Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis

Samia Bushnaq; Kamal Shah; Sana Tahir; Khursheed J. Ansari; Muhammad Sarwar; Thabet Abdeljawad; Samia Bushnaq; Kamal Shah; Sana Tahir; Khursheed J. Ansari; Muhammad Sarwar; Thabet Abdeljawad

doi:10.3934/math.2022610

AIMS Mathematics

2022, Volume 7, Issue 6: 10917-10938. doi: 10.3934/math.2022610

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Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis

1.
Department of Basic Sciences, Princess Sumaya University for Technology, King Abdullah II Faculty of Engineering, Amman 11941, Jordan
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, P.O.Box.11586, Saudi Arabia
3.
Department of Mathematics, University of Malakand, Chakadara Dir (Lower), Lower Dir 18800, Khyber Pakhtunkhwa, Pakistan
4.
Department of Mathematics, College of Science, King Khalid University, 61413 Abha, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 22 February 2022 Revised: 08 March 2022 Accepted: 24 March 2022 Published: 02 April 2022
MSC : 26A33, 34A08

In this work, we present some numerical results about variable order fractional differential equations (VOFDEs). For the said numerical analysis, we use Bernstein polynomials (BPs) with non-orthogonal basis. The method we use does not need discretization and neither collocation. Hence omitting the said two operations sufficient memory and time can be saved. We establish operational matrices for variable order integration and differentiation which convert the consider problem to some algebraic type matrix equations. The obtained matrix equations are then solved by Matlab 13 to get the required numerical solution for the considered problem. Pertinent examples are provided along with graphical illustration and error analysis to validate the results. Further some theoretical results for time complexity are also discussed.
- non-orthogonal basis,
- VOFDEs,
- operational matrices,
- algebraic matrix equation
Citation: Samia Bushnaq, Kamal Shah, Sana Tahir, Khursheed J. Ansari, Muhammad Sarwar, Thabet Abdeljawad. Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis[J]. AIMS Mathematics, 2022, 7(6): 10917-10938. doi: 10.3934/math.2022610

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Abstract

In this work, we present some numerical results about variable order fractional differential equations (VOFDEs). For the said numerical analysis, we use Bernstein polynomials (BPs) with non-orthogonal basis. The method we use does not need discretization and neither collocation. Hence omitting the said two operations sufficient memory and time can be saved. We establish operational matrices for variable order integration and differentiation which convert the consider problem to some algebraic type matrix equations. The obtained matrix equations are then solved by Matlab 13 to get the required numerical solution for the considered problem. Pertinent examples are provided along with graphical illustration and error analysis to validate the results. Further some theoretical results for time complexity are also discussed.

