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Existence of Solutions to a Class of Fractional Differential Equations
  
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KeyWord:Fractional differential equation; Monotone iteration method; Numerical simulation; Approximate solutions graphs
Author NameAffiliation
Ai Sun School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou, Jiangsu 221018, China
College of Science, Shenyang University of Technology, Shenyang, Liaoning 110142, China 
Youhui Su School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou, Jiangsu 221018, China 
Jianping Sun School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China 
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Abstract:
      In this paper, the existence of solutions to a class of fractional differential equations $D_{0+}^{\alpha}u(t)=h(t)f(t, u(t), D_{0+}^{\theta}u(t))$ 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.