Detailed error analysis for a fractional Adams method with graded meshes
The paper is concerned with the analysis of the fractional Adams method for the nonlinear fractional differential equation \[ D^{\alpha} y \left( t \right) = f \left( t, y \left( t \right) \right), \quad t \in \left(0,T\right), \quad \alpha>0, \] equipped with initial conditions. The fractional derivative \(D^{\alpha} y\) is understood in the Caputo sense. The authors focus on the case that the function \(D^{\alpha} y\) does not belong to the space \(C^2\left[0,T\right]\), because it is known that in this case the optimal order of convergence for the fractional Adams method can not be guaranteed on the uniform mesh. Therefore, they study the fractional Adams method with graded meshes and they prove that the optimal order of convergence is achieved under the assumption that \(D^{\alpha} y\) is twice continuously differentiable on \(\left( 0,T\right]\) and \(D^{\alpha} y\) behaves as \(t^{\sigma}\), \(0< \sigma <1\). The authors present several numerical experiments to show that the experimental orders of convergence under different assumptions on smoothness of \(D^{\alpha} y\) correspond to theoretical results.
scientific article; zbMATH DE number 6916369
scientific article