# On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data

Authors: Makhmud A. Sadybekov; Gulnar Dildabek; Marina B. Ivanova
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.


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## Body

## 1. Introduction and Statement of the Problem

The problems that imply the determination of coefficients or the right-hand side of a differential equation (together with its solution) are commonly referred to as inverse problems of mathematical physics. In this paper we consider one family of problems implying the determination of the density distribution and of heat sources from given values of initial and final distributions. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The mathematical statement of such problems leads to an inverse problem for the heat equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable.In this paper, we will consider the inverse problem close to that investigated in [1, 2]. Together with the solution it is necessary to find an unknown right-hand side of the equation. The equation contains the usual time derivative and an involution with respect to the spatial variable. In contrast to [1], we investigate the problem under nonlocal boundary conditions with respect to the spatial variable. The conditions for overdetermination are initial and final states.The second of the main differences in the investigated inverse problem being studied is that the unknown function enters, both in the right-hand side of the equation and in the conditions of the initial and final overdetermination.Let us consider a problem of modeling the thermal diffusion process which is close to that described in the report of Cabada and Tojo [2], where the example that describes a concrete situation in physics is given. Consider a closed metal wire (length 2π) wrapped around a thin sheet of insulation material in the manner shown in Figure 1.Figure 1
The closed metal wire wrapped around a thin sheet of insulation material.Assume that the positionx=0 is the lowest of the wire, and the insulation goes up to the left at -π and to the right up to π. Since the wire is closed, points -π and π coincide.The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified and to its right-hand side(∂2Φ/∂x2)(x,t) a third term ε(∂2Φ/∂x2)(-x,t) (where |ε|<1) is added. Here Φ(x,t) is the temperature at point x of the wire at time t.Thus, this process is described by the following:(1)Φtx,t-Φxxx,t+εΦxx-x,t=fxin the domain Ω=x,t:-π<x<π,0<t<T. Here f(x) is the influence of an internal source that does not change with time; t=0 is an initial time point and t=T is a final one.As the additional information we take values of the initial and final conditions of the temperature:(2)Φx,0=ϕx,Φx,T=ψx,x∈-π,π.Since the wire is closed, it is natural to assume that the temperature at the ends of the wire is the same at all times:(3)Φ-π,t=Φπ,t,t∈0,T.SinceΦ(x,t) is the temperature at point x of the wire at time t, average temperature along the entire length of the wire at time t can be calculated from formula Φ~(t)=∫-ππΦξ,tdξ. If additional heating (cooling) is applied through the boundary point x=-π, then the average temperature of the wire Φ~(t) increases (decreases). The greater the importance of this heating (cooling) is, the faster the average temperature of the wire Φ~(t) changes.Consider a process in which the temperature at one end at every time pointt is proportional to the change speed of the average value of the temperature throughout the wire. Then,(4)Φ-π,t=γddt∫-ππΦξ,tdξ,t∈0,T.Here γ is a proportionality coefficient.Thus the investigated process is reduced to the following mathematical inverse problem:Find the right-hand sidef(x)of the heat equation (1)and its solutionΦ(x,t)subject to the initial and final conditions (2),the boundary condition (3),and condition (4).


