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doi: 10.3934/dcdss.2020023

New aspects of time fractional optimal control problems within operators with nonsingular kernel

1. 

Department of Logistics Management, University of Turkish Aeronautical Association, 06790 Ankara, Turkey

2. 

Department of Electrical Engineering, University of Bojnord, Bojnord, Iran

3. 

Hurma Mah., 252. Sokak, 2/5, Konyaaltı, Antalya, Turkey

4. 

Department of Mathematics, Çankaya University, 06530, Ankara, Turkey

5. 

Institute of Soft Matter Mechanics, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu 210098, China

6. 

Institute of Space Sciences, Magurele-Bucharest 077125, Romania

* Corresponding author: Tuğba Akman Yıldız

PhD graduate from Department of Mathematics, Middle East Technical University, Ankara, Turkey

Received  June 2018 Revised  September 2018 Published  March 2019

This paper deals with a new formulation of time fractional optimal control problems governed by Caputo-Fabrizio (CF) fractional derivative. The optimality system for this problem is derived, which contains the forward and backward fractional differential equations in the sense of CF. These equations are then expressed in terms of Volterra integrals and also solved by a new numerical scheme based on approximating the Volterra integrals. The linear rate of convergence for this method is also justified theoretically. We present three illustrative examples to show the performance of this method. These examples also test the contribution of using CF derivative for dynamical constraints and we observe the efficiency of this new approach compared to the classical version of fractional operators.

Citation: Tuğba Akman Yıldız, Amin Jajarmi, Burak Yıldız, Dumitru Baleanu. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020023
References:
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[2]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, Journal of Inequalities and Applications, 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5. Google Scholar

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T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, Journal of Computational and Applied Mathematics, 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021. Google Scholar

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T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Advances in Difference Equations, 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5. Google Scholar

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T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, Journal of Nonlinear Sciences and Applications, 10 (2017), 1098-1107. doi: 10.22436/jnsa.010.03.20. Google Scholar

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T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos, Solitons & Fractals, 102 (2017), 106-110. doi: 10.1016/j.chaos.2017.04.006. Google Scholar

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T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Advances in Difference Equations, 2017 (2017), Paper No. 78, 9 pp. doi: 10.1186/s13662-017-1126-1. Google Scholar

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T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Reports on Mathematical Physics, 80 (2017), 11-27. doi: 10.1016/S0034-4877(17)30059-9. Google Scholar

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T. Abdeljawad and F. Madjidi, Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2, The European Physical Journal Special Topics, 226 (2017), 3355-3368. Google Scholar

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G. M. CocliteM. Garavello and L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete & Continuous Dynamical Systems-S, 11 (2018), 865-900. doi: 10.3934/dcdss.2018053. Google Scholar

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E. F. Doungmo Goufo and S. Mugisha, On analysis of fractional Navier-Stokes equations via nonsingular solutions and approximation, Mathematical Problems in Engineering, 2015 (2015), Art. ID 212760, 8 pp. doi: 10.1155/2015/212760. Google Scholar

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N. Ejlali and S. M. Hosseini, A pseudospectral method for fractional optimal control problems, Journal of Optimization Theory and Applications, 174 (2017), 83-107. doi: 10.1007/s10957-016-0936-8. Google Scholar

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R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000. doi: 10.1142/9789812817747. Google Scholar

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J. Hristov, Derivation of fractional Dodson's equation and beyond: Transient mass diffusion with a non-singular memory and exponentially fading–out diffusivity, Progress in Fractional Differentiation and Applications, 3 (2017), 255-270. Google Scholar

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J. Hristov, Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo–Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 757-762. Google Scholar

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show all references

References:
[1]

T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Advances in Difference Equations, 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0. Google Scholar

[2]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, Journal of Inequalities and Applications, 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5. Google Scholar

[3]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, Journal of Computational and Applied Mathematics, 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021. Google Scholar

[4]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Advances in Difference Equations, 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5. Google Scholar

[5]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, Journal of Nonlinear Sciences and Applications, 10 (2017), 1098-1107. doi: 10.22436/jnsa.010.03.20. Google Scholar

[6]

T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos, Solitons & Fractals, 102 (2017), 106-110. doi: 10.1016/j.chaos.2017.04.006. Google Scholar

[7]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Advances in Difference Equations, 2017 (2017), Paper No. 78, 9 pp. doi: 10.1186/s13662-017-1126-1. Google Scholar

[8]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Reports on Mathematical Physics, 80 (2017), 11-27. doi: 10.1016/S0034-4877(17)30059-9. Google Scholar

