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doi: 10.3934/naco.2019045

Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation

1. 

School of Mathematics and Statistics, Shaoguan University, Guangdong Shaoguan, 512005, P. R. China

2. 

Office of Party Committee, Foshan University, Guangdong Foshan 528000, P. R. China

* Corresponding author: Jie Song

Received  December 2018 Revised  July 2019 Published  September 2019

Fund Project: The second author is supported by NSF of Guangdong Province of China (S2012010010069).The third author is supported by the High-level talents Project of Guangdong Province Colleges and universities (2013-178)

In this paper, the existence and uniqueness of solution for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative are studied. The estimation of error between the approximate solution and the solution for such equation is presented by employing the quasilinear iterative method, and an example is given to demonstrate the application of our main result.

Citation: Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019045
References:
[1]

R. P. AgarwalM. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Georgian Mathe. J., 16 (2009), 401-411.  doi: 10.1007/s10440-008-9356-6.  Google Scholar

[2]

R. P. AgarwalY. Zhou and Y.-Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095-1100.  doi: 10.1016/j.camwa.2009.05.010.  Google Scholar

[3]

M. BenchohraS. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1-12.   Google Scholar

[4]

M. BenchohraS. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71 (2009), 2391-2396.  doi: 10.1016/j.na.2009.01.073.  Google Scholar

[5]

J. Brzdek and N. Eghbali, On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54 (2016), 31-35.  doi: 10.1016/j.aml.2015.10.004.  Google Scholar

[6]

J.-W. DengL.-J. Zhao and Y.-J. Wu, Efficient algorithms for solving the fractional ordinary differential equations, Appl. Math. Comput., 269 (2015), 196-216.  doi: 10.1016/j.amc.2015.07.048.  Google Scholar

[7]

A. M. A. El-Sayed, Fractional differential equations, Kyungpook Math. J., 28 (1988), 119-122.   Google Scholar

[8]

J.-H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (1999), 86-90.   Google Scholar

[9]

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

[10]

V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21 (2008), 828-834. doi: 10.1016/j.aml.2007.09.006.  Google Scholar

[11]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682.  doi: 10.1016/j.na.2007.08.042.  Google Scholar

[12]

V. Lakshmikantham and J. V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45.   Google Scholar

[13]

C. LiQ. Yi and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys., 316 (2016), 614-631.  doi: 10.1016/j.jcp.2016.04.039.  Google Scholar

[14]

H. Liang and M. Stynes, Collocation methods for general Caputo two-point boundary value problems, J. Sci. Comput., 76 (2018), 390-425.  doi: 10.1007/s10915-017-0622-5.  Google Scholar

[15]

P. LyuS. Vong and Z. Wang, A finite difference method for boundary value problems of a Caputo fractional differential equation, East. Asia. J. Appl. Math., 7 (2017), 752-766.  doi: 10.4208/eajam.181016.300517e.  Google Scholar

[16]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, INC., New York, 1993.  Google Scholar

[17] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974.   Google Scholar
[18] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, New York, 1999.   Google Scholar
[19]

M. Stynes and J.-L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal., 35 (2015), 698-721.  doi: 10.1093/imanum/dru011.  Google Scholar

[20]

Y.-F. Sun and P.-G. Wang, Quasilinear iterative scheme for a fourth-order differential equation with retardation and anticipation, Appl. Math. Comput., 217 (2010), 3442-3452.  doi: 10.1016/j.amc.2010.09.011.  Google Scholar

[21]

Y.-F. SunZ. Zeng and J. Song, Existence and uniqueness for the boundary value problems of nonlinear fractional differential equation, Appl. Math., 8 (2017), 312-323.   Google Scholar

[22]

P.-G. WangS.-H. Tian and Y.-H. Wu, Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions, Appl. Math. Comput., 203 (2008), 266-272.  doi: 10.1016/j.amc.2008.04.033.  Google Scholar

[23]

P.-G. WangH.-X. Wu and Y.-H. Wu, Higher even-order convergence and coupled solutions for second-order boundary value problems on time scales, Comput. Math. Appl., 55 (2008), 1693-1705.  doi: 10.1016/j.camwa.2007.06.026.  Google Scholar

show all references

References:
[1]

R. P. AgarwalM. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Georgian Mathe. J., 16 (2009), 401-411.  doi: 10.1007/s10440-008-9356-6.  Google Scholar

[2]

R. P. AgarwalY. Zhou and Y.-Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095-1100.  doi: 10.1016/j.camwa.2009.05.010.  Google Scholar

[3]

M. BenchohraS. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1-12.   Google Scholar

[4]

M. BenchohraS. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71 (2009), 2391-2396.  doi: 10.1016/j.na.2009.01.073.  Google Scholar

[5]

J. Brzdek and N. Eghbali, On approximate solutions of some delayed fractional differential equations, Appl. Math. Lett., 54 (2016), 31-35.  doi: 10.1016/j.aml.2015.10.004.  Google Scholar

[6]

J.-W. DengL.-J. Zhao and Y.-J. Wu, Efficient algorithms for solving the fractional ordinary differential equations, Appl. Math. Comput., 269 (2015), 196-216.  doi: 10.1016/j.amc.2015.07.048.  Google Scholar

[7]

A. M. A. El-Sayed, Fractional differential equations, Kyungpook Math. J., 28 (1988), 119-122.   Google Scholar

[8]

