Inverse source problems for positive operators. I. Hypoelliptic diffusion and subdiffusion equations

TL;DR

This paper investigates inverse problems for positive operators in parabolic and fractional diffusion equations, establishing existence and uniqueness results, and applies these findings to various hypoelliptic and subelliptic models on Lie groups.

Contribution

It provides new theoretical results on inverse problems for a broad class of positive operators, including hypoelliptic and subelliptic cases, with applications to diverse diffusion models.

Findings

Proved existence and uniqueness of solutions for inverse problems in fractional diffusion.

Applied results to models on graded Lie groups, including Heisenberg group.

Numerical example demonstrating the cooling problem with involution.

Abstract

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.

Equations

$$u_t(x,t)+\mathcal{L}u(x,t)=f(x),$$

$$u(x,0)=\phi (x),x\in \bar{Q},$$

$${\mathcal{D}}_{0+,t}^{\alpha }u\left(x,t\right)+\mathcal{L}u(x,t)=f\left(x\right),$$

$$u\left(x,0\right)=\phi \left(x\right),x\in \bar{Q},$$

$$u\left(x,T\right)=\psi \left(x\right),x\in \bar{Q},$$

$${\mathcal{D}}_{0+,t}^{\alpha }u\left(x,t\right)+\mathcal{R}u(x,t)=f\left(x\right),$$

$$u\left(x,0\right)=\phi \left(x\right),x\in \mathbb{G},$$

$$u\left(x,T\right)=\psi \left(x\right),x\in \mathbb{G},$$

$${\mathcal{D}}_{0+,t}^{\alpha }u\left(x,t\right)-{\Delta }_{{\mathbb{H}}^n}u(x,t)=f\left(x\right),$$

$$u\left(x,0\right)=\phi \left(x\right),x\in {\mathbb{H}}^n,$$

$$u\left(x,T\right)=\psi \left(x\right),x\in {\mathbb{H}}^n,$$

$$u_t(x,t)+\mathcal{L}u(x,t)=f(x),$$

$${\mathcal{D}}_{0+,t}^{\alpha }u\left(x,t\right)+\mathcal{L}u(x,t)=f(x).$$

$${\mathcal{D}}_{0+,t}^{\alpha }u\left(x,t\right)+\mathcal{R}u(x,t)=f(x),$$

$$u(x,T)=\psi (x),x\in \bar{Q}.$$

$$\Vert v{\Vert }_{{\mathcal{H}}^s}:=\Vert (I+\mathcal{L}{)}^{s}v{\Vert }_{\mathcal{H}},$$

$$u(x,t)=\phi (x)+\sum _{\xi \in \mathcal{I}}\frac{[(\phi ,e_{\xi }{)}_{\mathcal{H}}-(\psi ,e_{\xi }{)}_{\mathcal{H}}](e^{-{\lambda }_{\xi }t}-1)e_{\xi }(x)}{(1-e^{-{\lambda }_{\xi }T})},$$

$$f(x)=\mathcal{L}\phi (x)-\sum _{\xi \in \mathcal{I}}\frac{[(\mathcal{L}\phi ,e_{\xi }{)}_{\mathcal{H}}-(\mathcal{L}\psi ,e_{\xi }{)}_{\mathcal{H}}]e_{\xi }(x)}{(1-e^{-{\lambda }_{\xi }T})}.$$

$$u(x,t)=\sum _{\xi \in \mathcal{I}}{u}_{\xi }(t)e_{\xi }(x),$$

$$f(x)=\sum _{\xi \in \mathcal{I}}{f}_{\xi }{e}_{\xi }(x),$$

$$u_{\xi }^{{}^{\prime }}(t)+{\lambda }_{\xi }{u}_{\xi }(t)=f_{\xi }.$$

$$u_{\xi }(t)=\frac{f_{\xi }}{{\lambda }_{\xi }}+C_{\xi }{e}^{-{\lambda }_{\xi }t},$$

$${\phi }_{\xi }=(\phi ,e_{\xi }{)}_{\mathcal{H}},$$

$${\psi }_{\xi }=(\psi ,e_{\xi }{)}_{\mathcal{H}}.$$

$$u_{\xi }(0)=\frac{f_{\xi }}{{\lambda }_{\xi }}+C_{\xi }={\phi }_{\xi },$$

$$u_{\xi }(T)=\frac{f_{\xi }}{{\lambda }_{\xi }}+C_{\xi }{e}^{-{\lambda }_{\xi }T}={\psi }_{\xi },$$

$$C_{\xi }=\frac{{\phi }_{\xi }-{\psi }_{\xi }}{1-e^{-{\lambda }_{\xi }T}}.$$

$$f_{\xi }={\lambda }_{\xi }{\phi }_{\xi }-{\lambda }_{\xi }{C}_{\xi }.$$

$$u(x,t)=\phi (x)+\sum _{\xi \in \mathcal{I}}{C}_{\xi }(e^{-{\lambda }_{\xi }t}-1)e_{\xi }(x).$$

$$(\mathcal{L}\phi ,e_{\xi }{)}_{\mathcal{H}}=(\phi ,\mathcal{L}{e}_{\xi }{)}_{\mathcal{H}}$$

Full text

The first author was supported in parts by the FWO Odysseus Project, EPSRC grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. The second author was supported by the Ministry of Education and Science of the Republic of Kazakhstan Grant AP05130994. The third author was supported by the Ministry of Education and Science of the Republic of Kazakhstan Grant AP05131756. No new data was collected or generated during the course of research.

1. 1 Introduction

2. 2 Inverse problem for the heat equation

3. 2.1 Statement of the problem

4. 2.2 Proof of the existence result

5. 2.3 Proof of the uniqueness result

6. 3 Inverse problem for the time-fractional diffusion equation

7. 3.1 Preliminaries

8. 3.2 Proof of Theorem 3.6

9. 3.2.1 Existence of solution

10. 3.2.2 Uniqueness of solution

11. 4 Inverse time-fractional diffusion problem for the hypoelliptic operators

12. 4.1 Graded Lie groups

13. 4.2 Proof of Theorem 4.2

14. 4.2.1 Proof of the existence result.