[9]

T. Abdeljawad and F. Madjidi, Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2, The European Physical Journal Special Topics, 226 (2017), 3355-3368. Google Scholar

[10]

O. Agrawal, General formulation for the numerical solution of optimal control problems, International Journal of Control, 50 (1989), 627-638. doi: 10.1080/00207178908953385. Google Scholar

[11]

M. Al-Refai and T. Abdeljawad, Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel, Advances in Difference Equations, 2017 (2017), Paper No. 315, 12 pp. doi: 10.1186/s13662-017-1356-2. Google Scholar

[12]

B. S. AlkahtaniO. J. AlgahtaniR. S. Dubey and P. Goswam, The solution of modified fractional Bergman's minimal blood glucose-insulin model, Entropy, 19 (2017), 114. doi: 10.3390/e19050114. Google Scholar

[13]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar

[14]

R. L. Bagley and P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27 (1983), 201-210. doi: 10.1122/1.549724. Google Scholar

[15]

D. BaleanuA. Jajarmi and M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel, Journal of Optimization Theory and Applications, 175 (2017), 718-737. doi: 10.1007/s10957-017-1186-0. Google Scholar

[16]

R. K. Biswas and S. Sen, Fractional optimal control problems with specified final time, Journal of Computational and Nonlinear Dynamics, 6 (2011), 021009. doi: 10.1115/1.4002508. Google Scholar

[17]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 1-13. Google Scholar

[18]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11. doi: 10.18576/pfda/020101. Google Scholar

[19]

S. ChoiE. Jung and S.-M. Lee, Optimal intervention strategy for prevention tuberculosis using a smoking–tuberculosis model, Journal of Theoretical Biology, 380 (2015), 256-270. doi: 10.1016/j.jtbi.2015.05.022. Google Scholar

[20]

G. M. CocliteM. Garavello and L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete & Continuous Dynamical Systems-S, 11 (2018), 865-900. doi: 10.3934/dcdss.2018053. Google Scholar

[21]

E. F. Doungmo Goufo and S. Mugisha, On analysis of fractional Navier-Stokes equations via nonsingular solutions and approximation, Mathematical Problems in Engineering, 2015 (2015), Art. ID 212760, 8 pp. doi: 10.1155/2015/212760. Google Scholar

[22]

N. Ejlali and S. M. Hosseini, A pseudospectral method for fractional optimal control problems, Journal of Optimization Theory and Applications, 174 (2017), 83-107. doi: 10.1007/s10957-016-0936-8. Google Scholar

[23]

M. Enelund and P. Olsson, Damping described by fading memory–analysis and application to fractional derivative models, International Journal of Solids and Structures, 36 (1999), 939-970. doi: 10.1016/S0020-7683(97)00339-9. Google Scholar

[24]

J. Fujioka, A. Espinosa, R. F. Rodríguez and B. A. Malomed, Radiating subdispersive fractional optical solitons, Chaos, 24 (2014), 033121, 11pp. doi: 10.1063/1.4892616. Google Scholar

[25]

D.-p. Gao and N.-j. Huang, Optimal control analysis of a tuberculosis model, Applied Mathematical Modelling, 58 (2018), 47-64. doi: 10.1016/j.apm.2017.12.027. Google Scholar

[26]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000. doi: 10.1142/9789812817747. Google Scholar

[27]

J. Hristov, Derivation of fractional Dodson's equation and beyond: Transient mass diffusion with a non-singular memory and exponentially fading–out diffusivity, Progress in Fractional Differentiation and Applications, 3 (2017), 255-270. Google Scholar

[28]

J. Hristov, Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey's kernel to the Caputo–Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 757-762. Google Scholar

[29]

J. Hristov, Derivatives with non-singular kernels from the caputo–fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus. Sharjah: Bentham Science Publishers, 1 (2018), 269-341. Google Scholar

[30]

C. IonescuK. Desager and R. De Keyser, Fractional order model parameters for the respiratory input impedance in healthy and in asthmatic children, Computer Methods and Programs in Biomedicine, 101 (2011), 315-323. doi: 10.1016/j.cmpb.2010.11.010. Google Scholar

[31]

C. M. Ionescu and R. De Keyser, Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease, IEEE Transactions on Biomedical Engineering, 56 (2009), 978-987. doi: 10.1109/TBME.2008.2004966. Google Scholar

[32]

S. Jahanshahi and D. F. Torres, A simple accurate method for solving fractional variational and optimal control problems, Journal of Optimization Theory and Applications, 174 (2017), 156-175. doi: 10.1007/s10957-016-0884-3. Google Scholar

