Abstract
In this work, the solution regularity of a variable-order time-fractional diffusion equation with Mittag-Leffler kernel is analyzed and the uniqueness of determining the variable fractional order based on the observations of the solutions over a small time-space interval is proved. The main difficulty we overcome lies in the technical analysis of the variable-order Mittag-Leffler kernel and its derivatives via special functions, and the proved results provide theoretical supports for the applications of the proposed model in both the direct and inverse manner. Furthermore, the derived methods could be applied to analyze the analogous direct and inverse problems involving the variable-order Caputo–Fabrizio fractional operators.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Time-fractional reaction–diffusion equation attracts widely attentions in applications like viscoelastic behaviors and anomalous diffusion with extensive investigations for both the direct and inverse problems [4,5,6, 9, 11,12,13,14,15, 18, 21, 23]. In particular, the time-fractional problem with Mittag-Leffler kernel has been applied in various fields such as the groundwater pollution problem [3]. Nevertheless, a constant fractional order could not accommodate the variability of the memory with respect to time in practical problems, which motivates the application of the variable fractional order [16, 20, 27]. For this reason, in this work we consider the following variable-order time-fractional diffusion equation with Mittag-Leffler kernel
Here \(\Omega \subset {\mathbb {R}}^d\) (\(d=1,2,3\)) is a bounded domain with a smooth boundary \(\partial \Omega \), \(\varvec{x}=(x_1,\ldots ,x_d)\in \Omega \),K represents the diffusivity coefficient, f represents the source term, and the fractional derivative \({}_0\partial _t^{\alpha } z\) with variable-order Mittag-Leffler kernel of order \(0<\alpha _*\le \alpha (t)\le \alpha ^*<1\) for some \(0<\alpha _*,\alpha ^*<1\) is defined by [2, 7, 25]
where the function form of the variable-order Mittag-Leffler kernel is given as
Here \(M(\alpha (t))>0\) is a normalization function satisfying \(M(0)=M(1)=1\), and \(E_{\mu ,\nu }(t)\) (\(\mu > 0\) and \(\nu \in \mathbb R\)) represents the two-parametric Mittag-Leffler function given by [8]
Despite growing applications of the time-fractional problems involving the variable-order Mittag-Leffler kernel, the analysis for the corresponding forward and backward problems is still not well developed. In [25] the well-posedness of model (1) was studied. In this work, we continue to prove the solution regularity of model (1) and the corresponding inverse problem of unique determination of the variable fractional order in model (1) based on the observations of the solutions over a small time-space interval, which provide theoretical supports for applications of the model. The main difficulty we overcome lies in the technical analysis of the variable-order Mittag-Leffler function and its derivatives via special functions. The derived methods could also be applied to analyze the analogous problems involving the variable-order Caputo–Fabrizio fractional operators via a simpler manner.
The rest of the paper is organized as follows. In Sect. 2, we prove the solution regularity of model (1), based on which we prove the unique determination of the variable fractional order in model (1) in Sect. 3. Throughout this paper, \(Q>0\) represents a generic constant that may assume different values in difference circumstances.
2 Preliminaries
Let \(L^2(\Omega )\) be the space of square integrable functions on \(\Omega \) and \(H^{m}(\Omega )\) (\(m \in \mathbb {N}\)) be the Hilbert spaces of functions with weak derivatives of order m in \(L^2(\Omega )\). For non-integer \(s\ge 0\), the fractional Sobolev spaces \(H^s(\Omega )\) are defined via interpolation. All the spaces are equipped with the standard norms [1].
Let \(\{\lambda _i \}_{i=1}^\infty \) be the eigenvalues of Strum-Liouville problem
By [17], the eigenfunctions \(\{\phi _i \}_{i=1}^\infty \) form an orthonormal basis in \(L^2(\Omega )\) and \(0 <\lambda _1\le \lambda _2\le \ldots \rightarrow \infty \). Based on (2), we define the Sobolev space for \(\gamma \ge 0\) by [10]
Besides, for a Banach space \({\mathcal X}\), let \(C([0,T];\mathcal X)\) be the space of continuous functions on the interval [0, T] with respect to the norm \(\Vert \cdot \Vert _{\mathcal X}\) [1].
