Numerical Algorithms ( IF 2.1 ) Pub Date : 2021-01-14 , DOI: 10.1007/s11075-020-01032-2 Khadijeh Nedaiasl , Raziyeh Dehbozorgi
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 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 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.
中文翻译:
非线性分数阶微分方程的Galerkin有限元方法
在本文中,我们研究了一类非线性分数阶微分方程解的存在性,正则性和逼近。为此,针对具有Riemann-Liouville和Caputo分数导数的非线性边值问题以及齐次Dirichlet条件定义了合适的变分公式。我们研究了相应的弱解的适定性和规律性。然后,我们为弱公式的数值逼近开发了Galerkin有限元方法,并驱动了先验误差估计并证明了方案的稳定性。最后,通过一些数值实验证明了该方法的准确性。