## 2. Reduction to a Mathematical Problem

Condition (4) is significantly nonlocal (i.e., values of the unknown function over the entire domain are used). The integral along inner lines of the domain is present in this condition. Using the idea of A.A. Samarskii, we transform this condition. Taking into account (1) from (4), we get(5)Φ-π,t=γ∫-ππΦξξξ,t-εΦξξ-ξ,t+fξdξ,t∈0,T.Hence(6)Φ-π,t=γ1-εΦxπ,t-Φx-π,t+γ∫-ππfξdξ,t∈0,T.Let us introduce the following notations:(7)ux,t=Φx,t-γ∫-ππfξdξ.Then in terms of a new functionu(x,t) we get the following inverse problem:In the domainΩ=x,t:-π<x<π,0<t<Tfind the right-hand sidef(x)of the heat equation with involution,(8)utx,t-uxxx,t+εuxx-x,t=fx,and its solutionu(x,t)satisfying the initial conditions,(9)ux,0=ϕx-γ∫-ππfξdξ,x∈-π,π,and the final conditions,(10)ux,T=ψx-γ∫-ππfξdξ,x∈-π,π,and the boundary ones,(11)ux-π,t-uxπ,t-auπ,t=0,u-π,t-uπ,t=0,t∈0,T,where ϕ(x) and ψ(x) are given sufficiently smooth functions; ε is a nonzero real number such that |ε|<1; and a=1/γ(ε-1).In the physical sense, the second of conditions (11) means the equality of the distribution densities at the ends of the interval. And the first of conditions (11) means the proportionality of the difference of fluxes across opposite boundaries to the density value at the boundary. We note that in [1] the Dirichlet boundary conditions u-π,t=uπ,t=0 were used instead of condition (11).The well-posedness of direct and inverse problems for parabolic equations with involution was considered in [3–5].The solvability of various inverse problems for parabolic equations was studied in papers of Anikonov Yu.E. and Belov Yu.Ya., Bubnov B.A., Prilepko A.I. and Kostin A.B., Monakhov V.N., Kozhanov A.I., Kaliev I. A., Sabitov K.B., and many others. These citations can be seen in our papers [6, 7]. In [1] there are good references to publications on related issues. We note [3, 8–28] from recent papers close to the theme of our article. In these papers different variants of direct and inverse initial-boundary value problems for evolutionary equations are considered, including problems with nonlocal boundary conditions and problems for equations with fractional derivatives.In this paper we shall use a spectral problem for an ordinary differential operators with involution. Such and similar spectral problems were considered in [29–40].Definition 1.
Bya regular solution of the inverse problem (8)-(11) we mean a pair of functions u(x,t),f(x) of the class u(x,t)∈Cx,t2,1(Ω¯), f(x)∈C[-π,π] that inverts (8) and conditions (9)-(11) into an identity.Definition 2.
Bya generalized solution of the inverse problem (8)-(11) we mean a pair of functions (u(x,t),f(x)) of the class u(x,t)∈W22,1(Ω)∩CΩ¯, f(x)∈L2-π,π that satisfies (8) and conditions (9)-(11) almost everywhere.When one uses the method of separation of variables to solve the problem, a spectral problem appears, which is mentioned in the next section.


## 3. Spectral Problem

The use of the Fourier method for solving problem (8)–(11) leads to a spectral problem for an operator L given by the differential expression,(12)LXx≡-X′′x+εX′′-x=λXx,-π<x<π,and the boundary conditions,(13)X′-π-X′π-aXπ=0,X-π-Xπ=0,where λ is a spectral parameter.The spectral problems for (12) were first considered, apparently in [37]. There was considered a case of Dirichlet and Neumann boundary conditions and cases of conditions in the form (13) for a=0. Here we consider the case a≠0. We assume that a>0.We represent a general solution of (12) in the following form:(14)Xx=Asin⁡μ1x+Bcos⁡μ2x,μ1=λ1+ε,μ2=λ1-ε,where A and B are arbitrary complex numbers. Satisfying the boundary conditions (13) for finding eigenvalues, we obtain the following:(15)sin⁡μ1π=0,tan⁡μ2π=a2μ2.Therefore, the spectral problem (12)-(13) has two series of eigenvalues:(16)λk,1=1+εk2,k∈N;λk,2=1-εk+δk2, k∈N0≡N∪0, where δk = Oa/(k+1)>0, as k→∞, with corresponding normalized eigenfunctions given by(17)Xk,1x=1πsinkx,k∈N;Xk,2x=νkcos⁡k+δkx,k∈N0.Here νk is a normalization coefficient: (18)νk-2=cos⁡k+δkxL2-π,π2=π+a2k+δka2+k+δk2π2.Lemma 3.
The system of functions (17) is complete and orthonormal in L2-π,π.Proof.
We note that the system of eigenfunctions (17) does not depend on the parameter ε. Only the eigenvalues of problem (12)-(13) depend on ε.
In the case whenε=0, system (17) is a system of eigenfunctions of the classical Strum-Liouville problem:(19)-X′′x=λXx,-π<x<π,with the self-adjoint boundary conditions (13).
Consequently, system (17) forms the complete orthonormal system in L2-π,π; that is, it is the orthonormal basis of the space.


