -0.0870988 $ $ -0.0865881 $ $ -0.0870909 $ $ -0.0870909 $ $ -0.0870982 $
0.2 $ -0.0336738 $ $ -0.0335485 $ $ -0.0336745 $ $ -0.0336745 $ $ -0.0336737 $
0.4 $ \phantom{+}0.0218134 $ $ \phantom{+}0.0219177 $ $ \phantom{+}0.0218149 $ $ \phantom{+}0.0218149 $ $ \phantom{+}0.0218134 $
0.6 $ \phantom{+}0.0774030 $ $ \phantom{+}0.0773811 $ $ \phantom{+}0.0774012 $ $ \phantom{+}0.0774012 $ $ \phantom{+}0.0774030 $
0.8 $ \phantom{+}0.1301664 $ $ \phantom{+}0.1298957 $ $ \phantom{+}0.1301676 $ $ \phantom{+}0.1301676 $ $ \phantom{+}0.1301664 $
1 $ \phantom{+}0.1758728 $ $ \phantom{+}0.1757065 $ $ \phantom{+}0.1758696 $ $ \phantom{+}0.1758696 $ $ \phantom{+}0.1758731 $
1.2 $ \phantom{+}0.2205516 $ $ \phantom{+}0.2079875 $ $ \phantom{+}0.2205631 $ $ \phantom{+}0.2205631 $ $ \phantom{+}0.2205505 $
1.4 $ \phantom{+}0.2751223 $ $ \phantom{+}0.2785798 $ $ \phantom{+}0.2751004 $ $ \phantom{+}0.2751004 $ $ \phantom{+}0.2751218 $
1.6 $ \phantom{+}0.3417751 $ $ \phantom{+}0.3464972 $ $ \phantom{+}0.3417959 $ $ \phantom{+}0.3417959 $ $ \phantom{+}0.3417753 $
1.8 $ \phantom{+}0.4231851 $ $ \phantom{+}0.4093533 $ $ \phantom{+}0.4231753 $ $ \phantom{+}0.4231753 $ $ \phantom{+}0.4231855 $
2 $ \phantom{+}0.5226194 $ $ \phantom{+}0.5872923 $ $ \phantom{+}0.5226835 $ $ \phantom{+}0.5226835 $ $ \phantom{+}0.5226206 $
Table 2.  $ J^* $ for some $ \alpha $ in (94), Example 1, $ k = 2 $
$ \alpha $ CW, Type I, $ \xi=2 $, $ M=7 $ LW, Type I, $ \xi=2 $, $ M=7 $ LW, Type I, $ \xi=4 $, $ M=8 $
2 0.1974785 0.1974785 0.1974785
1.99 0.1934399 0.1934481 0.1934459
1.98 0.1894355 0.1894379 0.1894288
1.97 0.1854667 0.1854495 0.1854282
1.96 0.1815345 0.1814848 0.1814454
1.95 0.1776403 0.1775456 0.1774818
1.94 0.1737852 0.1736341 0.1735389
1.93 0.1699705 0.1697526 0.1696184
1.92 0.1661976 0.1659034 0.1657223
1.91 0.1624677 0.1620891 0.1618525
1.9 0.1587825 0.1583125 0.1580112
$ \alpha $ CW, Type I, $ \xi=2 $, $ M=7 $ LW, Type I, $ \xi=2 $, $ M=7 $ LW, Type I, $ \xi=4 $, $ M=8 $
2 0.1974785 0.1974785 0.1974785
1.99 0.1934399 0.1934481 0.1934459
1.98 0.1894355 0.1894379 0.1894288
1.97 0.1854667 0.1854495 0.1854282
1.96 0.1815345 0.1814848 0.1814454
1.95 0.1776403 0.1775456 0.1774818
1.94 0.1737852 0.1736341 0.1735389
1.93 0.1699705 0.1697526 0.1696184
1.92 0.1661976 0.1659034 0.1657223
1.91 0.1624677 0.1620891 0.1618525
1.9 0.1587825 0.1583125 0.1580112
Table 3.  Comparison of $ J^* $ for Example 2, $ k = 2 $
$ \mathfrak{a} $ $ \alpha $ CW2, $ \xi=4, M=8 $ CW3, $ \xi=2, M=7 $ [29], $ \xi=2, M=7 $ [27], $ \xi=2, M=7 $
0 1 4.79679791916 4.79679791916 4.79679791913
0 0.999 4.79684648427 4.79697117915
0 0.99 4.79728438115 4.79853220259