J.-H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (1999), 86-90.   Google Scholar

[9]

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

[10]

V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21 (2008), 828-834. doi: 10.1016/j.aml.2007.09.006.  Google Scholar

[11]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682.  doi: 10.1016/j.na.2007.08.042.  Google Scholar

[12]

V. Lakshmikantham and J. V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45.   Google Scholar

[13]

C. LiQ. Yi and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys., 316 (2016), 614-631.  doi: 10.1016/j.jcp.2016.04.039.  Google Scholar

[14]

H. Liang and M. Stynes, Collocation methods for general Caputo two-point boundary value problems, J. Sci. Comput., 76 (2018), 390-425.  doi: 10.1007/s10915-017-0622-5.  Google Scholar

[15]

P. LyuS. Vong and Z. Wang, A finite difference method for boundary value problems of a Caputo fractional differential equation, East. Asia. J. Appl. Math., 7 (2017), 752-766.  doi: 10.4208/eajam.181016.300517e.  Google Scholar

[16]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, INC., New York, 1993.  Google Scholar

[17] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974.   Google Scholar
[18] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, New York, 1999.   Google Scholar
[19]

M. Stynes and J.-L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal., 35 (2015), 698-721.  doi: 10.1093/imanum/dru011.  Google Scholar

[20]

Y.-F. Sun and P.-G. Wang, Quasilinear iterative scheme for a fourth-order differential equation with retardation and anticipation, Appl. Math. Comput., 217 (2010), 3442-3452.  doi: 10.1016/j.amc.2010.09.011.  Google Scholar

[21]

Y.-F. SunZ. Zeng and J. Song, Existence and uniqueness for the boundary value problems of nonlinear fractional differential equation, Appl. Math., 8 (2017), 312-323.   Google Scholar

[22]

P.-G. WangS.-H. Tian and Y.-H. Wu, Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions, Appl. Math. Comput., 203 (2008), 266-272.  doi: 10.1016/j.amc.2008.04.033.  Google Scholar

[23]

P.-G. WangH.-X. Wu and Y.-H. Wu, Higher even-order convergence and coupled solutions for second-order boundary value problems on time scales, Comput. Math. Appl., 55 (2008), 1693-1705.  doi: 10.1016/j.camwa.2007.06.026.  Google Scholar

Table 1.  The numerical results
t $ x_{1}(t) $ $ x_{2}(t) $ $ x_{3}(t) $ $ x_{4}(t) $ ...... $ x_{9}(t) $ $ x_{10}(t) $ $ z^{*}(t) $
0.0000 0.0000 0.0000 0.0000 0.0000 ...... 0.0000 0.0000 0.0000
0.1111 -0.0083 -0.0072 -0.0075 -0.0074 ...... -0.0074 -0.0074 -0.0074
0.2222 -0.0164 -0.0140 -0.0146 -0.0145 ...... -0.0145 -0.0145 -0.0145
0.3333 -0.0241 -0.0202 -0.0211 -0.0210 ...... -0.0210 -0.0210 -0.0210
0.4444 -0.0315 -0.0258 -0.0269 -0.0267 ...... -0.0267 -0.0267 -0.0267
0.5556 -0.0383 -0.0305 -0.0318 -0.0317 ...... - 0.0317 -0.0317 -0.0317
0.6667 -0.0447 -0.0344 -0.0359 -0.0357 ...... -0.0357 -0.0357 -0.0357
0.7778 -0.0505 -0.0372 -0.0389 -0.0387 ...... -0.0387 -0.0387 -0.0387
0.8889 -0.0557 -0.0390 -0.0407 -0.0406 ...... -0.0406 -0.0406 -0.0406
1.0000 -0.0602 -0.0396 -0.0414 -0.0412 ...... -0.0412 -0.0412 -0.0412
t $ x_{1}(t) $ $ x_{2}(t) $ $ x_{3}(t) $ $ x_{4}(t) $ ...... $ x_{9}(t) $ $ x_{10}(t) $ $ z^{*}(t) $
0.0000 0.0000 0.0000 0.0000 0.0000 ...... 0.0000 0.0000 0.0000
0.1111 -0.0083 -0.0072 -0.0075 -0.0074 ...... -0.0074 -0.0074 -0.0074
0.2222 -0.0164 -0.0140 -0.0146 -0.0145 ...... -0.0145 -0.0145 -0.0145
0.3333 -0.0241 -0.0202 -0.0211 -0.0210 ...... -0.0210 -0.0210 -0.0210
0.4444 -0.0315 -0.0258 -0.0269 -0.0267 ...... -0.0267 -0.0267 -0.0267
0.5556 -0.0383 -0.0305 -0.0318 -0.0317 ...... - 0.0317 -0.0317 -0.0317
0.6667 -0.0447 -0.0344 -0.0359 -0.0357 ...... -0.0357 -0.0357 -0.0357
0.7778 -0.0505 -0.0372 -0.0389 -0.0387 ...... -0.0387 -0.0387 -0.0387
0.8889 -0.0557 -0.0390 -0.0407 -0.0406 ...... -0.0406 -0.0406 -0.0406
1.0000 -0.0602 -0.0396 -0.0414 -0.0412 ...... -0.0412 -0.0412 -0.0412
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