15. 4.2.2 Convergence of the formal solution.

16. 5 Appendix: Non-harmonic analysis of operators with discrete spectrum

17. 6 Examples

18. 7 Numerical illustrations

19. 7.1 Conclusion

1. Introduction

Many instances are known in which the practical needs lead to the problems of determining the coefficients or the right-hand-side of a differential equation from some available data about the solution. These are called the inverse problems of mathematical physics. Inverse problems arise in various areas of human activity such as seismology, mineral exploration, biology, medicine, quality control of industrial goods, etc. All these circumstances place inverse problems among the important problems of modern mathematics.

The first purpose of this paper is to study inverse problems for the heat equation. We consider the heat equation

[TABLE]

with Cauchy condition

[TABLE]

for $(x,t)\in\Omega_{t}=\{x\in Q\subset\mathbb{R}^{d},\,d\geq 1,\,\,t\in[0,T]\}$, where $\mathcal{L}$ is a linear self-adjoint positive operator with a discrete spectrum $\{\lambda_{\xi}>0:\xi\in\mathcal{I}\}$ on a separable Hilbert space $\mathcal{H}$. Here $Q$ is a bounded domain with a smooth boundary or unbounded domain. Respectively $\lambda_{\xi}$, the operator $\mathcal{L}$ has the system of orthonormal eigenfunctions $\{e_{\xi}:\xi\in\mathcal{I}\}$ on the separable Hilbert function space $\mathcal{H}$.

The problem of determination of temperature at interior points of a region when the initial and boundary conditions along with diffusion source term are specified are known as direct diffusion conduction problems. In many physical problems, determination of coefficients or right hand side (the source term, in case of the diffusion equation) in a differential equation from some available information is required; these problems are known as inverse problems. These kind of problems are ill posed in the sense of Hadamard. A number of articles address the analytical and numerical solvability of the inverse problems for the diffusion and anomalous diffusion equations (see [CD98, CNYY09, JR15, KS10, OS12a, OS12b, TT17, FIK14, IC16, KM11, KST17, NLN16, ZX11, ZW13, WYH13] and references therein).

The setting of a general operator $\mathcal{L}$ as in this paper allows one to include many models. A number of physical examples are discussed in Section 6, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group.

Section 3 is dedicated to finding the couple of functions $\left(u({x,t}),f(x)\right)$ satisfying the equation

[TABLE]

in the domain $(x,t)\in\Omega_{t}=\{x\in Q\subset\mathbb{R}^{d},\,d\geq 1,\,\,t\in[0,T]\},$ under the conditions

[TABLE]

[TABLE]

where $\mathcal{D}_{0+,t}^{\alpha}$ is a Caputo fractional derivative of order $0<\alpha\leq 1,$ $\varphi\left(x\right)$ and $\psi\left(x\right)$ are sufficiently smooth functions, and $\mathcal{L}$ is a linear positive operator with a discrete spectrum.

In many contexts, for example, in the sub-Riemannian settings, the (fractional) time diffusion equation for subelliptic operators arises naturally. However, such operators have a non-discrete, but continuous spectrum. Following Rothschild and Stein [RS76] and further developments, many of such operators can be modelled by the so-called Rockland operators (homogeneous left-invariant hypoelliptic differential operators) on graded Lie groups. By using the proceeding analysis, we will employ the group Fourier transform to reduce such problems to those with the discrete spectrum, for which the above established results would be applicable.

Thus, let $\mathcal{R}$ be a positive self–adjoint Rockland operator acting on $L^{2}({\mathbb{G}})$, where ${\mathbb{G}}$ is a graded Lie group of homogeneous dimension $Q\geq 3$. In a domain $\Omega=\{(t,x):\,(0,T)\times{\mathbb{G}}\}$ we seek a couple of functions $\left(u(t),f\right)$ satisfying the equation

[TABLE]

under the conditions

[TABLE]

[TABLE]

where $\mathcal{D}_{0+,t}^{\alpha}$ is a Caputo fractional derivative, $0<\alpha\leq 1$.

We seek a solution $u\in C([0,T];L^{2}({\mathbb{G}}))$ of the problem (1.4)–(1.6) such that $\mathcal{D}_{0+,t}^{\alpha}u\in C([0,T];L^{2}({\mathbb{G}}))$, $\mathcal{R}u\in C([0,T];L^{2}({\mathbb{G}}))$, and $f\in C([0,T];L^{2}({\mathbb{G}}))$. We are able to solve this problem by employing the natural global Fourier analysis on ${\mathbb{G}}$, allowing one to use the solution to the problem (1.3) applied to the infinitesimal representations of the operator $\mathcal{R}$, which is in turn known to have the discrete spectrum.

We note that the inverse source problems for Rockland operators (1.4)–(1.6) cover also the Heisenberg case, namely, the equation

[TABLE]

with the conditions

[TABLE]

[TABLE]

where $\mathcal{D}_{0+,t}^{\alpha}$ is a Caputo fractional derivative, $0<\alpha\leq 1$. Here, $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian on the Heisenberg group $\mathbb{H}^{n}$. As the physical application, the relevance of the Heisenberg group $\mathbb{H}^{n}$ for quantum mechanics has been established. In 1931 Weyl [Wey31] recognized that the Heisenberg algebra generated by the momentum and position operators originates from the representation of the Lie algebra associated with the corresponding group, namely, the Heisenberg group, or the Weyl group as the physicists call it. For the recent studies of the heat equation on the Heisenberg groups, we refer to the publications [BC17, DM05, Jui14, TW18, RTT19].