0 0.95 4.79931928713 4.80544625813
0 0.9 4.80232045821 4.81377758646
1 0.9 5.26128658218 5.27371052428
1 0.95 5.24909761641 5.25573149155
1 0.99 5.23983310060 5.24118253395
1 0.999 5.23778132890 5.23791614199
1 1 5.23755370619 5.23755370619 5.23755370619
1 1.001 5.53898632105 5.49646031081 5.49646031081
1 1.01 5.53527054351 5.49452438269 5.49452438269
1 1.1 5.49587484403 5.46836520817 5.46836520817
1 1.2 5.44737670256 5.42892272878 5.42892272878
1 1.3 5.39396724356 5.38170076246 5.38170076246
$ \mathfrak{a} $ $ \alpha $ CW2, $ \xi=4, M=8 $ CW3, $ \xi=2, M=7 $ [29], $ \xi=2, M=7 $ [27], $ \xi=2, M=7 $
0 1 4.79679791916 4.79679791916 4.79679791913
0 0.999 4.79684648427 4.79697117915
0 0.99 4.79728438115 4.79853220259
0 0.95 4.79931928713 4.80544625813
0 0.9 4.80232045821 4.81377758646
1 0.9 5.26128658218 5.27371052428
1 0.95 5.24909761641 5.25573149155
1 0.99 5.23983310060 5.24118253395
1 0.999 5.23778132890 5.23791614199
1 1 5.23755370619 5.23755370619 5.23755370619
1 1.001 5.53898632105 5.49646031081 5.49646031081
1 1.01 5.53527054351 5.49452438269 5.49452438269
1 1.1 5.49587484403 5.46836520817 5.46836520817
1 1.2 5.44737670256 5.42892272878 5.42892272878
1 1.3 5.39396724356 5.38170076246 5.38170076246
Table 4.  $ J^* $ for Example 3
$ \mathfrak{a} $ $ \alpha_1 $ $ \alpha_2 $ This work, LW This work, CW [29]
0 1 1.56224137366 1.56224137355 1.56224137355
1 1 0.999 1.41013313297 1.41013507255 1.41013747158
1 1 0.99 1.41080207527 1.41094921928 1.41106062244
1 1 0.95 1.41374036510 1.41394156982 1.41378641071
1 1 0.91 1.41655889207 1.41695140257 1.41668866306
1 1 0.9 1.41729512016 1.41766964017 1.41740062973
1 1 0.8 1.42440244581 1.42467128854 1.42442691113
$ \mathfrak{a} $ $ \alpha_1 $ $ \alpha_2 $ This work, LW This work, CW [29]
0 1 1.56224137366 1.56224137355 1.56224137355
1 1 0.999 1.41013313297 1.41013507255 1.41013747158
1 1 0.99 1.41080207527 1.41094921928 1.41106062244
1 1 0.95 1.41374036510 1.41394156982 1.41378641071
1 1 0.91 1.41655889207 1.41695140257 1.41668866306
1 1 0.9 1.41729512016 1.41766964017 1.41740062973
1 1 0.8 1.42440244581 1.42467128854 1.42442691113
Table 5.  $ J^* $ for Case 1 of Example 4
$ \alpha $ This work, LW This work, CW [31]
1 2.54807 2.54807 2.54807
0.999 2.54887 2.54884 2.54887
0.99 2.55613 2.55586 2.55616
0.95 2.59051 2.58935 2.59081
0.91 2.62858 2.62681 2.62918
0.9 2.63869 2.63682 2.63937
0.8 2.75391 2.75189 2.75520
$ \alpha $ This work, LW This work, CW [31]
1 2.54807 2.54807 2.54807
0.999 2.54887 2.54884 2.54887
0.99 2.55613 2.55586 2.55616
0.95 2.59051 2.58935 2.59081
0.91 2.62858 2.62681 2.62918
0.9 2.63869 2.63682 2.63937
0.8 2.75391 2.75189 2.75520