## 4. Uniqueness of the Solution of the Problem

Let the pair of functionsu(x,t),f(x) be a solution of the inverse problem (8)-(11). Let us introduce notations:(20)uk,it=∫-ππux,tXk,ixdx,fk,i=∫-ππfxXk,ixdx,i=1,2.We apply the operatord/dt to uk,it. Then, using (8), by integrating by parts, we obtain the following problem:(21)uk,i′t+λk,iuk,it=fk,i,0<t<T,i=1,2.(22)uk,i0=ϕk,i-σk,i,i=1,2.(23)uk,iT=ψk,i-σk,i,i=1,2.Here we use the following notations:(24)ϕk,i=∫-ππϕxXk,ixdx,ψk,i=∫-ππψxXk,ixdx,σk,i=γ∫-ππfξdξ∫-ππXk,ixdx.It is easy to see that the functionu~k,1t=λk,i-1fk,i is a partial solution of the inhomogeneous equation (21). Therefore, we obtain a unique solution of the Cauchy problem (21)-(22) in the following form:(25)uk,it=ϕk,i-σk,i-fk,iλk,ie-λk,it+fk,iλk,i.Since|ε|<1, a>0, we have λk,i>0 for all k,i. Consequently,(26)1-e-λk,iT≥1-e-λ0,2T>0for  all  the  values  of  the  indexesk,i.Therefore, using the condition of "final overdetermination" (23), we get(27)fk,i=λk,iψk,i-ϕk,ie-λk,iT1-e-λk,iT-λk,iσk,i.Lemma 4.
Ifa>0, then the generalized solution (u(x,t),f(x)) of the inverse problem (8)-(11) is unique.Proof.
Suppose that there are two generalized solutions of the inverse problem (8)-(11): (u1(x,t),f1(x)) and (u2(x,t),f2(x)). Denote(28)ux,t=u1x,t-u2x,t,fx=f1x-f2x.Then the functions u(x,t) and f(x) satisfy (8), the boundary conditions (11), and the homogeneous conditions (9) and (10): (29)ux,0=-γ∫-ππfξdξ,x∈-π,π,ux,T=-γ∫-ππfξdξ,x∈-π,π.
Therefore, by using the notations (20) from (27), we find(30)fk,i=-λk,iσk,i≡-λk,iγ∫-ππfξdξ∫-ππXk,ixdx.
Since(31)λk,i∫-ππXk,ixdx=∫-ππ-Xk,i′′x+εXk,i′′-xdx=1-εXk,i′-π-Xk,i′π=a1-εXk,iπ,this gives(32)fk,i=-a1-εγ∫-ππfξdξXk,iπ.
SinceXk,1π=0, then for i=1 from (32), we have fk,1=0.
For the generalized solution of the problem (8)-(11) it is necessary that f∈L2-π,π. From this, by virtue of Parseval’s equality for orthogonal bases, we have that(33)∑k=1∞fk,22≤f2<∞is necessary.
Sincea(1-ε)≠0,(34)Xk,2π=νkcos⁡k+δkπ,limk→∞⁡νk=1π,limk→∞⁡δk=0,then (32) and inequality (33) are possible only at γ∫-ππfξdξ=0. Hence we obtain fk,2=0.
Therefore, using this result, from (25) and (27), we find(35)uk,it≡∫-ππux,tXk,ixdx=0,fk,i≡∫-ππfxXk,ixdx=0for all values of the indexes k∈N for i=1 and k∈N0 for i=2. Further, by the completeness of system (17) in L2-π,π, we obtain(36)ux,t≡0,fx≡0∀x,t∈Ω¯.The uniqueness of the generalized solution of the inverse problem (8)-(11) is proved.Corollary 5.
Ifa>0, then the regular solution (u(x,t),f(x)) of the inverse problem (8)-(11) is unique.