Thus, let us briefly summarise the results of this paper:

• •

Existence and uniqueness for the inverse diffusion problem

[TABLE]

for general positive operators $\mathcal{L}$ with discrete spectrum and the basis of eigenfunctions.

• •

Existence and uniqueness for the inverse time-fractional subdiffusion problem

[TABLE]

• •

Existence and uniqueness for the inverse diffusion and subdiffusion problems

[TABLE]

for $0<\alpha\leq 1$, and for general homogeneous left-invariant hypoelliptic differential operators $\mathcal{R}$ on graded Lie groups.

• •

Numerical analysis of the obtained formulae in the case of the inverse heat problems for differential operators with involution.

2. Inverse problem for the heat equation

2.1. Statement of the problem

The Section is concerned with inverse problem for the heat equation (1.1). We obtain existence and uniqueness results for this problem, based on the ${\mathcal{L}}$–Fourier method. An introduction and some basic definitions of the ${\mathcal{L}}$–Fourier analysis are given in [KRT17, DRT17, RT16, RT17a, RT18b].

Problem 2.1**.**

Find the couple of functions $(u(x,t),f(x))$ satisfying the Cauchy problem (1.1)-(1.2), under the condition

[TABLE]

A generalised solution of Problem 2.1 is the pair of functions $(u(x,t),f(x)),$ where $u=u(x,t)\in C^{1}([0,T],\mathcal{H})\cap C([0,T],\mathcal{H}^{1})$, and $f=f(x)\in\mathcal{H}$. Here $\mathcal{H}^{s}$ is the closure of $\mathcal{H}_{\mathcal{L}}^{\infty}$ (see, Appendix 5) under the norm

[TABLE]

for all $s\in\mathbb{R}_{+}$.

For the considered Problem 2.1, the following theorem holds true.

Theorem 2.2**.**

Let $\varphi,\psi\in\mathcal{H}^{1}$. Then the generalised solution of Problem 2.1 exists, is unique, and can be written in the form

[TABLE]

[TABLE]

2.2. Proof of the existence result

We want to find a generalised solution by the Fourier method, we have the eigenvalues $\lambda_{\xi}$ and eigenfunctions system $e_{\xi}(x)$ of operator $\mathcal{L}$ on the space $\mathcal{H}$. Eigenfunctions system $e_{\xi}(x)$ is an orthonormal basis in $\mathcal{H}$, the functions $u(x,t)$ and $f(x)$ can be expanded as follows:

[TABLE]

and

[TABLE]

where $f_{\xi},u_{\xi}(t)$ are unknown. Substituting Equations (2.2) and (2.3) into Equation (1.1), we obtain the following equation for the function $u_{\xi}(t)$ and the constant $f_{\xi}$:

[TABLE]

Solving this equation, we obtain

[TABLE]

where the constants $C_{\xi},f_{\xi}$ are unknown. To find these constants, we use conditions (1.2) and (2.1). Let $\varphi_{\xi},\psi_{\xi}$ be the coefficients of the expansions of $\varphi(x)$ and $\psi(x)$

[TABLE]

and

[TABLE]

We first find $C_{\xi}$:

[TABLE]

and

[TABLE]

then

[TABLE]

The constant $f_{\xi}$ is represented as

[TABLE]

Substituting $f_{\xi},u_{\xi}(t)$ into formulas (2.2) and (2.3), we find

[TABLE]

Using the self-adjointness property of the operator $\mathcal{L},$ we have

[TABLE]

we know that $\mathcal{L}e_{\xi}=\lambda_{\xi}e_{\xi}$ using this we obtain

[TABLE]

and for $\psi(x)$ we can write it in a similar way. Substituting these equality into formula of $C_{\xi}$ we can get that

[TABLE]

Then

[TABLE]

Similarly,

[TABLE]

Now, for the convergence of the series, using $\|e_{\xi}\|_{\mathcal{H}}=1$ and $\varphi,\psi\in\mathcal{H}^{1}$, we have the following estimate

[TABLE]

and

[TABLE]

From this and $\varphi,\psi\in\mathcal{H}^{1}$, we obtain

[TABLE]

Similarly for $f(x)$, we obtain the estimate

[TABLE]

Existence of the solution of Problem 2.1 is proved.

2.3. Proof of the uniqueness result

Suppose that there are two solutions $\{u_{1}(x,t),f_{1}(x)\}$ and $\{u_{2}(x,t),f_{2}(x)\}$ of Problem 2.1. Denote

[TABLE]

and

[TABLE]

Then the functions $u(x,t)$ and $f(x)$ satisfy Equation (1.1) and homogeneous conditions (1.2) and (2.1). We also have

[TABLE]

and

[TABLE]

Applying the operator $\frac{d}{dt}$ to (2.4), we have

[TABLE]

Thus we get

[TABLE]

an ordinary first-order differential equation. The general solution of this equation is:

[TABLE]

where $A_{\xi}$ and $f_{\xi}$ are unknown constants. Using the homogeneous conditions (1.2) and (2.1) we obtain following conditions:

[TABLE]

Using this we can find unknown constants $A_{\xi}$ and $f_{\xi}$. We first find $A_{\xi}$:

[TABLE]

Similarly, from

[TABLE]

we obtain

[TABLE]

this implies

[TABLE]

Further, by the completeness of the system $e_{\xi}$ in $\mathcal{H}$, we obtain $f(x)\equiv 0,u(x,t)\equiv 0.$ Uniqueness of the solution of the Problem 2.1 is proved.

3. Inverse problem for the time-fractional diffusion equation

The section deals with an inverse problem concerning the time-fractional diffusion equation.

3.1. Preliminaries

Now, to formulate the problem, we need to define fractional differentiation operators.

Definition 3.1** ([KST06]).**

The left and right Riemann–Liouville fractional integrals $I_{a+}^{\alpha}$ and $I_{b-}^{\alpha}$ of order $\alpha\in\mathbb{R}$ ($\alpha>0$) are given by

[TABLE]

and

[TABLE]

respectively. Here $\Gamma$ denotes the Euler gamma function.