We then refer to the Gronwall inequality for future use [22].
Lemma 1
Let \(0 \le C_0(t) \in L_{loc}(a,b)\) and \(C_1\) be a non-negative constant. Suppose \(0 \le g(t) \in L_{loc}(a,b)\) satisfies
where the fractional integral operator \({}_aI_t^\gamma \) is defined as
Then g can be bounded by
In particular, if \(C_0(t)\) is non-decreasing, then
We finally refer the well-posedness result of model (1) from [25].
Theorem 1
Suppose \(\alpha \in C[0,T]\), \(u_0\in \check{H}^\gamma (\Omega )\), \(f\in H^\kappa (0,T;\check{H}^\gamma (\Omega ))\) for some \(\kappa >1/2\) and \(\gamma \ge 2\), and \(-K\Delta u_0(\varvec{x})=f(\varvec{x},0)\). Then problem (1) has a unique mild solution in \(C([0,T];\check{H}^\gamma (\Omega ))\) with the following stability estimate
3 Solution regularity
We prove the regularity of the solution in the following theorem.
Theorem 2
Suppose M and \(\alpha \) are continuously differentiable functions and \(\partial _t f\in H^\kappa (0,T;\check{H}^{s-2}(\Omega ))\) for some \(s\ge 2\) and \(\kappa >1/2\). Then the following regularity estimate holds
Proof
Let \(\{u_i(t),f_i(t)\}_{i=1}^\infty \) be the Fourier coefficients of u and f, respectively, expanded under the eigenfunctions \(\{\phi _i\}_{i=1}^\infty \), that is,
Then problem (1) could be reduced to the following system of ordinary differential equations
Differentiating this equation with respect to t leads to a second-kind Volterra integral equation
The key component in \(\partial _t\mathcal Q(\alpha (t),t-s)\) is the following derivative of variable-order Mittag-Leffler function
Direct calculation shows that
the right-hand side of which is continuous with respect to s and t (cf. the proof of Lemma 2.2 in [25]). Thus all but the last right-hand side terms of (5) could be reformulated as
which could be bounded as \(Q(t-s)^{\alpha (t)-1}\) for some constant \(Q>0\). To estimate the last right-hand side term of (5), we apply the definition of the Polygamma function \(\psi (\cdot ):=\Gamma '(\cdot )/\Gamma (\cdot )\) and the asymptotic properties of \(\psi (\cdot )\) and \(\Gamma (\cdot )\) [19]
to obtain for i large enough
Consequently the ith summand (with i large enough) in the last right-hand side term of (5) could be bounded by
It is clear that for i large enough, the following estimate holds
Then we apply the Weierstrass M-test to conclude that the last right-hand side term of (5) is continuous and bounded. We incorporate the preceding estimates in (4) to obtain
where we used the estimate
Then an application of the Gronwall inequality with singular kernel leads to
which immediately yields
Thus the proof is completed. \(\square \)
4 Uniqueness of determining variable fractional order
We prove the uniqueness of determining the variable fractional order in model (1) based on the proved regularity results and the observation on the solution u over a small time-space interval. Here we require that the normalization function M(z) with \(\alpha _*\le z\le \alpha ^*\) satisfies
for some constant \(M_0>0\). In most literature \(M(z)\equiv 1\) while in, e.g. [7], a specific form \(M(z)=\frac{2}{2-z}\) is selected. It could be verified directly that for both cases the condition (8) is satisfied since for the later case
Theorem 3
Suppose M and \(\alpha \) are continuously differentiable functions with the condition (8), \(u_0\in \check{H}^{2+d/2+\varepsilon }(\Omega )\) and \(f\in H^\kappa (0,T;\check{H}^{2+d/2+\varepsilon }(\Omega ))\) for some \(0<\varepsilon \ll 1\) and \(\kappa >1/2\) with \(\partial _t f\in H^\kappa (0,T;L^2(\Omega ))\). Let u and \(\bar{u}\) be solutions to model (1) with respect to \(\alpha (t)\) and \(\bar{\alpha }(t)\), respectively, i.e. u and \(\bar{u}\) satisfy
and
Suppose \(\alpha _*\le \alpha (t),\bar{\alpha }(t)\le \alpha ^*\) are analytic functions over [0, T] and \(u=\bar{u}\) over \(\Omega _0\times [0,\tau ]\) for some open set \(\Omega _0\subset \Omega \) and for some \(0<\tau \ll 1\) such that \(\partial _t u(\varvec{x}_0,0)\ne 0\) for some \(\varvec{x}_0\in \Omega _0\). Then \(\alpha (t)\equiv \bar{\alpha }(t)\) over [0, T].