## 5. Construction of a Formal Solution of the Problem

As the eigenfunctions system (17) forms the orthonormal basis in L2-π,π, then the unknown functions u(x,t) and f(x) can be represented as(37)ux,t=∑k=1∞uk,1tXk,1x+∑k=0∞uk,2tXk,2x,(38)fx=∑k=1∞fk,1Xk,1x+∑k=0∞fk,2Xk,2x,where uk,1t and uk,2t are unknown functions; fk,1 and fk,2 are unknown constants.Substituting (37) and (38) into (8), we obtain the inverse problems (21)-(23). If the constants σk,i are assumed to be given, then the solutions of these inverse problems exist, are unique, and are represented by formulas (25) and (27). Substituting (25) and (27) into series (37) and (38), we obtain a formal solution of the inverse problem (8)-(11).Letϕ(x),ψ(x)∈W22-π,π and satisfy the boundary conditions (13). Since λk,i>0, then(39)∑k=1∞λk,iψk,i-ϕk,ie-λk,iT1-e-λk,iT2≤CϕW22-π,π2+ψW22-π,π2,i=1,2.Therefore, from the analysis of formula (27), it is easy to see that, in order for the formal solution (37) of the problem (8)-(11) to be a generalized solution, it is necessary that(40)limk→∞⁡λk,iσk,i=0,i=1,2holds.As above, we calculate(41)λk,iσk,i≡a1-εγ∫-ππfξdξXk,iπ,where Xk,1π=0 and(42)Xk,2π=νkcos⁡k+δkπ,limk→∞νk=1π,limk→∞δk=0.Thus, (40) holds if and only if σk,i=0 for all values of the indexes k,i. This is possible only in the case(43)∫-ππfξdξ=0.Thus, condition (43) is a necessary condition for the existence of the generalized solution of the inverse problem (8)-(11). In this case, problems (8)-(11) and (1)-(4) coincide. Indeed, from (43) and (1) we have(44)0=∫-ππfξdξ=∫-ππΦtξ,tdξ-∫-ππΦξξξ,t+εΦξξ-ξ,tdξ.For the first integral we apply condition (4) and calculate the second integral. Then we obtain(45)0=1-εΦx-π,t-Φxπ,t+1γ1-εΦ-π,t.This means that the boundary conditions (4) and (11) coincide. Hence, the problems (8)-(11) and (1)-(4) also coincide.Thus, in what follows, we shall consider problem (1)-(3) with the boundary condition:(46)Φx-π,t-Φxπ,t-aΦ-π,t=0.Thus, in what follows, we will consider the inverse problem (1)-(3), (46).Similarly, as before, the formal solution of this problem can be constructed in the form of series(47)Φx,t=∑k=1∞Φk,1tXk,1x+∑k=0∞Φk,2tXk,2x,(48)fx=∑k=1∞fk,1Xk,1x+∑k=0∞fk,2Xk,2x,where(49)Φk,it=ϕk,i-fk,iλk,ie-λk,it+fk,iλk,i,(50)fk,i=λk,iψk,i-ϕk,ie-λk,iT1-e-λk,iT.In order to complete our study, it is necessary, as in the Fourier method, to justify the smoothness of the resulting formal solutions and the convergence of all appearing series.