Definition 3.2** ([KST06]).**

The left Riemann–Liouville fractional derivative $D_{a+}^{\alpha}$ of order $\alpha\in\mathbb{R}$ ($0<\alpha<1$) is defined by

[TABLE]

Similarly, the right Riemann–Liouville fractional derivative $D_{b-}^{\alpha}$ of order $\alpha\in\mathbb{R}$ ($0<\alpha<1$) is given by

[TABLE]

Definition 3.3** ([KST06]).**

The left and right Caputo fractional derivatives of order $\alpha\in\mathbb{R}$ ($0<\alpha<1$) are defined by

[TABLE]

and

[TABLE]

respectively.

Now, we are in a way to state our problem.

Problem 3.4**.**

Find the couple of functions $\left(u\left({x,t}\right),f\left(x\right)\right)$ satisfying the equation

[TABLE]

in the domain $(x,t)\in\Omega_{t}=\{x\in Q\subset\mathbb{R}^{d},\,d\geq 1,\,\,t\in[0,T]\},$ under the conditions

[TABLE]

[TABLE]

where $\varphi\left(x\right)$ and $\psi\left(x\right)$ are sufficiently smooth functions, $\mathcal{L}$ is a linear self-adjoint operator with a discrete spectrum.

If $\alpha=1,$ then equation (3.1) coincides with the classical heat equation. The heat equation also describes the diffusion process. So, the equation of the form (3.1) with fractional derivatives with respect to the time variable is called the sub-diffusion equation [Uch13]. This equation describes the slow diffusion. When $\alpha=\frac{1}{2}$ the equation was interpreted by Nigmatullin [Nig86] within a percolation (pectinate) model. The solution (in an unbounded domain in the space variable) was investigated by Mainardi [Mai00] and others by means of integral transformations.

Definition 3.5** ([CF18]).**

Let $X$ be a Banach space. We say that $u\in C^{\alpha}([0,T],X)$ if $u\in C([0,T],X)$ and $\mathcal{D}^{\alpha}_{t}u\in C([0,T],X).$

Observe that, viewed as a subspace of $C([0,T],X),$ the space $C^{\alpha}([0,T],X)$ is a Banach space.

A generalised solution of Problem 3.4 is the pair of functions $(u(x,t),f(x)),$ where $u=u(x,t)\in C^{\alpha}([0,T],\mathcal{H})\cap C([0,T],\mathcal{H}^{1})$ and $f=f(x)\in\mathcal{H}$.

For the considered Problem 3.4, the following theorem holds true.

Theorem 3.6**.**

Let $\varphi,\psi\in\mathcal{H}^{1}.$ Then the generalised solution of the Problem 3.4, exists, is unique, and can be written in the form

[TABLE]

[TABLE]

where $E_{\alpha,1}(z)$ is the Mittag-Leffler function:

[TABLE]

For different properties of the Mittag-Leffler function $E_{\alpha,1}(z)$ see e.g. [KST06].

3.2. Proof of Theorem 3.6

We give the full proof for Problem 2.1.

3.2.1. Existence of solution

We want to find a generalised solution by Fourier method, we have the eigenvalues $\lambda_{\xi}$ and eigenfunctions system $e_{\xi}(x)$ of operator $\mathcal{L}$ on the space $\mathcal{H}$. Eigenfunctions system $e_{\xi}(x)$ is an orthonormal basis in $\mathcal{H}$, so that the functions $u(x,t)$ and $f(x)$ can be expanded as follows:

[TABLE]

[TABLE]

where $u_{\xi}\left(t\right),f_{\xi}$ are unknown. Substituting (3.4) and (3.5) into (3.1), we obtain the following equations for the unknown functions $u_{\xi}\left(t\right)$ and the constants $f_{\xi}:$

[TABLE]

By solving this equation (see [KST06]), we obtain

[TABLE]

where the constants $C_{\xi}$ and $f_{\xi}$ are unknown and $E_{\alpha,1}(z)$ is the Mittag-Leffler function [KST06]:

[TABLE]

To find these constants, we use conditions (3.2). Let $\varphi_{\xi},\psi_{\xi}$ be the coefficients of the expansions of $\varphi\left(x\right)$ and $\psi\left(x\right)$

[TABLE]

where $(\cdot,\cdot)_{\mathcal{H}}$ is the inner product of the Hilbert space $\mathcal{H}.$

Then for $C_{\xi}$ we have

[TABLE]

[TABLE]

[TABLE]

Thus, we get

[TABLE]

The unknowns $f_{\xi}$ can be represented as

[TABLE]

Substituting $u_{\xi}\left(t\right),$ $f_{\xi}$ into (3.4) and (3.5), we find

[TABLE]

Using the self-adjointness property of the operator $\mathcal{L},$ we have

[TABLE]

we know that $\mathcal{L}e_{\xi}=\lambda_{\xi}e_{\xi}$ using this we obtain

[TABLE]

and for $\psi(x)$ we can write it in a similar way. Substituting these equalities into formula of $C_{\xi}$ we can get that

[TABLE]

Then

[TABLE]

Similarly,

[TABLE]

The following Mittag-Leffler function’s estimate is known [Sim14]:

[TABLE]

From this inequality it follows that

[TABLE]

Now, for the convergence of the series, using $\|e_{\xi}\|_{\mathcal{H}}=1$ and $\varphi,\psi\in\mathcal{H}^{1}$, we have the following estimate

[TABLE]

We also have

[TABLE]

From this and $\varphi,\psi\in\mathcal{H}^{1}$, we obtain

[TABLE]

Similarly for $f(x)$, we obtain the estimate

[TABLE]

Existence of the solution of the Problem 3.4 is proved.

