In this paper, we study the existence, regularity, and approximation of the solution for a class of nonlinear fractional differential equations. {In order to do this}, suitable variational formulations are defined for a nonlinear boundary value problems with Riemann-Liouville and Caputo fractional derivatives together with the homogeneous Dirichlet condition. We {investigate} the well-posedness and also the regularity of the corresponding weak solutions. Then, we develop a Galerkin finite element approach {\color{blue}for} the numerical approximation of the weak formulations and {drive a priori error estimates and prove the stability of the schemes}. Finally, some numerical experiments are provided to {demonstrate} the accuracy of the proposed method.
翻译:在本文中,我们研究了非线性分数方程类别解决方案的存在、规律性和近似性。 {为了做到这一点},我们为Riemann-Liouville和Caputo分数衍生物的非线性边界值问题以及同质dirichlet条件确定了适当的变式配方。 我们{ Investigate} 充分性和相应的薄弱解决方案的规律性。 然后,我们开发了一种 Galerkin 有限元素法 {color{blue}, 用于} 微弱配方的数值近似值, 和{驱动先验错误估计, 并证明方案的稳定性}。 最后,我们提供了一些数字实验, 以证明拟议方法的准确性 。