## 6. Main Results

Here we present the existence and uniqueness results for our inverse problem.Theorem 6 (leta>0).
(A) Letϕ(x),ψ(x)∈W22-π,π and let the functions ϕ(x),ψ(x) satisfy the boundary conditions (13). Then for a real number ε such that |ε|<1 the inverse problem (1)-(3), (46) has a unique generalized solution, which is stable in norm:(51)ΦtL2Ω2+ΦxxL2Ω2+fL2-π,π2≤CϕW22-π,π2+ψW22-π,π2,where the constant C does not depend on ϕ(x) and ψ(x).
(B) Letϕ(x),ψ(x)∈C4-π,π and the functions ϕ(x),ψ(x) and ϕ′′(x), ψ′′(x) satisfy the boundary conditions (13), then for a real number ε such that |ε|<1 the inverse problem (1)-(3), (46) has a unique regular solution.Proof.
Since|ε|<1, a>0, we have λk,i>0 for all k,i. Consequently, there exists a constant β>0, such that for all the values of the indexes k,i we have(52)1-e-λk,iT≥1-e-λ0,2T≥β>0.
Sinceλk,i>0, then e-λk,iT≤e-λk,it<1. Therefore, from representations (49) and (50), we get estimates(53)fk,i≤λk,iβϕk,i+ψk,i,(54)Φk,it≤1+2βϕk,i+ψk,i,where constant β does not depend on the indexes k,i and on the functions ϕ(x) and ψ(x).
As proved in Lemma3, the eigenfunctions system (17) forms the orthonormal basis in L2-π,π. Therefore, we have an expansion:(55)ϕx=∑k=1∞ϕk,1Xk,1x+∑k=0∞ϕk,2Xk,2x.Taking into account (12), we obtain(56)-ϕ′′x+εϕ′′-x=∑k=1∞λk,1ϕk,1Xk,1x+∑k=0∞λk,2ϕk,2Xk,2x.
From this, by Parseval’s equality, it is easy to obtain an estimate:(57)∑k=1∞λk,12ϕk,12+∑k=1∞λk,22ϕk,22≤21+ε2ϕ′′L2-π,π2.
Similarly, for the coefficients of the expansion,(58)ψx=∑k=1∞ψk,1Xk,1x+∑k=0∞ψk,2Xk,2xwe have the inequality(59)∑k=1∞λk,12ψk,12+∑k=1∞λk,22ψk,22≤21+ε2ψ′′L2-π,π2.
Now, from (53), (57), and (59), we see that the series (48) converges and the estimate(60)fL2-π,π2≤21+ε2β2ϕ′′L2-π,π2+ψ′′L2-π,π2holds.
Taking into account (12), from (47), we obtain(61)-Φxxx,t+εΦxx-x,t=∑k=1∞λk,1Φk,1tXk,1x+∑k=0∞λk,2Φk,2tXk,2x.Substituting the value (-x) instead of x, we find(62)-Φxx-x,t+εΦxxx,t=-∑k=1∞λk,1Φk,1tXk,1x+∑k=0∞λk,2Φk,2tXk,2x.Here we use the properties Xk,1-x=-Xk,1x, Xk,2-x=Xk,2x obtained from the explicit form (12) of the eigenfunctions.
Multiplying (62) by ε and summing with (61), we have(63)-1+ε2Φxxx,t=1-ε∑k=1∞λk,1Φk,1tXk,1x+1+ε∑k=0∞λk,2Φk,2tXk,2x.
Since|ε|<1, from this, by Parseval’s equality, we find(64)Φxx·,tL2-π,π2=11+ε2∑k=1∞λk,12Φk,1t2+11-ε2∑k=0∞λk,22Φk,2t2for each t∈[0,T]. Integrating this equation with respect to t∈[0,T], taking (54) into account, we have(65)ΦxxL2Ω2≤8T1-ε221+2β2∑i=12∑k=1∞λk,i2ϕk,i2+ψk,i2.
Hence, using (57) and (59), we obtain(66)ΦxxL2Ω2≤16T1+ε21-ε221+2β2ϕ′′L2-π,π2+ψ′′L2-π,π2.
Now we can easily obtain an estimate forΨtx,t from (1). This fact together with (60) and (66) gives the necessary estimate (51) for the solution.
From the obtained estimates it also follows that in the constructed by us formal solution of the inverse problem all the series converge, they can be term-by-term differentiated, and the series obtained during differentiation also converge in sense of the metricsL2.
From (47) and (54), by using the Holder’s inequality, it is easy to justify the inequality(67)maxx,t∈Ω¯⁡Φx,t2≤Cϕ′′L2-π,π2+ψ′′L2-π,π2,which justifies the continuity of Φx,t in the closed domain Ω¯.
From the representation of the solution in the form of series (37), (38) and inequalities (53), (54) it is easy to justify the estimates(68)Φxxx,t+Φtx,t+fx≤C∑k=1∞λk,i2ϕk,i+ψk,i.Let ϕ(x),ψ(x)∈C4-π,π and let the functions ϕ(x), ψ(x) and ϕ′′(x), ψ′′(x) satisfy the boundary conditions (13), then the number series in the right-hand side of (68) converges. Therefore, in such case the constructed by us formal solution gives the regular solution of the inverse problem (1)-(3), (46).
The uniqueness of this constructed solution follows from Lemma4.
The theorem is completely proved.



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