3.2.2. Uniqueness of solution

Hence the obtained solution satisfies the equation (3.1) point-wise; by construction, it satisfies the conditions (3.2)-(3.3).

Suppose that there are two solutions $\left\{{u_{1}\left({x,t}\right),f_{1}\left(x\right)}\right\}$ and $\left\{{u_{2}\left({x,t}\right),f_{2}\left(x\right)}\right\}$ of Problem 3.4. Denote

[TABLE]

and

[TABLE]

Then the functions $u\left({x,t}\right)$ and $f\left(x\right)$ satisfy (3.1) and homogeneous conditions (3.2) and (3.3).

Let

[TABLE]

and

[TABLE]

Applying the operator $\mathcal{D}^{\alpha}_{0+}$ to the equation (3.6), we have

[TABLE]

By self-adjointness and taking into account the homogeneous conditions (3.2) and (3.3), we obtain

[TABLE]

Consequently, $f_{\xi}\equiv 0,u_{\xi}\left(t\right)\equiv 0.$

Further, by the completeness of the system $\left\{e_{\xi}(x)\right\}_{\xi\in\mathcal{I}}$ in $\mathcal{H}$ we obtain

[TABLE]

Hence, uniqueness of the solution of Problem 3.4 is proved.

4. Inverse time-fractional diffusion problem for the hypoelliptic operators

In this section we show how the obtained results can be applied for the analysis of the corresponding inverse problems for a large class of hypoelliptic diffusions and subdiffusions.

4.1. Graded Lie groups

We start by recalling following Folland and Stein [FS82] or [FR16, Section 3.1] and give some definitions and notations. A Lie algebra $\mathfrak{g}$ is graded if it is endowed with a vector space decomposition

[TABLE]

where all but finitely many of $V_{j}$’s are zero. Consequently, we call that a connected simply connected Lie group ${\mathbb{G}}$ is graded if its Lie algebra $\mathfrak{g}$ is graded. When the first stratum $V_{1}$ generates $\mathfrak{g}$ as an algebra, we get a stratified case of ${\mathbb{G}}$.

Graded Lie groups are homogeneous Lie groups with dilations. Define the operator $A$ by setting $AX=\nu_{j}X$ for $X\in V_{j}$. Then the dilations on $\mathfrak{g}$ are defined by

[TABLE]

The homogeneous dimension $Q$ of the graded Lie group ${\mathbb{G}}$ is defined by

[TABLE]

In what follows, we assume that ${\mathbb{G}}$ is a graded Lie group. Rockland operators are firstly defined in [Roc78] through the representations. By following [FR16, Definition 4.1.1], we call that $\mathcal{R}$ is a Rockland operator on the graded Lie group ${\mathbb{G}}$ if $\mathcal{R}$ is a left-invariant differential operator which is homogeneous of a positive order $\nu\in\mathbb{N}$ and satisfies the Rockland condition: for all representations $\pi\in\widehat{{\mathbb{G}}}$, excluding the trivial one, $\pi(\mathcal{R})$ is injective on $\mathcal{H}^{\infty}_{\pi}$, i.e., from

[TABLE]

it follows that $v=0$ for arbitrary $v\in\mathcal{H}^{\infty}_{\pi}$.

We denote by $\widehat{{\mathbb{G}}}$ the unitary dual of ${\mathbb{G}}$, $\mathcal{H}^{\infty}_{\pi}$ is the space of smooth vectors of the representation $\pi\in\widehat{{\mathbb{G}}}$, and $\pi(\mathcal{R})$ is the infinitesimal representation of $\mathcal{R}$, see [FR16, Definition 1.7.2]. For more information on graded Lie groups and Rockland operators the readers are referred to [FR16, Chapter 4].

Let $\pi$ be a representation of the graded Lie group ${\mathbb{G}}$ on the separable Hilbert space $\mathcal{H}_{\pi}$. We say that $v\in\mathcal{H}_{\pi}$ is smooth if the function

[TABLE]

is of class $C^{\infty}$. We denote by $\mathcal{H}_{\pi}^{\infty}$ the space of all smooth vectors of a representation $\pi$.

In the paper [HJL85] Hulanicki, Jenkins and Ludwig showed that the spectrum of $\pi(\mathcal{R})$ is purely discrete and positive. Hence we can choose an orthonormal basis for $\mathcal{H}_{\pi}$ such that the infinite matrix related to the (self-adjoint) operator $\pi(\mathcal{R})$ has the following form

[TABLE]

where $\pi\in\widehat{{\mathbb{G}}}\backslash\{1\}$ and $\pi_{j}\in\mathbb{R}_{>0}$.

For $f\in L^{1}({\mathbb{G}})$ and $\pi\in\widehat{G}$, let us define the group Fourier transform of $f$ at $\pi$ by

[TABLE]

with the integration against the biinvariant Haar measure on the graded Lie group ${\mathbb{G}}$, which implies that a linear mapping $\widehat{f}(\pi)$ from $\mathcal{H}_{\pi}$ to itself can be represented by an infinite matrix once we choose a basis for $\mathcal{H}_{\pi}$. Consequently, we obtain

[TABLE]

From now on, when we write $\widehat{f}(\pi)_{m,k}$, we will be using the same basis in $\mathcal{H}_{\pi}$ as the one giving (4.1).

In [CG90] by Kirillov’s orbit method, the authors showed that the Plancherel measure $\mu$ on $\widehat{{\mathbb{G}}}$ can be constructed explicitly. This means that we can have the Fourier inversion formula. Furthermore, $\pi(f)=\widehat{f}(\pi)$ is the Hilbert-Schmidt operator, i.e.

