# American Institute of Mathematical Sciences

June  2020, 10(2): 157-164. 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, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045
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he 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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