Many results of the approximation theory to abstract differential equations in Banach spaces simplify the design of concrete numerical approaches. Thus, an approximation theory of differential equations has attracted much attention due to its wide application in recent years.

In [1], Guidetti, Karasözen and Piskarev investigated the general approximation theory for differential equations with first-order derivatives in Banach spaces. Using the approximation theory, they analyzed the numerical problems of homogeneous differential equations and semilinear differential equations, respectively. In [2,3], Li, Morozov and Piskarev considered the approximation theory for derivatives of integrated semigroups. For other papers on the approximation of first-order differential equations, we suggest that readers consult [4,5,6,7,8,9].

Recently, fractional Cauchy problems and their approximation have become an important topic due to their broad application in engineering, physics and biology. A large number of findings on this topic have been reported in the literature [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. Among these, in [22], Liu, Li and Piskarev considered the fulldiscretization approximation for solutions of the following equation with fractional time derivative $\alpha \in (0,1)$ in abstract space E, by virtue of finite differences and projection methods. In the same year, by discussing the relations of compact convergence of resolvents and semidiscrete approximation, the authors [23] studied the semidiscretization approximation of semilinear fractional problems where $0<\alpha <1$. They demonstrated that the semidiscrete approximation to the solution is convergent if the corresponding resolvents are compactly convergent. However, in [23], the authors did not consider the fulldiscretization of the nonlinear term $J^{1-\alpha }f(t,u(t))$. In [20], the authors discussed the well-posedness and maximal regularity of fractional semilinear differential equations in Hölder space, and derived the existence and stability of an implicit difference scheme for the fractional systems. We refer to [11,15,20,21,24,25,27,32] and the references therein for the approximation of various differential equations in Banach spaces.

Motivated by above papers, we investigate the fulldiscrete approximation of nonhomogeneous fractional equation in abstract space E, where operator A is the generator of $C_0$-semigroup $exp(tA)$, $0<\alpha \le 1$, f is a smooth enough function, the Caputo fractional-order derivative $D_t^{\alpha }$ with order $\alpha$ is defined by and the Riemann–Liouville fractional order integral $J^{1-\alpha }f(t)$ with order $1-\alpha$ is defined by if the above two integrals exist.

The general discretization scheme for problem (3) in Banach space $E_n$ is with a series of smooth enough functions $f_n(·)$.

In this paper, we find new iteration formulas of solutions to the implicit scheme and explicit scheme for the nonhomogeneous Cauchy problem (3) using the methods of iteration, finite differences and projection. At the same time, we discuss the stability for the two schemes using the Trotter–Kato theorem.

Define $E_n$ and E as Banach spaces, $p_n\in \mathcal{B}(E,E_n)$, $B_n\in \mathcal{B}(E_n)$ and $B\in \mathcal{B}(E)$ with $n\in \mathbb{N}$, where $\mathcal{B}(E,E_n)$ denotes the space of all continuous linear operators from E to $E_n$, $\mathcal{B}(E_n)$ denotes $\mathcal{B}(E_n,E_n)$. Now, we introduce some notations and definitions of approximation theory, as follows.

By [9], we always assume that $\{p_n\}$, $p_n\in \mathcal{B}(E,E_n)$, satisfies that $\parallel p_n{x\parallel }_{E_n}$ goes to ${\parallel x\parallel }_E$ when n tends to infinity for each $x\in E$.

Use $\mathcal{C}(E)$ to denote the space of all densely defined closed linear operators on E. One version of the Trotter–Kato theorem [1], which is essential in the investigation of the approximation theory for differential equations, is shown as follows.

The main purpose of the paper is to investigate the fulldiscrete approximation of the Equation (4). Therefore, the difference schemes for the general approximation to the problem (3) are needed.

Let $t_m=m{\tau }_n$, $m=0,1,2,\dots$; we approximate the fractional derivative $(D_t^{\alpha }{x}_n)(t_m)$ of functions $x_n:[0,L]\to E_n$ by the finite difference scheme ${▵}_{t_m}^{\alpha }{x}_n(·)$, where and

In view of [24], the solution of the homogeneous equation of problem (4) can be expressed by $u_n(t)=S_{\alpha }(t,A_n)u_n^0$ for any smooth initial value $u_n^0\in D(A_n^{l+1})$ with the smallest integer l, such that $(l+1)\alpha \ge 2$. In this situation, they proved the following relation regarding the order of convergence

On the other hand, we approximate $J^{1-\alpha }f(t_m)$ by $J_{t_m}^{1-\alpha }{f}_n(·)$, where and

Now, we can approximate problem (3) using the implicit difference scheme and the explicit scheme respectively.

Now, we present the proofs of the iteration formulas that solve the two difference schemes through the method of induction, and discuss the stability of the solutions under the condition (B) with $\omega =0$ in the Trotter–Kato theorem.

Let $b_j={(j+1)}^{1-\alpha }-j^{1-\alpha }$ in the sequel. The two iteration formulas of solutions for implicit and explicit difference schemes are presented as follows.