[33]

F. JaradT. Abdeljawad and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Applied Mathematics and Computation, 218 (2012), 9234-9240. doi: 10.1016/j.amc.2012.02.080. Google Scholar

[34]

Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems, Struct. Multidiscip. Optim., 38 (2009), 571–581, URL http://dx.doi.org/10.1007/s00158-008-0307-7. doi: 10.1007/s00158-008-0307-7. Google Scholar

[35]

T. Kaczorek, Reachability of fractional continuous-time linear systems using the Caputo–Fabrizio derivative, in ECMS, 2016, 53–58. doi: 10.7148/2016-0053. Google Scholar

[36]

T. Kaczorek and K. Borawski, Fractional descriptor continuous–time linear systems described by the Caputo–Fabrizio derivative, International Journal of Applied Mathematics and Computer Science, 26 (2016), 533-541. doi: 10.1515/amcs-2016-0037. Google Scholar

[37]

C. K. KwuimyG. Litak and C. Nataraj, Nonlinear analysis of energy harvesting systems with fractional order physical properties, Nonlinear Dynamics, 80 (2015), 491-501. doi: 10.1007/s11071-014-1883-2. Google Scholar

[38]

C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149-179. doi: 10.1080/01630563.2012.706673. Google Scholar

[39]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus, vol. 24, CRC Press, 2015. Google Scholar

[40]

J. Liouville, Mémoire: Sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, J l'Ecole Polytéch, 13 (1832), 1-66. Google Scholar

[41]

A. LotfiM. Dehghan and S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers & Mathematics with Applications, 62 (2011), 1055-1067. doi: 10.1016/j.camwa.2011.03.044. Google Scholar

[42]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Redding, 2006.Google Scholar

[43]

J. P. MateusP. RebeloS. RosaC. M. Silva and D. F. Torres, Optimal control of non-autonomous SEIRS models with vaccination and treatment, Discrete & Continuous Dynamical Systems-S, 11 (2018), 1179-1199. doi: 10.3934/dcdss.2018067. Google Scholar

[44]

V. Morales-DelgadoJ. Gómez-Aguilar and M. Taneco-Hernandez, Analytical solutions for the motion of a charged particle in electric and magnetic fields via non-singular fractional derivatives, The European Physical Journal Plus, 132 (2017), 527. doi: 10.1140/epjp/i2017-11798-7. Google Scholar

[45]

