Detailed error analysis for a fractional Adams method with graded meshes

Abstract

We consider a fractional Adams method for solving the nonlinear fractional differential equation C 0 D α t y(t) = f (t, y(t)), α > 0, equipped with the initial conditions y (k) Here, α may be an arbitrary positive number and α denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 2004) introduced a fractional Adams method with the uniform meshes t n = T (n/N), n = 0, 1, 2, . . . , N and proved that this method has the optimal convergence order uniformly in t n , that is, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C m [0, T ] for some m ∈ N and 0 < α < m, the Caputo fractional derivative C 0 D α t y(t) takes the form " C 0 D α t y(t) = ct α −α + smoother terms" (Diethelm et al. Numer. Algor. 36, 2004), which implies that C 0 D α t y behaves as t α −α which is not in C 2 [0, T ]. By using the graded meshes t n = T (n/N) r , n = 0, 1, 2, . . . , N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t n even if C 0 D α t y behaves as

On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

“…Setting sup t∈[0,T] |f(t, 0, 0, 0)| � M 1 , sup t∈[0,T] |g(t, 0)| � M 2 , and sup t∈[0,T] |h(t, 0)| � M 3 . For a positive number r, let B r � u ∈ C: ‖u‖ ≤ r { } and r ≥ r 2 /(1 − r 1 ), with r 1 is given by (23), we will show that ΦB r ⊂ B r , where Φ is defined by (19), and…”

“…Moreover, the existence and uniqueness of solutions for fractional differential equations have been mathematically studied from different methods [10][11][12][13][14][15], yielding methods for solving fractional differential equations [16][17][18][19]. As we all know, boundary value problems of fractional differential equations have been investigated for many years.…”

“…Proof. In view of the fixed point problem (19), we just need to prove the existence of at least one solution u ∈ R satisfying (19). Define a suitable ball B S ∈ C with radius S > 0 as…”

A High Order Numerical Method for Solving Nonlinear Fractional Differential Equation with Non-uniform Meshes

“…Recently, Liu et al [9] designed a predictor-corrector numerical method for solving (1) with graded meshes and the detailed error estimates are provided. Liu et al [10] also introduced a numerical method with non-uniform meshes for solving (1) and the detailed error estimates are considered. This paper is the continuation of the works in [9], [10] and we will introduce a high order numerical method for solving (1) with non-uniform meshes.…”

“…Liu et al [10] also introduced a numerical method with non-uniform meshes for solving (1) and the detailed error estimates are considered. This paper is the continuation of the works in [9], [10] and we will introduce a high order numerical method for solving (1) with non-uniform meshes. More precisely, we first approximate the integral in (2) with the piecewise quadratic interpolation polynomials with non-uniform meshes.…”