[TABLE]

and $\widehat{{\mathbb{G}}}\ni\pi\mapsto\|\pi(f)\|^{2}_{{\rm HS}}$ is an integrable function with respect to $\mu$. Moreover, the Plancherel formula holds (see e.g. [CG90] or [FR16]):

[TABLE]

In [HN79] Helffer and Nourrigat showed that a left-invariant differential operator $\mathcal{R}$ of homogeneous positive degree $\nu\in\mathbb{N}$ satisfies the Rockland condition if and only if it is hypoelliptic. Such operators we call Rockland operators.

The Sobolev spaces $H^{s}_{\mathcal{R}}({\mathbb{G}})$, $s\in\mathbb{R}$, associated to positive Rockland operators $\mathcal{R}$ have been analysed in [FR17] (see also [FR16]). One of the definitions of Sobolev spaces is

[TABLE]

with the norm $\|f\|_{H^{s}_{\mathcal{R}}({\mathbb{G}})}:=\|(I+\mathcal{R})^{s/\nu}f\|_{L^{2}({\mathbb{G}})},$ for a positive Rockland operator of homogeneous degree $\nu$. It is known that these spaces do not depend on a particular choice of a Rockland operator used in the above definition. We refer to [FR16] for details of the Fourier analysis on graded Lie groups.

Now we state the main problem of this section.

Problem 4.1**.**

Let ${\mathbb{G}}$ be a graded Lie group of homogeneous dimension $Q\geq 3$ and let $\mathcal{R}$ be a positive self–adjoint Rockland operator acting on $L^{2}({\mathbb{G}})$. In a domain $\Omega=\{(t,x):\,(0,T)\times{\mathbb{G}}\}$ we seek a couple of functions $\left(u(t),f\right)$ satisfying the equation

[TABLE]

under the conditions

[TABLE]

[TABLE]

where $\mathcal{D}_{0+,t}^{\alpha}$ is a Caputo fractional derivative, $0<\alpha\leq 1$.

We seek a solution $u\in C([0,T];L^{2}({\mathbb{G}}))$ of the problem (4.3)–(4.5) such that $\mathcal{D}_{0+,t}^{\alpha}u\in C([0,T];L^{2}({\mathbb{G}}))$, $\mathcal{R}u\in C([0,T];L^{2}({\mathbb{G}}))$, and $f\in C([0,T];L^{2}({\mathbb{G}}))$.

Theorem 4.2**.**

Assume that $\mathcal{R}$ is a positive Rockland operator of homogeneous order $\nu$. Let $\varphi,\psi\in H^{\nu}({\mathbb{G}}).$ Then there exists a unique solution $(u,f)$ such that $f\in L^{2}({\mathbb{G}})$, $u\in C([0,T];H^{\nu}({\mathbb{G}}))$, and $\mathcal{D}_{0+,t}^{\alpha}u\in C([0,T];L^{2}({\mathbb{G}}))$ of Problem 4.1, and can be written in the form

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

for all $\pi\in\widehat{{\mathbb{G}}}$ and $l,k\in\mathbb{N}$, where $E_{\alpha,1}(z)$ is the Mittag-Leffler function.

4.2. Proof of Theorem 4.2

4.2.1. Proof of the existence result.

We give a full proof of Problem 4.1. Let us take the group Fourier transform of (4.3) with respect to $x\in{\mathbb{G}}$ for all $\pi\in\widehat{G}$, that is,

[TABLE]

Taking into account (4.1), we rewrite the matrix equation (4.7) componentwise as an infinite system of equations of the form

[TABLE]

for all $\pi\in\widehat{{\mathbb{G}}}$, and any $l,k\in\mathbb{N}$. Now let us decouple the system given by the matrix equation (4.7). For this, we fix an arbitrary representation $\pi$, and a general entry $(l,k)$ and we treat each equation given by (4.8) individually.

According to [LG99], the solutions of the equations (4.8) satisfying initial conditions

[TABLE]

can be represented in the form

[TABLE]

[TABLE]

for all $\pi\in\widehat{{\mathbb{G}}}$ and any $l,k\in\mathbb{N}$, where $E_{\alpha,1}(z)$ is the Mittag-Leffler function [KST06]:

[TABLE]

Then there exists a solution of Problem 4.1, and it can be written as

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

for all $\pi\in\widehat{{\mathbb{G}}}$ and $l,k\in\mathbb{N}$.

We note that the above expressions ares well-defined in view of $\varphi,\psi\in\mathcal{H}^{\nu}({\mathbb{G}})$. Finally, based on (4.10), we rewrite our formal solution as (4.11).

4.2.2. Convergence of the formal solution.

Here, we prove convergence of the obtained infinite series corresponding to functions $u(x,t)$, $\mathcal{D}_{0+,t}^{\alpha}u(x,t)$, and $\mathcal{R}u(x,t)$.

Thus, since for any Hilbert-Schmidt operator $A$ one has

[TABLE]

for any orthonormal basis $\{\phi_{1},\phi_{2},\ldots\}$, then we can consider the infinite sum over $l,k$ of the inequalities provided by (4.10), we have

[TABLE]

[TABLE]

and

[TABLE]

Thus, integrating both sides of (4.14) against the Plancherel measure $\mu$ on $\widehat{{\mathbb{G}}}$, then using the Plancherel identity (4.2) we obtain

[TABLE]

[TABLE]

and

[TABLE]

Similarly for $f$, we obtain the estimate

[TABLE]

The uniqueness result can be proved in analogy to the previous arguments.

