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

Yu-Feng Sun; Zheng Zeng; Jie Song

doi:10.3934/naco.2019045

Article Contents

2020, Volume 10, Issue 2: 157-164. Doi: 10.3934/naco.2019045

This issue Previous Article Linear optimal control of time delay systems via Hermite wavelet Next Article Quadratic optimization with two ball constraints

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
^* Corresponding author: Jie Song

Received: December 2018

Revised: July 2019

Published: April 2020

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 26A33, 34A08, 34B15; Secondary: 47J25.

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Table 1. 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

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