TY - JOUR
T1 - Existence of Solutions to a Class of Fractional Differential Equations
AU - Sun , Ai
AU - Su , Youhui
AU - Sun , Jianping
JO - Journal of Nonlinear Modeling and Analysis
VL - 3
SP - 409
EP - 442
PY - 2022
DA - 2022/06
SN - 4
DO - http://doi.org/10.12150/jnma.2022.409
UR - https://global-sci.org/intro/article_detail/jnma/20715.html
KW - Fractional differential equation, Monotone iteration method, Numerical simulation, Approximate solutions graphs.
AB - <p style="text-align: justify;">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.</p>