Proof
By the smoothness assumptions on the data, we apply Theorems 1 and 2 and the Sobolev embedding \(H^{m+d/2+\varepsilon }(\Omega )\hookrightarrow C^{m}(\Omega )\) to conclude that \(u,\bar{u}\in C([0,T];C^2(\Omega ))\) with \(\partial _t u,\partial _t\bar{u}\in C([0,T];C(\Omega ))\). Since \(\partial _t u(\varvec{x}_0,0)(=\partial _t\bar{u}(\varvec{x}_0,0))\ne 0\) at \(\varvec{x}_0\in \Omega _0\), we assume \(\partial _t u(\varvec{x}_0,0)> 0\) on \(\Omega _0\) without loss of generality. By the smoothness of u and \(\bar{u}\), there exist \(0<\tau _1\le \tau \) and \(\Omega _1\subset \Omega _0\) such that \(\partial _t u=\partial _t \bar{u}\ge \beta >0\) over \(\Omega _1\times [0,\tau _1]\) with \(\Delta u=\Delta \bar{u}\) on \(\Omega _1\times [0,\tau _1]\). Thus we obtain by subtracting (9) from (10) that
over \(\Omega _1\times [0,\tau _1]\). As \(\partial _t u=\partial _t \bar{u}\ge \beta >0\) over \(\Omega _1\times [0,\tau _1]\), we get from this equation that
that is,
where \(\hat{\alpha }(s,t)\) lies in between \(\alpha (t)\) and \(\bar{\alpha }(t)\). Differentiating \(\mathcal Q(z,t-s)\) with respect to z and using a similar derivation as (6) yield
By the monotonicity of the Gamma function, we have
for some constant \(Q>0\), the following estimate holds for \(t\ll 1\)
By the Weierstrass M-test, the right-hand side series of this equation is continuous with respect to \(t-s\), and consequently close to 0 for t small enough. We incorporate this with condition (8) to conclude that there exists a \(0<\tau _2\le \tau _1\) such that the first right-hand side term of (12) has a positive lower bound for \(t\le \tau _2\)
For the other right-hand side terms of (12), similar analysis indicates that since they are exactly 0 for \(t-s=0\), there exists a \(0<\tau _3\le \tau _2\) such that these terms are bounded by \(C_0/2\) for \(t\le \tau _3\). We combine these estimates to conclude that
Since \(\partial _t u\ge \beta >0\) over \(\Omega _1\times [0,\tau _1]\), (11) implies that \(\alpha (t)=\bar{\alpha }(t)\) for \(t\in [0,\tau _3]\). Based on this result, we could follow the same proof around [26, Equation 13] to apply the analyticity assumption of \(\alpha (t)\) and \(\bar{\alpha }(t)\) in this theorem to reach \(\alpha (t)=\bar{\alpha }(t)\) for \(t\in [0,T]\).Thus the proof is completed. \(\square \)
5 Concluding remarks
In this work, we analyze the solution regularity of a variable-order time-fractional diffusion equation with Mittag-Leffler kernel, based on which we derive the uniqueness of determining the variable fractional order based on the observations of the solutions over a small time-space interval. These theoretical results provide concrete supports for the applications of the proposed model in both the forward and backward manner.
It is worth mentioning that by similar and simper calculations, one could also analyze the solution regularity and the same inverse problem for the variable-order time-fractional diffusion equation with Caputo–Fabrizio derivative, which replaces the definition of the kernel \(\mathcal Q\) as [24]
More generally, one could extend the developed ideas and methods to analyze more complicated nonlocal problems that will be investigated in the near future.