5. Appendix: Non-harmonic analysis of operators with discrete spectrum

In this section we recall the necessary elements of the global Fourier analysis that has been developed in [RT16], and its applications to the spectral properties of operators in [DRT17]. The space

[TABLE]

is called the space of test functions for $\mathcal{L}.$ Here we define

[TABLE]

where $\textrm{Dom}\left(\mathcal{L}^{k}\right)$ is the domain of the operator $\mathcal{L}^{k},$ in turn defined as

[TABLE]

The Fréchet topology of $\mathcal{H}^{\infty}_{\mathcal{L}}$ is given by the family of semi-norms

[TABLE]

Analogously to the operator $\mathcal{L}^{*}$ ($\mathcal{H}$-conjugate to $\mathcal{L}$), we introduce the space

[TABLE]

of test functions for $\mathcal{L^{*}},$ and we define

[TABLE]

where $\textrm{Dom}\left(\left(\mathcal{L^{*}}\right)^{k}\right)$ is the domain of the operator $\left(\mathcal{L^{*}}\right)^{k},$ in turn defined as

[TABLE]

The Fréchet topology of $\mathcal{H}^{\infty}_{\mathcal{L}^{*}}$ is given by the family of semi-norms

[TABLE]

Now the space

[TABLE]

of linear continuous functionals on $\mathcal{H}^{\infty}_{\mathcal{L}}$ is called the space of $\mathcal{L}^{*}$-distributions. We can understand the continuity here in terms of the topology (5.1). For $w\in\mathcal{H}^{-\infty}_{\mathcal{L}^{*}}$ and $w\in\mathcal{H}^{\infty}_{\mathcal{L}},$ we shall also write

[TABLE]

For any $\psi\in\mathcal{H}^{\infty}_{\mathcal{L}^{*}},$ the functional

[TABLE]

is an ${\mathcal{L}^{*}}$-distribution, which gives an embedding $\psi\in\mathcal{H}^{\infty}_{\mathcal{L}^{*}}\hookrightarrow\mathcal{H}^{-\infty}_{\mathcal{L}^{*}}.$

Since the system of eigenfunctions $\{e_{\xi}:\xi\in\mathcal{I}\}$ of the operator $\mathcal{L}$ is a Riesz basis in $\mathcal{H}$ then its biorthogonal system $\{e^{*}_{\xi}:\xi\in\mathcal{I}\}$ is also a Riesz basis in $\mathcal{H}$ (see e.g. Bari [Bar51], as well as Gelfand [Gel63]). Note that the function $e^{*}_{\xi}$ is an eigenfunction of the operator $\mathcal{L}^{*}$ corresponding to the eigenvalue $\bar{\lambda}_{\xi}$ for each $\xi\in\mathcal{I}.$ They satisfy the orthogonality relations

[TABLE]

where $\delta_{\xi,\eta}$ is the Kronecker delta.

If $\mathcal{L}$ is self-adjoint, we clearly have $e^{*}_{\xi}=e_{\xi}.$

6. Examples

Now as an illustration we give several examples of the settings where our inverse problems are applicable. Of course, there are many other examples, here we collect the ones for which different types of partial differential equations have particular importance.

• •

**Sturm-Liouville problem.

**First, we describe the setting of the Sturm-Liouville operator. Let $l$ be an ordinary second order differential operator in $L^{2}(a,b)$ generated by the differential expression

[TABLE]

and one of the boundary conditions

[TABLE]

or

[TABLE]

where $a^{2}_{1}+a^{2}_{2}>0,\,b_{1}^{2}+b_{2}^{2}>0$ and $\alpha_{j},\,\beta_{j},\,j=1,2$ some real numbers.

It is known [Nai68] that the Sturm-Liouville problem for equation (6.1) with boundary conditions (6.2) or with boundary conditions (6.3) is self-adjoint in $L^{2}(a,b).$ It is known that the self-adjoint problem has real eigenvalues and their eigenfunctions form a complete orthonormal basis in $L^{2}(a,b).$

• •

**Differential operator with involution.

**As a next example, we consider the differential operator with involution in $L^{2}(0,\pi)$ generated by the expression

[TABLE]

and homogeneous Dirichlet conditions

[TABLE]

where $|\varepsilon|<1$ some real number.

The nonlocal functional-differential operator (6.4)-(6.5) is self-adjoint [KST17, TT17]. For $|\varepsilon|<1,$ the operator (6.4)-(6.5) has the following eigenvalues

[TABLE]

and corresponding eigenfunctions

[TABLE]

• •

**Fractional Sturm-Liouville operator.

**We consider the operator generated by the integro-differential expression

[TABLE]

and the conditions

[TABLE]

where $\mathcal{D}_{a+}^{\alpha}$ is the left Caputo derivative of order $\alpha\in\left({1/2,1}\right],$ $D_{b-}^{\alpha}$ is the right Riemann-Liouville derivative of order $\alpha\in\left({1/2,1}\right]$ and $I_{b-}^{\alpha}$ is the right Riemann-Liouville integral of order $\alpha\in\left({1/2,1}\right]$ (see. Subsection 3.1 and [KST06]). The fractional Sturm-Liouville operator (6.6)-(6.7) is self-adjoint and positive in $L^{2}(a,b)$ (see [TT16]). The spectrum of fractional Sturm-Liouville operator (6.6)-(6.7) is discrete, positive and real valued, and the system of eigenfunctions is a complete orthogonal basis in $L^{2}(a,b).$ For more properties of the operator generated by the problem (6.6)-(6.7) we refer to [TT18, TT18d, TT18a].

• •

**Harmonic oscillator.

**For any dimension $d\geq 1$, let us consider the harmonic oscillator,

[TABLE]

${\mathcal{L}}$ is an essentially self-adjoint operator on $C_{0}^{\infty}(\mathbb{R}^{d})$. It has a discrete spectrum, consisting of the eigenvalues

[TABLE]

and with the corresponding eigenfunctions

[TABLE]

which are an orthogonal basis in $L^{2}(\mathbb{R}^{d})$. We denote by $P_{l}(\cdot)$ the $l$–th order Hermite polynomial, and

[TABLE]

where $\xi\in\mathbb{R}$, and

[TABLE]

For more information, see for example [NR10].

• •

**Anharmonic oscillator.

**Another class of examples – anharmonic oscillators (see for instance [HR82]), operators on $L^{2}(\mathcal{R})$ of the form

[TABLE]

for integers $k,l\geq 1$ and with $p(x)$ being a polynomial of degree $\leq 2l-1$ with real coefficients. Other examples are a large class of harmonic and anharmonic oscillators in all dimension, see e.g. [CDR18].

