Year: 2022
Author: Ai Sun, Youhui Su, Jianping Sun
Journal of Nonlinear Modeling and Analysis, Vol. 4 (2022), Iss. 3 : pp. 409–442
Abstract
In this paper, the existence of solutions to a class of fractional differential equations $D^α_{0+}u(t) = h(t)f(t, u(t), D^θ_{0+}u(t))$ is obtained by an efficient and simple monotone iteration method. At first, the existence of a solution to the problem above is guaranteed by finding a bounded domain $D_M$ on functions $f$ and $g.$ Then, sufficient conditions for the existence of monotone solution to the problem are established by applying monotone iteration method. Moreover, two efficient iterative schemes are proposed, and the convergence of the iterative process is proved by using the monotonicity assumption on $f$ and $g.$ In particular, a new algorithm which combines Gauss-Kronrod quadrature method with cubic spline interpolation method is adopted to achieve the monotone iteration method in Matlab environment, and the high-precision approximate solution is obtained. Finally, the main results of the paper are illustrated by some numerical simulations, and the approximate solutions graphs are provided by using the iterative method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.12150/jnma.2022.409
Journal of Nonlinear Modeling and Analysis, Vol. 4 (2022), Iss. 3 : pp. 409–442
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 34
Keywords: Fractional differential equation Monotone iteration method Numerical simulation Approximate solutions graphs.