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Elsevier, San Diego (2003)
Almeida, R., Tavares, D., Torres, D.F.M.: The Variable Order Fractional Calculus of Variations. Springer Briefs in Applied Sciences and Technology. Springer, Cham (2019)
Atangana, A., Alqahtani, R.: Numerical approximation of the space-time Caputo–Fabrizio fractional derivative and application to groundwater pollution equation. Adv. Differ. Equ. 156, 2016 (2016)
Coclite, G., Coclite, M.: On a model for the evolution of morphogens in a growing tissue III: \( \theta < \) log 2. J. Differ. Equ. 263, 1079–1124 (2017)
Coclite, G., Coclite, M.: Long time behavior of a model for the evolution of morphogens in a growing tissue II: \(\theta <\) log 2. J. Differ. Equ. 272, 1015–1049 (2021)
Coclite, G., Dipierro, S., Maddalena, F., Valdinoci, E.: Singularity formation in fractional Burgers equations. J. Nonlinear Sci. 30, 1285 (2020)
Ganji, R.M., Jafari, H., Baleanu, D.: A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel. Chaos Solitons Fractals 130, 109405 (2020)
Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)
Inc, M., Yusuf, A., Aliyu, A., Baleanu, D.: Investigation of the logarithmic-KdV equation involving Mittag-Leffler type kernel with Atangana–Baleanu derivative. Phys. A 506, 520–531 (2018)
Jin, B.: Fractional differential Equations: An Approach via Fractional Derivatives. Applied Mathematical Sciences vol. 206, Springer Cham (2021)
Jin, B., Rundell, W.: An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl. 28, 7501075028 (2012)
Kian, Y., Oksanen, L., Soccorsi, E., Yamamoto, M.: Global uniqueness in an inverse problem for time fractional diffusion equations. J. Differ. Equ. 264, 1146–1170 (2018)
Li, Z., Yamamoto, M.: Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation. Appl. Anal. 94, 570–579 (2015)
Li, G., Zhang, D., Jia, X., Yamamoto, M.: Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation. Inverse Probl. 29, 065014 (2013)
Liu, Y., Yamamoto, M.: Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by inexact data. arXiv:2206.02108 (2022)
Lorenzo, C., Hartley, T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)
Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)
Sekerci, Y., Ozarslan, R.: Respiration effect on Plankton-Oxygen dynamics in view of non-singular time fractional derivatives. Phys. A 553, 123942 (2020)
Srivastava, H., Choi, J.: 1 - Introduction and preliminaries, Editor(s): H.M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, pp.1–140 (2012)
Sun, H., Chang, A., Zhang, Y., Chen, W.: A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22, 27–59 (2019)
Van Bockstal, K., Hendy, A., Zaky, M.: Space-dependent variable-order time-fractional wave equation: existence and uniqueness of its weak solution. Quaest. Math. (2022). https://doi.org/10.2989/16073606.2022.2110959
Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)
Zaky, M., Bockstal, K., Taha, T., Suragan, D., Hendy, A.: An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion-reaction equations with fixed delay. J. Comput. Appl. Math. 420, 114832 (2023)
Zheng, X., Wang, H., Fu, H.: Well-posedness of fractional differential equations with variable-order Caputo–Fabrizio derivative. Chaos Solitons Fractals 138, 109966 (2020)
Zheng, X., Wang, H., Fu, H.: Analysis of a physically-relevant variable-order time-fractional reaction-diffusion model with Mittag-Leffler kernel. Appl. Math. Lett. 112, 106804 (2021)
Zheng, X., Wang, H.: Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains. Inverse Prob. Sci. Eng. 29, 1401–1411 (2021)
Zheng, X., Wang, H.: Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J. Numer. Anal. 41, 1522–1545 (2021)
Author information
Authors and Affiliations
Contributions
XG: writing—reviewing and editing, funding acquisition. XZ: conceptualization, methodology, formal analysis, writing—original draft.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was funded in part by the National Natural Science Foundation of China under Grant 12271303, by the Natural Science Foundation of Shandong Province for Excellent Youth Scholars ZR2020YQ02, and by Taishan Scholars Program of Shandong Province of China (tsqn201909044).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Guo, X., Zheng, X. Variable-order time-fractional diffusion equation with Mittag-Leffler kernel: regularity analysis and uniqueness of determining variable order. Z. Angew. Math. Phys. 74, 64 (2023). https://doi.org/10.1007/s00033-023-01959-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-023-01959-1