References

[1]	J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculas, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amester Dam: Elesvier, 2006.
[3]	R. Hilfer, Threefold introduction to fractional derivatives, In: Anomalous transport: foundations and applications, Berlin, Germany, 2008, 17–73. https://doi.org/10.1002/9783527622979.ch2
[4]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: John Wiley and Sons, 1993.
[5]	K. S. Miller, Derivatives of noninteger order, Mathematics Magazine, 68 (1995), 183–192. https://doi.org/10.1080/0025570X.1995.11996309 doi: 10.1080/0025570X.1995.11996309
[6]	M. ur Rahman, M. Arfan, K. Shah, J. F. Gómez-Aguilar, Investigating a nonlinear dynamical model of COVID-19 disease under fuzzy caputo, random and ABC fractional order derivative, Chaos Soliton. Fract., 140 (2020), 110232. https://doi.org/10.1016/j.chaos.2020.110232 doi: 10.1016/j.chaos.2020.110232
[7]	D. A. Tvyordyj, Hereditary Riccati equation with fractional derivative of variable order, J. Math. Sci., 253 (2021), 564–572. https://doi.org/10.1007/s10958-021-05254-0 doi: 10.1007/s10958-021-05254-0
[8]	R. Agrawal, M. Belmekki, M. Benchohra, A survey on Semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differ. Equ., 2009 (2009), 981728. https://doi.org/10.1155/2009/981728 doi: 10.1155/2009/981728
[9]	L. Suarez, A. shokooh, An eigenvector expansion method for the solution of motion containing fractional derivatives, J. Appl. Mech., 64 (1997), 629–635. https://doi.org/10.1115/1.2788939 doi: 10.1115/1.2788939
[10]	O. Abdulaziz, I. Hashim, S. Momani, Solving systems of fractional differential equations by homotopy-perturbation method, Phys. Lett. A, 372 (2008), 451–459. https://doi.org/10.1016/j.physleta.2007.07.059 doi: 10.1016/j.physleta.2007.07.059
[11]	Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 15–27. https://doi.org/10.1515/IJNSNS.2006.7.1.27 doi: 10.1515/IJNSNS.2006.7.1.27
[12]	V. S. Erturk, S. Momani, Solving systems of fractional differential equations using differential transform method, J. Comput. Appl. Math., 215 (2008), 142–151. https://doi.org/10.1016/j.cam.2007.03.029 doi: 10.1016/j.cam.2007.03.029
[13]	V. Daftardar-Gejji, H. Jafari, Solving a multi-order fractional differentail equation using Adomian decomposition, Appl. Math. Comput., 189 (2007), 541–548. https://doi.org/10.1016/j.amc.2006.11.129 doi: 10.1016/j.amc.2006.11.129
[14]	E. Sousa, C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative, Appl. Numer. Math., 90 (2015), 22–37. https://doi.org/10.1016/j.apnum.2014.11.007 doi: 10.1016/j.apnum.2014.11.007
[15]	E. Ziada, Numerical solution for multi-term fractional delay differential equations, Journal of Fractional Calculus and Nonlinear Systems, 2 (2021), 1–12.
[16]	A. Saadatmandi, M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1135–1142. https://doi.org/10.1016/j.camwa.2011.04.014 doi: 10.1016/j.camwa.2011.04.014
[17]	M. Rahman, M. Arfan, Z. Shah, P. Kumam, M. Shutaywi, Nonlinear fractional mathematical model of tuberculosis (TB) disease with incomplete treatment under Atangana-Baleanu derivative, Alex. Eng. J., 60 (2021), 2845–2856. https://doi.org/10.1016/j.aej.2021.01.015 doi: 10.1016/j.aej.2021.01.015
[18]	K. Yadav, J. P. Jaiswal, On the operational matrix for fractional integration and its application for solving Abel integral equation using Bernoulli wavelets, Global Journal of Pure and Applied Mathematics, 15 (2019), 81–101.
[19]	F. Mirzaee, N. Samadyar, S. Alipour, Numerical solution of high order linear complex differential equations via complex operational matrix method, SeMA, 76 (2019), 1–13. https://doi.org/10.1007/s40324-018-0151-7 doi: 10.1007/s40324-018-0151-7
[20]	F. Mirzaee, N. Samadyar, Numerical solution of two dimensional stochastic Volterra-Fredholm integral equations via operational matrix method based on hat functions, SeMA, 77 (2020), 227–241. https://doi.org/10.1007/s40324-020-00213-2 doi: 10.1007/s40324-020-00213-2
[21]	F. Mirzaee, S. Alipour, A hybrid approach of nonlinear partial mixed integro-differential equations of fractional order, Iran. J. Sci. Technol. Trans. Sci., 44 (2020), 725–737. https://doi.org/10.1007/s40995-020-00859-7 doi: 10.1007/s40995-020-00859-7
[22]	F. Mirzaee, N. Samadyar, Explicit representation of orthonormal Bernoulli polynomials and its application for solving Volterra–Fredholm–Hammerstein integral equations, SeMA, 77 (2020), 81–96. https://doi.org/10.1007/s40324-019-00203-z doi: 10.1007/s40324-019-00203-z
[23]	F. Mirzaee, N. Samadyar, Numerical solution of time fractional stochastic Korteweg–de Vries equation via implicit meshless approach, Iran. J. Sci. Technol. Trans. Sci., 43 (2019), 2905–2912. https://doi.org/10.1007/s40995-019-00763-9 doi: 10.1007/s40995-019-00763-9
[24]	F. Mirzaee, S. Alipour, Solving two-dimensional non-linear quadratic integral equations of fractional order via operational matrix method, Multidiscipline Modeling in Materials and Structures, 15 (2019), 1136–1151. https://doi.org/10.1108/MMMS-10-2018-0168 doi: 10.1108/MMMS-10-2018-0168