Table 6.  $ J^* $ for Example 5; for both wavelets, we set $ k = 2, \xi = 3 $, $ M = 8 $
This work , CW This work , LW [32]
$ \alpha $ $ J_{QP}^* $ $ J^* $ $ J_{QP}^* $ $ J^* $ $ J^* $
1 0.592319871 0.758986537 0.592319871 0.758986537 0.833609761
0.999 0.592609998 0.759276665 0.592610281 0.759276948
0.99 0.595224540 0.761891207 0.595227150 0.761893817
0.98 0.598136364 0.764803031 0.598141037 0.764807703
0.97 0.601054736 0.767721403 0.601060817 0.767727484
0.96 0.603979032 0.770645699 0.603985765 0.770652432
0.95 0.606908605 0.773575272 0.606915143 0.773581809
0.94 0.609842787 0.776509454 0.609848197 0.776514863
0.93 0.612780884 0.779447551 0.612784163 0.779450830
0.92 0.615722181 0.782388847 0.615722265 0.782388932
0.91 0.618665935 0.785332602 0.618661716 0.785328382
0.9 0.621611381 0.788278047 0.621601717 0.788268384
0.8 0.650973598 0.817640265 0.650850308 0.817516974
0.7 0.679517928 0.846184595 0.679227593 0.845894260
0.6 0.706304327 0.872970994 0.705895247 0.872561914
This work , CW This work , LW [32]
$ \alpha $ $ J_{QP}^* $ $ J^* $ $ J_{QP}^* $ $ J^* $ $ J^* $
1 0.592319871 0.758986537 0.592319871 0.758986537 0.833609761
0.999 0.592609998 0.759276665 0.592610281 0.759276948
0.99 0.595224540 0.761891207 0.595227150 0.761893817
0.98 0.598136364 0.764803031 0.598141037 0.764807703
0.97 0.601054736 0.767721403 0.601060817 0.767727484
0.96 0.603979032 0.770645699 0.603985765 0.770652432
0.95 0.606908605 0.773575272 0.606915143 0.773581809
0.94 0.609842787 0.776509454 0.609848197 0.776514863
0.93 0.612780884 0.779447551 0.612784163 0.779450830
0.92 0.615722181 0.782388847 0.615722265 0.782388932
0.91 0.618665935 0.785332602 0.618661716 0.785328382
0.9 0.621611381 0.788278047 0.621601717 0.788268384
0.8 0.650973598 0.817640265 0.650850308 0.817516974
0.7 0.679517928 0.846184595 0.679227593 0.845894260
0.6 0.706304327 0.872970994 0.705895247 0.872561914
Table 7.  Comparison of results for Example 6
Method $ i_{max} $ $ J^{*} $ $ \mathbf{K}_{x} $ $ \mathbf{K}_{x_h} $
[31] 30 46.190119 $ [1.114775 \;\; 11.230139] $ $ [8.331317 \;\; 1.595857] $
This work, CW 29 46.190067 $ [1.144505 \;\; 11.220603] $ $ [8.297858 \;\; 1.589278] $
This work, LW 29 46.190014 $ [1.144826 \;\; 11.220515] $ $ [8.297552 \;\; 1.589329] $
Method $ i_{max} $ $ J^{*} $ $ \mathbf{K}_{x} $ $ \mathbf{K}_{x_h} $
[31] 30 46.190119 $ [1.114775 \;\; 11.230139] $ $ [8.331317 \;\; 1.595857] $
This work, CW 29 46.190067 $ [1.144505 \;\; 11.220603] $ $ [8.297858 \;\; 1.589278] $
This work, LW 29 46.190014 $ [1.144826 \;\; 11.220515] $ $ [8.297552 \;\; 1.589329] $
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