14 research outputs found
Detailed error analysis for a fractional adams method with graded meshes
The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-017-0419-5We consider a fractional Adams method for solving the nonlinear fractional differential equation \, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0, equipped with the initial conditions . Here may be an arbitrary positive number and denotes the smallest integer no less than and the differential operator is the Caputo derivative. Under the assumption \, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T], Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes and proved that this method has the optimal convergence order uniformly in , that is if and if . They also showed that if \, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T], the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for for some and , we show that the optimal convergence order of this method can be recovered uniformly in even if \, ^{C}_{0}D^{\alpha}_{t} y behaves as . Numerical examples are given to show that the numerical results are consistent with the theoretical results
A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line
Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions
A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative
In this paper, we consider the nonlinear boundary value problems involving the Caputo fractional derivatives of order α∈ (1 , 2) on the interval (0, T). We present a Legendre spectral collocation method for the Caputo fractional boundary value problems. We derive the error bounds of the Legendre collocation method under the L2- and L∞-norms. Numerical experiments are included to illustrate the theoretical results.MOE (Min. of Education, S’pore