Inverse source problem for a space-time fractional diffusion equation

Abstract

This paper is concerned with an inverse source problem for a space-time fractional diffusion equation. We aim to identify an unknown source term from partially observed data. The employed model involves the Caputo fractional derivative in time and the non-local fractional Laplacian operator in space. The well-posedness of the forward problem is discussed. The considered ill-posed inverse source problem is formulated as a minimization one. The existence, uniqueness and stability of the solution of the minimization problem are examined. An iterative process is developed for identifying the unknown source term. A numerical implementation of the proposed approach is performed. The convergence of the discretized fractional derivatives is analyzed. The efficiency and accuracy of the proposed identification algorithm are confirmed by some numerical experiments.

Apprendix A

In this appendix, we provide the proofs of Lemma 3.1, Proposition 3.2, Corollary 3.3 and Proposition 3.4

1.1 Proof of Lemma 3.1

Firstly, let us introduce the following set $A:=\left\{ {\mathcal {J}}(h), f \in {\mathcal {H}}\right\}$. Using the fact that the function ${\mathcal {J}}$ is non-negative by definition, then, one can derive that the set A is reduced by zero. It follows that $\displaystyle \inf _{h \in {\mathcal {H}}}{\mathcal {J}}(h)$ is finite. Exploiting the definition of the minimum, then for each $\varepsilon >0$,

$$\begin{aligned} \exists {\mathcal {Y}}^{\varepsilon } \in A \text { such that } \inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h) \le {\mathcal {Y}}^{\varepsilon } \le \inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h)+\varepsilon , \end{aligned}$$

which implies that for each $\varepsilon >0$ there exists $h^{\varepsilon } \in {\mathcal {H}}$, such that ${\mathcal {J}}\left( h^{\varepsilon }\right) ={\mathcal {Y}}^{\varepsilon }$, satisfying

$$\begin{aligned} \inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h) \le {\mathcal {J}} \left( h^{\varepsilon }\right) \le \inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h)+\varepsilon . \end{aligned}$$

Replacing $\varepsilon$ by $\frac{1}{n}$, i.e. for each $n \ge 1,$ there exists $h^{n} \in {\mathcal {H}}$ satisfies

$$\begin{aligned} \inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h) \le {\mathcal {J}} \left( h^{n}\right) \le \inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h)+\frac{1}{n}, \end{aligned}$$

(25)

tending n to infinity, one can get

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathcal {J}}\left( h^{n}\right) =\inf _{h \in {\mathcal {H}}} {\mathcal {J}}(h). \end{aligned}$$

1.2 Proof of Proposition 3.2

By construction, the functional ${\mathcal {G}}$ is non-negative. Then, by applying Lemma 3.1, there exists a minimizing sequence $\lbrace f^n \rbrace _{n \ge 1}$ in $L^{2}({\mathcal {D}})$ satisfies

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathcal {G}}\left( f^{n}\right) =\inf _{f \in L^{2}({\mathcal {D}})} {\mathcal {G}}(f). \end{aligned}$$

Using relation (25), one can get

$$\begin{aligned} 0 \le {\mathcal {G}}\left( f^{n}\right) \le {\mathcal {G}}(0)+c =\left\| {\mathcal {O}}^{\delta }\right\| _{L^{2}({\mathcal {S}} \times (0, T))}+c, \end{aligned}$$

here $c \ge 1$ be an arbitrary constant. It follows,

$$\begin{aligned} \exists B>0, \text { such that }\left\| f^{n}\right\| _{L^{2}({\mathcal {D}})} \le B, \forall n \in {\mathbb {N}}^{*}. \end{aligned}$$

It implies that the sequence $\left\{ f^{n}\right\} _{n \ge 1},$ is uniformly bounded in $L^{2}({\mathcal {D}})$. Then, there exist a $f^{\star } \in L^{2}({\mathcal {D}})$ and a subsequence of $\left\{ f^{n}\right\} _{n \ge 1},$ still denoted by $\left\{ f^{n}\right\} _{n \ge 1},$ such that

$$\begin{aligned} f^{n} \rightharpoonup f^{\star } \text { in } L^{2}({\mathcal {D}}) \text { as } n \longrightarrow \infty . \end{aligned}$$