[25]	N. Samadyar, F. Mirzaee, Numerical scheme for solving singular fractional partial integro-differential equation via orthonormal Bernoulli polynomials, Int. J. Numer. Model. El., 32 (2019), e2652. https://doi.org/10.1002/jnm.2652 doi: 10.1002/jnm.2652
[26]	F. Mirzaee, N. Samadyar, On the numerical method for solving a system of nonlinear fractional ordinary differential equations arising in HIV infection of CD4 $^{+}$ T Cells, Iran. J. Sci. Technol. Trans. Sci., 43 (2019), 1127–1138. https://doi.org/10.1007/s40995-018-0560-6 doi: 10.1007/s40995-018-0560-6
[27]	F. Mirzaee, S. Alipour, Fractional-order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro-differential equations, Math. Method. Appl. Sci., 42 (2019), 1870–1893. https://doi.org/10.1002/mma.5481 doi: 10.1002/mma.5481
[28]	F. Mirzaee, N. Samadyar, Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order, Appl. Math. Comput., 344 (2019), 191–203. https://doi.org/10.1016/j.amc.2018.10.020 doi: 10.1016/j.amc.2018.10.020
[29]	A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Appl. Math. Model., 38 (2014), 1365–1372. https://doi.org/10.1016/j.apm.2013.08.007 doi: 10.1016/j.apm.2013.08.007
[30]	K. Shah, J. Wang, A numerical scheme based on nondiscretization of data for boundary value problems of fractional order differential equations, RACSAM, 113 (2019), 2277–2294. https://doi.org/10.1007/s13398-018-0616-7 doi: 10.1007/s13398-018-0616-7
[31]	Y. Feng, M. Yagoubi, Robust control of linear descriptor systems, Singapore: Springer, 2017. https://doi.org/10.1007/978-981-10-3677-4
[32]	F. Mirzaee, S. Rezaei, N. Samadyar, Application of combination schemes based on radial basis functions and finite difference to solve stochastic coupled nonlinear time fractional sine-Gordon equations, Comp. Appl. Math., 41 (2022), 10. https://doi.org/10.1007/s40314-021-01725-x doi: 10.1007/s40314-021-01725-x
[33]	S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf. Spec. Funct., 1 (1993), 277–300. https://doi.org/10.1080/10652469308819027 doi: 10.1080/10652469308819027
[34]	C. Han, Y. Chen, D. Y. Liu, D. Boutat, Numerical analysis of viscoelastic rotating beam with variable fractional order model using shifted Bernstein–Legendre Polynomial collocation Algorithm, Fractal Fract., 5 (2021), 8. https://doi.org/10.3390/fractalfract5010008 doi: 10.3390/fractalfract5010008
[35]	J. Jiang, J. L. G. Guirao, T. Saeed, The existence of the extremal solution for the boundary value problems of variable fractional order differential equation with causal operator, Fractals, 28 (2020), 2040025. https://doi.org/10.1142/S0218348X20400253 doi: 10.1142/S0218348X20400253
[36]	Y. Xu, Z. He, Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations, J. Appl. Math. Comput., 43 (2013), 295–306. https://doi.org/10.1007/s12190-013-0664-2 doi: 10.1007/s12190-013-0664-2
[37]	A. Razminiaa, A. F. Dizajib, V. J. Majda, Solution existence for non-autonomous variable-order fractional differential equations, Math. Comput. Model., 55 (2012), 1106–1117. https://doi.org/10.1016/j.mcm.2011.09.034 doi: 10.1016/j.mcm.2011.09.034
[38]	C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys.-Berlin, 12 (2003), 692–703. https://doi.org/10.1002/andp.200310032 doi: 10.1002/andp.200310032
[39]	G. Diaz, C. F. M. Coimbra, Nonlinear dynamics and control of a variable order oscillator with application to the Van der Pol equation, Nonlinear Dyn., 56 (2009), 145–157. https://doi.org/10.1007/s11071-008-9385-8 doi: 10.1007/s11071-008-9385-8
[40]	J. F. Gómez-Aguilar, Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Physica A, 494 (2018), 52–75. https://doi.org/10.1016/j.physa.2017.12.007 doi: 10.1016/j.physa.2017.12.007
[41]	C. J. Zúniga-Aguilar, H. M. Romero-Ugalde, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos Soliton. Fract., 103 (2017), 382–403. https://doi.org/10.1016/j.chaos.2017.06.030 doi: 10.1016/j.chaos.2017.06.030
[42]	A. Dabiri, B. P. Moghaddam, J. T. Machado, Optimal variable-order fractional PID controllers for dynamical systems, J. Comput. Appl. Math., 339 (2018), 40–48. https://doi.org/10.1016/j.cam.2018.02.029 doi: 10.1016/j.cam.2018.02.029
[43]	B. S. T. Alkahtani, S. Jain, Numerical analysis of COVID-19 model with constant fractional order and variable fractal dimension, Results Phys., 20 (2021), 103673. https://doi.org/10.1016/j.rinp.2020.103673 doi: 10.1016/j.rinp.2020.103673
[44]	A. Leblanc, On estimating distribution functions using Bernstein polynomials, Ann. Inst. Stat. Math., 64 (2012), 919–943. https://doi.org/10.1007/s10463-011-0339-4 doi: 10.1007/s10463-011-0339-4
[45]	B.-Y. Guo, Spectral methods and their applications, Singapore: World Scientific, 1998. https://doi.org/10.1142/3662
[46]	J. Shen, T. Tang, Spectral and high-order methods with applications, Beijing: Science Press, 2006.
[47]	D. G. Zill, M. R. Cullen, Solutions manual for Zill/Cullen's differential equations with boundary-value problems, Boston: Cengage Learning, 2016.
[48]	I. Podlubny, Fractional differential equations, New York: Acadmic Press, 1998.

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