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Figure 1.  Example 1: Comparative results of $u(t)$ and $u^*(t)$ for $M = 800$ and $\alpha = \{0.6, 0.7, 0.8, 0.9\}$
Figure 2.  Example 1: Comparative results of $x(t)$ and $x^*(t)$ for $M = 800$ and $\alpha = \{0.6, 0.7, 0.8, 0.9\}$
Figure 3.  Example 1: The absolute error plots for $x(t)$ (left) and $u(t)$ (right) with $M = 800$ and $\alpha = \{0.6, 0.7, 0.8, 0.9\}$
Figure 4.  Example 2: Numerical results of $x(t)$ (left) and $u(t)$ (right)
Figure 5.  Example 2: Numerical results of $x(t)$ for Caputo and CF derivatives
Figure 6.  Example 3: Numerical results of $x_1(t)$, $x_2(t)$ and $u(t)$
Figure 7.  Example 3: Numerical results of $x_1(t)$ for Caputo and CF derivatives
Figure 8.  Example 3: Numerical results of $x_2(t)$ for Caputo and CF derivatives
Table 1.  Example 1: The values of J, absolute error, order of convergence and computational time (CT) for α = {0.6, 0.7}
$\alpha=0.6$ $\alpha=0.7$
$M$ $J$ $e_J$ $r_J$ CT $J$ $e_J$ $r_J$ CT
50 4.2795 0.0116 - 0.33 6.1951 0.0309 - 0.29
100 4.2818 0.0058 1.00 0.43 6.2030 0.0154 1.00 0.41
200 4.2838 0.0029 1.00 0.74 6.2088 0.0077 1.00 0.77
400 4.2850 0.0015 0.95 2.97 6.2122 0.0039 0.98 2.88
800 4.2857 0.00073 1.03 18.91 6.2140 0.0019 1.03 19.74
$\alpha=0.6$ $\alpha=0.7$
$M$ $J$ $e_J$ $r_J$ CT $J$ $e_J$ $r_J$ CT
50 4.2795 0.0116 - 0.33 6.1951 0.0309 - 0.29
100 4.2818 0.0058 1.00 0.43 6.2030 0.0154 1.00 0.41
200 4.2838 0.0029 1.00 0.74 6.2088 0.0077 1.00 0.77
400 4.2850 0.0015 0.95 2.97 6.2122 0.0039 0.98 2.88
800 4.2857 0.00073 1.03 18.91 6.2140 0.0019 1.03 19.74
Table 2.  Example 1: The values of $J$, absolute error, order of convergence and computational time (CT) for $\alpha = \{0.8, 0.9\}$.
$\alpha=0.8$ $\alpha=0.9$
$M$ $J$ $e_J$ $r_J$ CT $J$ $e_J$ $r_J$ CT
50 9.9900 0.0863 - 0.31 20.2108 0.2844 - 0.27
100 10.0091 0.0432 0.99 0.38 20.1666 0.1425 0.99 0.40
200 10.0247 0.0216 1.00 0.76 20.1912 0.0713 0.99 0.74
400 10.0339 0.0108 1.00 2.43 20.2152 0.0356 1.00 2.90
800 10.0390 0.0054 1.00 18.85 20.2301 0.0178 1.00 19.18
$\alpha=0.8$ $\alpha=0.9$
$M$ $J$ $e_J$ $r_J$ CT $J$ $e_J$ $r_J$ CT
50 9.9900 0.0863 - 0.31 20.2108 0.2844 - 0.27
100 10.0091 0.0432 0.99 0.38 20.1666 0.1425 0.99 0.40
200 10.0247 0.0216 1.00 0.76 20.1912 0.0713 0.99 0.74
400 10.0339 0.0108 1.00 2.43 20.2152 0.0356 1.00 2.90
800 10.0390 0.0054 1.00 18.85 20.2301 0.0178 1.00 19.18
Table 3.  Example 1: The values of absolute error for $x(t)$ and the order of convergence for $\alpha = \{0.6, 0.7, 0.8, 0.9\}$
$\alpha=0.6$ $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$
$M$ $e_x$ $r_x$ $e_x$ $r_x$ $e_x$ $r_x$ $e_x$ $r_x$
50 0.0090 - 0.0085 - 0.0146 - 0.0531 -
100 0.0045 1.00 0.0043 0.98 0.0072 1.01 0.0260 1.03
200 0.0022 1.03 0.0021 1.03 0.0036 1.00 0.0129 1.01
400 0.0011 1.00 0.0011 0.93 0.0018 1.00 0.0064 1.01
800 5.57e-04 0.98 5.34e-04 1.04 8.91e-04 1.01 0.0032 1.00
$\alpha=0.6$ $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$
$M$ $e_x$ $r_x$ $e_x$ $r_x$ $e_x$ $r_x$ $e_x$ $r_x$
50 0.0090 - 0.0085 - 0.0146 - 0.0531 -
100 0.0045 1.00 0.0043 0.98 0.0072 1.01 0.0260 1.03
200 0.0022 1.03 0.0021 1.03 0.0036 1.00 0.0129 1.01
400 0.0011 1.00 0.0011 0.93 0.0018 1.00 0.0064 1.01
800 5.57e-04 0.98 5.34e-04 1.04 8.91e-04 1.01 0.0032 1.00
Table 4.  Example 1: The values of absolute error for $u(t)$ and the order of convergence for $\alpha = \{0.6, 0.7, 0.8, 0.9\}$
$\alpha=0.6$ $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$
$M$ $e_u$ $r_u$ $e_u$ $r_u$ $e_u$ $r_u$ $e_u$ $r_u$
50 0.0240 - 0.0417 - 0.0770 - 0.1914 -
100 0.0120 1.00 0.0207 1.01 0.0381 1.01 0.0939 1.02