1.3 Proof of Corollary 3.3

The existence of a minimizing sequence $\left\{ f^{n}\right\} _{n \ge 1}$ of the functional ${\mathcal {G}}$ is guaranteed by Proposition 3.2. And, for each $n \ge 1$, the function $u\left( f^{n}\right)$ is solution to (1)–(4) with spatial component in source term $f=f^{n}$. It follows immediately from Theorem 2.3 and the uniformly boundless of the sequence $\left\{ f^{n}\right\} _{n \ge 1}$ that the sequence $\left\{ u\left( f^{n}\right) \right\} _{n \ge 1}$ is uniformly bounded in ${\mathcal {X}}^{\alpha , s}\left( 0, T; L^{2}({\mathcal {D}})\right)$ which leads to the existence of :

• $u^{\star } \in {\mathcal {X}}^{\alpha , s}\left( 0, T ; L^{2}({\mathcal {D}})\right)$

• a subsequence of $\left\{ u\left( f^{n}\right) \right\} _{n \ge 1}$ again still denoted by $\left\{ u\left( f^{n}\right) \right\} _{n \ge 1}$ such that

  $$\begin{aligned} u\left( f^{n}\right) \rightharpoonup u^{\star } \text { in } {\mathcal {X}}^{\alpha , s} \left( 0, T ; L^{2}({\mathcal {D}})\right) . \end{aligned}$$

Its implies

$$\begin{aligned} \partial _{t}^{\alpha } u\left( f^{n}\right) \rightharpoonup \partial _{t}^{\alpha } u^{\star } \text { and }(-\Delta )^{s} u\left( f^{n}\right) \rightharpoonup (-\Delta )^{s} u^{\star }, \text { as } n \longrightarrow \infty . \end{aligned}$$

Using the fact that $u\left( f^{n}\right)$ satisfies the following weak formulation

$$\begin{aligned} \int _{0}^{T} \int _{{\mathcal {D}}}\left[ \partial _{t}^{\alpha } u\left( f^{n}\right) +(-\Delta )^{s} u\left( f^{n}\right) \right] \varphi d x d t=\int _{0}^{T} \int _{{\mathcal {D}}} f^{n} \varphi d x d t, \quad \forall \varphi \in L^{2} \left( 0, T ; \tilde{H}^{s}({\mathcal {D}})\right) , \end{aligned}$$

by passing $n \longrightarrow \infty ,$ one can get

$$\begin{aligned} \int _{0}^{T} \int _{{\mathcal {D}}}\left[ \partial _{t}^{\alpha } u^{\star }+(-\Delta )^{s} u^{\star }\right] \varphi d x d t=\int _{0}^{T} \int _{{\mathcal {D}}} f^{\star } \varphi d x d t, \quad \forall \varphi \in L^{2}\left( 0, T ; \tilde{H}^{s}({\mathcal {D}})\right) . \end{aligned}$$

Following the same line as in [28], more precisely in proof of Theorem 2.1, to prove that $u^{\star }$ coincides with $u\left( f^{\star }\right)$ we will just have to show that $u^{\star }(., 0)=u\left( f^{\star }\right) (., 0).$ From Lemma 2.2, the following integral hold

$$\begin{aligned} \int _{{\mathcal {D}}} \int _{0}^{T} \partial _{t}^{\alpha } u\left( f^{n}\right) \psi v ~\mathrm {d} t ~\mathrm {d} x= & {} -\int _{{\mathcal {D}}} u\left( f^{n}\right) (\cdot , 0) J_{T^{-}}^{1-\alpha } \psi (0) v ~\mathrm {d} x\\&-\int _{{\mathcal {D}}}\int _{0}^{T} u\left( f^{n}\right) \frac{\mathrm {d}}{\mathrm {d} t} J_{T^{-}}^{1-\alpha } \psi v ~\mathrm {d} t ~\mathrm {d} x, \end{aligned}$$