• •

**Landau Hamiltonian in 2D.

**The next example is one of the simplest and most interesting models of the Quantum Mechanics, that is, the Landau Hamiltonian.

The Landau Hamiltonian in 2D is given by

[TABLE]

acting on the Hilbert space $L^{2}(\mathbb{R}^{2})$, where $B>0$ is some constant. The spectrum of ${\mathcal{L}}$ consists of infinite number of eigenvalues (see [Foc28, Lan30]) with infinite multiplicity of the form

[TABLE]

and the corresponding system of eigenfunctions (see [ABGM15, HH13]) is

[TABLE]

where $L_{n}^{(\alpha)}$ are the Laguerre polynomials given by

[TABLE]

Note that in [RT17b, RT18, RT19a] the wave equation for the Landau Hamiltonian with a singular magnetic field was studied.

The reader is referred to [MRT19a, MRT19b, RT17c, RT18c, RT19b] for more examples of operators ${\mathcal{L}}$ and its applications.

• •

**The restricted fractional Laplacian.

**On the other hand, one can define a fractional Laplacian operator by using the integral representation in terms of hypersingular kernels already mentioned

[TABLE]

where $s\in(0,1).$

In this case we materialize the zero Dirichlet condition by restricting the operator to act only on functions that are zero outside bounded domain $\Omega\subset\mathbb{R}^{n}.$ Caffarelli and Siro [CS17] called the operator defined in such a way the restricted fractional Laplacian $\left(-\Delta_{\Omega}\right)^{s}.$ As defined, $\left(-\Delta_{\Omega}\right)^{s}$ is a self-adjoint operator on $L^{2}(\Omega),$ with a discrete spectrum $\lambda_{s,k}>0,\,\,k\in\mathbb{N}.$ The corresponding set of eigenfunctions $\{V_{s,k}(x)\}_{k\in\mathbb{N}},$ normalized in $L^{2}(\Omega).$

• •

**Hesienberg Harmonic and Anharmonic Oscillators.

**In [RR18] the authors studied classes of operators yielding harmonic and anharmonic oscillators on the Heisenberg group $\mathbf{H}_{n}$. These operators share one main feature: Every gradable Lie algebra equipped with a specific gradation admits a non-empty set of positive Rockland forms, consequently, one can associate to each positive Rockland form $P\in\mathfrak{u}(\mathfrak{h}_{n,2})$ a family of anharmonic oscillators $\{\mathfrak{A}_{\mathbf{H}_{n}}^{k}\}_{k\in\mathbb{R}\setminus\{0\}}$; the family of harmonic oscillators $\{\mathfrak{O}_{\mathbf{H}_{n}}^{k}\}_{k\in\mathbb{R}\setminus\{0\}}$ was included as the special case in which the Dynin-Folland Lie algebra $\mathfrak{h}_{n,2}$ was equipped with the canonical homogeneous structure related to the natural stratification and $P=-(X_{1}^{2}+\ldots X_{2n}^{2}+Y_{2n+1}^{2})$. Such operators have a discrete spectrum and also fall within the scope of this paper.

7. Numerical illustrations

In this section we provide some numerical simulations. We deal with the case of operator (6.4)-(6.5). Namely, in $L^{2}(0,\pi)$ consider the inverse source problem for the heat equation

[TABLE]

with boundary conditions

[TABLE]

and with the Cauchy problem

[TABLE]

with some extra information

[TABLE]

where $-1<\varepsilon<1$ is some real number.

By Theorem 2.2, from the Cauchy data $\varphi\in\mathcal{H}^{2}:=\dot{W}_{2}^{4}(0,\pi)$, that is, $\dot{W}_{2}^{4}(0,\pi):=\{\varphi,\varphi^{(4)}\in L^{2}(0,\pi)\,\,\,\hbox{with}\,\,\,\varphi(0)=\varphi(\pi)=\varphi^{\prime\prime}(0)=\varphi^{\prime\prime}(\pi)=0\},$ and from the observation function $\psi\in\dot{W}_{2}^{4}(0,\pi)$ we restore a unique solution $u\in C^{2}([0,T],L^{2}(0,\pi))\cap C([0,T],\dot{W}_{2}^{4}(0,\pi))$ and a source term $f\in L^{2}(0,\pi)$ by formulas

[TABLE]

and

[TABLE]

Here, we denote

[TABLE]

From Example (6.4)-(6.5), we have a representation for the eigenvalues

[TABLE]

and for the eigenfunctions

[TABLE]

In what follows, we consider and compare the cases $\varepsilon=0$ and $\varepsilon=0.9$. For numerical simulations, we choose

[TABLE]

for all $x\in[0,\pi]$. Our problem is to cool (make $u(x,T)=0$ at some time $T$) a rod with the initial temperature

[TABLE]

Now, a question stands as follows: how should we choose a cooling function (a source term) $f$ to succeed the goal?

For our calculations we use the following intermediate formulas:

[TABLE]

and

[TABLE]

Here $l\in\mathbb{N}$ means a number of terms taken into account in the series of the formulas (7.5)–(7.6). Here, we analyse the convergence of the series in (7.8)–(7.9) for different $l$.

In Figures 2, 3, 4 we compare the pair of solutions $(u,f)$ of the inverse problem (7.1)–(7.4) in the cases $\varepsilon=0$ and $\varepsilon=0.9$.

7.1. Conclusion

By analysing Figures 2, 3, 4, we observe that in the case of the problem with an involution (when $\varepsilon\neq 0$) we need more energy (see, Figure 3) to cool the rod with the initial temperature shown in Figure 1. In Figure 3 we can see the “cooling speed” of the rod. As the result, the absence of the involution guarantees relatively rapid cooling.