Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations

Abstract

In this paper, we consider the well-posedness and approximation for nonhomogeneous fractional differential equations in Banach spaces E. Firstly, we get the necessary and sufficient condition for the well-posedness of nonhomogeneous fractional Cauchy problems in the spaces $C_0^{\beta }([0,T];E).$ Secondly, by using implicit difference scheme and explicit difference scheme, we deal with the full discretization of the solutions of nonhomogeneous fractional differential equations in time variables, get the stability of the schemes and the order of convergence.

Keywords:

• α-times resolvent family
• discretization methods
• explicit scheme
• fractional Cauchy problem
• implicit scheme
• nonhomogeneous fractional equations

MATHEMATICS SUBJECT CLASSIFICATION:

• 45L05
• 65M06
• 65M12

Appendix

In this part, we recall the following version of Trotter-Kato’s Theorem [28, 29] on general approximation scheme.

Theorem 4.1.

[4] (Theorem ABC) Assume that $A\in \mathcal{C}(E),A_n\in \mathcal{C}(E_n)$ and they generate C₀-semigroups. The following conditions (A) and (B) are equivalent to condition $(C).$

(A) Consistency. There exists $\lambda \in \rho (A)\cap \underset{n}{\cap }\rho (A_n)$ such that the resolvents converge ${(\lambda I_n-A_n)}^{-1}\stackrel{\mathcal{P}\mathcal{P}}{\to }{(\lambda I-A)}^{-1};$

(B) Stability. There are some constants $M\ge 1$ and $\omega ,$ which are not depending on n and such that $\left|\right|\hspace{0.17em}\text{exp}\hspace{0.17em}(t{A}_n)\left|\right|\le M\hspace{0.17em}\text{exp}\hspace{0.17em}(\omega t)$ for $t\ge 0$ and any $n\in \mathbb{N};$

(C) Convergence. For any finite $T>0$ one has $${\max }_{t\in [0,T]} \left|\right|\hspace{0.17em}\text{exp}\hspace{0.17em}(t{A}_n)u_n^0-p_n\hspace{0.17em}\text{exp}\hspace{0.17em}(tA)u^0\left|\right|\to 0$$ as $n\to \infty ,$ whenever $u_n^0\stackrel{\mathcal{P}}{\to }{u}^0$ for any $u_n^0\in E_n,u^0\in E.$

Remark 4.1.

In case of approximation of analytic semigroups they have some changes in formulation of Theorem 4.1:

$(B_1)$ Stability. There exist constants $M\ge 1$ and ω independent of n such that for any $Re\lambda >\omega ,$ $\left|\right|{(\lambda I_n-A_n)}^{-1}\left|\right|\le \frac{M}{|\lambda -\omega |} \mathrm{ }\text{for}\mathrm{ }\text{all}\mathrm{ }n\in \mathbb{N};$

$(C_1)$ Convergence. For any finite $\mu >0$ and some $0<\theta <\frac{\pi }{2}$ we have ${\max }_{\eta \in \Sigma (\theta ,\mu )}\left|\right|\hspace{0.17em}\text{exp}\hspace{0.17em}(\eta A_n)u_n^0-p_n\hspace{0.17em}\text{exp}\hspace{0.17em}(\eta A)u^0\left|\right|\to 0\mathrm{ }\text{as}\mathrm{ }n\to \infty \mathrm{ }\text{whenever}\mathrm{ }{u}_n^0\stackrel{\mathcal{P}}{\to }{u}^0.$ Here we denote $\Sigma (\theta ,\mu )=\{z\in \Sigma (\theta ):|z|\le \mu \}$ and $\Sigma (\theta )=\{z\in \mathbb{C} : |\mathit{arg} z|\le \theta \}.$

For the semidiscrete approximation of α-times resolvent family, we have the following ABC Theorems:

Theorem 4.2.

[16] Suppose that $0<\alpha \le 2$ and A, A_n generate exponentially bounded α-times resolvent families $S_{\alpha }(·,A),S_{\alpha }(·,A_n)$ in the Banach spaces E, E_n, respectively. The following conditions (A) and $(\tilde{B})$ are equivalent to condition $(\tilde{C}).$

(A) Consistency. There exists $\lambda \in \rho (A)\cap \underset{n}{\cap }\rho (A_n)$ such that the resolvents converge ${(\lambda I_n-A_n)}^{-1}\stackrel{\mathcal{P}\mathcal{P}}{\to }{(\lambda I-A)}^{-1};$

$(\tilde{B})$ Stability. There are some constants $M\ge 1$ and ω, which are not depending on n and such that $\left|\right|S_{\alpha }(t,A_n)|{|}_{B(E_n)}\le M{e}^{\omega t}\mathrm{ }\text{for}\mathrm{ }t\ge 0,n\in \mathbb{N};$

$(\tilde{C})$ Convergence. For some finite ${\omega }_1>0$ one has ${\max }_{t\in [0,\infty )}{e}^{-{\omega }_{1}t}\left|\right|S_{\alpha }(t,A_n)x_n-p_n{S}_{\alpha }(t,A)x|{|}_{E_n}\to 0$ as $n\to \infty ,$ whenever $x_n\stackrel{\mathcal{P}}{\to }x$ for any $x_n\in E_n,x\in E.$

Theorem 4.3.

[17] Suppose that $0<\alpha \le 2$ and A, A_n generate exponentially bounded analytic α-times resolvent families $S_{\alpha }(·,A),S_{\alpha }(·,A_n)$ in the Banach spaces E, E_n, respectively. The following conditions (A) and $(B\prime )$ are equivalent to condition $(C\prime ).$

(A) Consistency. There exists $\lambda \in \rho (A)\cap \underset{n}{\cap }\rho (A_n)$ such that the resolvents converge ${(\lambda I_n-A_n)}^{-1}\stackrel{\mathcal{P}\mathcal{P}}{\to }{(\lambda I-A)}^{-1};$

$(B\prime )$ Stability. There are some constants $M\ge 1,$ $0<\theta \le \pi /2$ and ω which are independent of n, such that the sector ${(\omega +{\Sigma }_{\theta +\pi /2})}^{\alpha }$ is included in $\rho (A_n)$ and $$\underset{\lambda \in \omega +{\Sigma }_{\beta +\pi /2}}{\sup }\left|\right|{\lambda }^{\alpha -1}R({\lambda }^{\alpha };A_n)|{|}_{B(E_n)}\le M/|\lambda -\omega |\mathrm{ }\mathit{\text{for}}\mathrm{ }\mathit{\text{any}}\mathrm{ }n\in \mathbb{N}\mathrm{ }\mathit{\text{and}}\mathrm{ }\mathit{\text{for}}\mathrm{ }\mathit{\text{any}}0<\beta <\theta .$$

$(C\prime )$ Convergence. For some finite ${\omega }_1>0$ one has $$\underset{z\in {\Sigma }_{\beta }}{\sup }{e}^{-{\omega }_{1}Rez}\left|\right|S_{\alpha }(z,A_n)x_n-p_n{S}_{\alpha }(z,A)x|{|}_{E_n}\to 0\mathrm{ }\text{as}\mathrm{ }n\to \infty ,$$ whenever $x_n\stackrel{\mathcal{P}}{\to }x$ for any $x_n\in E_n,x\in E$ and for any $0<\beta <\theta .$

Additional information

Funding

The first author was supported by Scientific Research Starting Foundation (Chengdu University, No. 2081915055); the second author was partially supported by the Russian Science Foundation (RSF), No. 20-11-20085.