for all $v \in L^{2}({\mathcal {D}})$ and $\psi \in C^{1}[0, T]$ such that $J_{T^{-}}^{1-\alpha } \psi (T)=0.$ Now, using the fact that $u\left( f^{n}\right) (., 0)=$ in ${\mathcal {D}}$ and passing $n \longrightarrow \infty$ in the previous equation, one can get

$$\begin{aligned} \int _{{\mathcal {D}}} \int _{0}^{T} \partial _{t}^{\alpha } u^{\star } \psi v ~\mathrm {d} t ~\mathrm {d} x=-\int _{{\mathcal {D}}} \int _{0}^{T} u^{\star } \frac{\mathrm {d}}{\mathrm {d} t} J_{T^{-}}^{1-\alpha } \psi v ~\mathrm {d} t ~\mathrm {d} x. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \int _{{\mathcal {D}}} \int _{0}^{T} \partial _{t}^{\alpha } u^{\star } \psi v ~\mathrm {d} t ~\mathrm {d} x=-\int _{{\mathcal {D}}} u^{\star }(\cdot , 0) J_{T^{-}}^{1-\alpha } \psi (0) v ~\mathrm {d} x-\int _{{\mathcal {D}}} \int _{0}^{T} u^{\star } \frac{\mathrm {d}}{\mathrm {d} t} J_{T^{-}}^{1-\alpha } \psi v ~\mathrm {d} t ~\mathrm {d} x, \end{aligned}$$

it implies that $u^{\star }(\cdot , 0)=0$ in ${\mathcal {D}}$. Then, from the uniqueness of the weak solution for the forward problem (1)–(4) in Theorem 2.3 , one can check that $u^{\star }$ coincides with the unique solution to (1)–(4) with spatial component in source term $f=f^{\star },$ i.e. $u^{\star }=u\left( f^{\star }\right)$.

1.4 Proof of Proposition 3.4

Thanks to Corollary 3.3 which guaranties the existence of $u^\star \in {\mathcal {X}}^{\alpha , s}\left( 0, T; L^{2}({\mathcal {D}})\right)$ and a sub-sequence $\{ h^n\}_{n \ge 1} \subset {\mathcal {X}}^{\alpha , s}\left( 0, T; L^{2}({\mathcal {D}})\right)$ satisfying

$$\begin{aligned} u(h^{n}) \rightharpoonup u^{\star }\; \; \text {in}\; \; {\mathcal {X}}^{\alpha , s} \left( 0, T; L^{2}({\mathcal {D}})\right) \; \; \text {as}\; \; n \longrightarrow \infty , \end{aligned}$$

where $u(h^n)$ is the solution to the forward problem (1)–(4) with spatial component $h^n$ in the source term. Now, by employing the semi-continuity of the $L^2$-norm, we get

$$\begin{aligned} \left\| u\left( h^{\star }\right) -{\mathcal {O}}^{\delta }\right\| _{L^{2} ((0, T)\times {\mathcal {S}})}^{2}\le & {} \liminf _{n \rightarrow \infty } \left\| u\left( h^{n}\right) -{\mathcal {O}}_{k}^{\delta }\right\| _{L^{2} ({\mathcal {S}}\times (0, T))}^{2} \; \; \text {and}\; \; \Vert h^{\star } \Vert _{L^{2}({\mathcal {D}})} \\\le & {} \liminf _{n \rightarrow \infty } \left\| h^n\right\| _{L^{2}({\mathcal {D}})}^{2}, \end{aligned}$$

it follows

$$\begin{aligned} {\mathcal {G}}(h^\star ) \le \liminf _{n \rightarrow \infty } \left\| u\left( h^{n}\right) -{\mathcal {O}}_{k}^{\delta }\right\| _{L^{2} ({\mathcal {S}}\times (0, T))}^{2} +\mu \liminf _{n \rightarrow \infty } \left\| h^n\right\| _{L^{2}({\mathcal {D}})}^{2}, \end{aligned}$$

which leads to

$$\begin{aligned} {\mathcal {G}}(h^\star ) \le \liminf _{n \rightarrow \infty } {\mathcal {G}}(h^n). \end{aligned}$$