200 0.0060 1.00 0.0103 1.00 0.0190 1.00 0.0465 1.01
400 0.0030 1.00 0.0051 1.01 0.0095 1.00 0.0232 1.00
800 0.0015 1.00 0.0026 0.97 0.0047 1.01 0.0116 1.00
$\alpha=0.6$ $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$
$M$ $e_u$ $r_u$ $e_u$ $r_u$ $e_u$ $r_u$ $e_u$ $r_u$
50 0.0240 - 0.0417 - 0.0770 - 0.1914 -
100 0.0120 1.00 0.0207 1.01 0.0381 1.01 0.0939 1.02
200 0.0060 1.00 0.0103 1.00 0.0190 1.00 0.0465 1.01
400 0.0030 1.00 0.0051 1.01 0.0095 1.00 0.0232 1.00
800 0.0015 1.00 0.0026 0.97 0.0047 1.01 0.0116 1.00
Table 5.  Example 2: The values of $J$ and computational time (CT)
$\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$
$M$ $J$ CT $J$ CT $J$ CT $J$ CT
50 0.2048 0.08 0.1912 0.08 0.1817 0.08 0.1754 0.08
100 0.2008 0.15 0.1909 0.15 0.1856 0.14 0.1842 0.14
200 0.2019 0.41 0.1945 0.40 0.1920 0.39 0.1940 0.41
400 0.2031 1.70 0.1971 1.67 0.1963 1.71 0.2002 1.61
800 0.2039 17.17 0.1986 17.55 0.1986 16.96 0.2035 17.18
$\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$
$M$ $J$ CT $J$ CT $J$ CT $J$ CT
50 0.2048 0.08 0.1912 0.08 0.1817 0.08 0.1754 0.08
100 0.2008 0.15 0.1909 0.15 0.1856 0.14 0.1842 0.14
200 0.2019 0.41 0.1945 0.40 0.1920 0.39 0.1940 0.41
400 0.2031 1.70 0.1971 1.67 0.1963 1.71 0.2002 1.61
800 0.2039 17.17 0.1986 17.55 0.1986 16.96 0.2035 17.18
Table 6.  Example 2: The comparative values of $J$ with $M = 800$
$J$
FD $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$
Caputo 0.2301 0.2073 0.2002 0.2035
CF 0.2039 0.1986 0.1986 0.2035
$J$
FD $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$
Caputo 0.2301 0.2073 0.2002 0.2035
CF 0.2039 0.1986 0.1986 0.2035
Table 7.  Example 3: The values of $J$, rate of convergence and computational time (CT) for $\alpha = \{0.7, 0.8\}$
$\alpha=0.7$ $\alpha=0.8$
$N$ $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT
50 12.1411 - - 0.63 13.8799 - - 0.64
100 7.9860 4.1551 - 1.32 8.6678 5.2121 - 1.25
200 6.6614 1.3246 1.65 4.31 6.9545 1.7133 1.61 4.33
400 6.1602 0.5012 1.40 17.18 6.3194 0.6351 1.43 16.97
800 5.9455 0.2147 1.22 146.48 6.0524 0.2670 1.25 144.69
$\alpha=0.7$ $\alpha=0.8$
$N$ $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT
50 12.1411 - - 0.63 13.8799 - - 0.64
100 7.9860 4.1551 - 1.32 8.6678 5.2121 - 1.25
200 6.6614 1.3246 1.65 4.31 6.9545 1.7133 1.61 4.33
400 6.1602 0.5012 1.40 17.18 6.3194 0.6351 1.43 16.97
800 5.9455 0.2147 1.22 146.48 6.0524 0.2670 1.25 144.69
Table 8.  Example 3: The values of $J$, rate of convergence and computational time (CT) for $\alpha = \{0.9, 1\}$
$\alpha=0.9$ $\alpha=1$
$N$ $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT
50 15.6616 - - 0.73 18.3533 - - 0.69
100 9.9329 5.7287 - 1.39 12.3690 5.9843 - 1.73
200 7.7685 2.1644 1.40 4.30 9.7979 2.5711 1.22 3.81
400 6.9493 0.8192 1.40 18.26 8.7483 1.0496 1.29 18.53
800 6.6071 0.3422 1.26 148.13 8.3018 0.4465 1.23 167.40
$\alpha=0.9$ $\alpha=1$
$N$ $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT $J_{N}$ $J_N - J_{N/2}$ $\rho$ CT
50 15.6616 - - 0.73 18.3533 - - 0.69
100 9.9329 5.7287 - 1.39 12.3690 5.9843 - 1.73
200 7.7685 2.1644 1.40 4.30 9.7979 2.5711 1.22 3.81
400 6.9493 0.8192 1.40 18.26 8.7483 1.0496 1.29 18.53
800 6.6071 0.3422 1.26 148.13 8.3018 0.4465 1.23 167.40
Table 9.  Example 3: The comparative values of $J$ with $M = 800$
$J$
FD $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$
Caputo 6.6976 6.6329 7.1959 8.3018
CF 5.9455 6.0524 6.6071 8.3018
$J$
FD $\alpha=0.7$ $\alpha=0.8$ $\alpha=0.9$ $\alpha=1$
Caputo 6.6976 6.6329 7.1959 8.3018
CF 5.9455 6.0524 6.6071 8.3018
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