# Moving Heat Source Reconstruction from the Cauchy Boundary Data

Authors: Nilson C. Roberty; Marcelo L. S. Rainha
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

We consider the problem of reconstruction of an unknown characteristic transient
thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert
spaces, we can treat the problem with methods similar to those used in the analysis of the stationary
source reconstruction problem. Further, the finite differenceθ scheme applied to the
transient heat conduction equation leads to a model based on a sequence of modified Helmholtz
equation solutions. For each modified Helmholtz equation the characteristic star-shape source
function may be reconstructed uniquely from the Cauchy boundary data. Using representation
formula, we establish reciprocity functional mapping functions that are solutions of the modified
Helmholtz equation to their integral in the unknown characteristic support.


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## Body

## 1. Introduction

Inverse source transient heat problem has been studied by a huge number of authors. In relation to books with specific chapters in the subject, we can give special attention to Anger [1] and Isakov [2]. Those gives specific results for the problem of source reconstruction in models with different operators and overspecification of boundary conditions, and specifically demonstrates an uniqueness theorem related with the moving characteristic source studied in this work. Early works by Cannon and Pérez Esteva [3] studied stationary support reconstruction under hypotheses of a known intensity function, f(t,x)=f(t)χω(x). Some years later, Cannon and Pérez Esteva studied the same problem in a three-dimensional case [4]. More recently, Lefevre and Le Niliot [5] used the boundary elements method for identification of static and moving point sources. Also, it is important to mention the authors Hettlich and Rundell [6] who model the unmoving characteristic domain and El Badia and Ha Duong [7], whose stationary source reconstruction and the transient point sources reconstructions have a fundamental influence in the present work. The present work has come from the investigation of the stationary source reconstruction by the fundamental solution method in Alves et al. [8]. The adoption of the reciprocity gap functional method to solve the stationary source in the Laplace Poisson equation, Roberty and Alves [9], and the solution of the full identification of sources with the Helmholtz Poisson model, Alves et al. [10], have developed to the modeling adopted to the transient heat transient characteristic source reconstruction in this work. The model is based on the modified Helmholtz Poisson equation that is obtained from the transient equation through the θ-scheme related to time finite differences discretization. Analysis of the related mathematical and computational work involved has been presented by the author in national conferences, Roberty and Alves [11, 12], Roberty and Sousa [13], and Roberty and Rainha [14, 15].This paper will be structured as follows. Some definitions and mathematical results extracted from Lions and Magenes [16] that are important to the understanding of the inverse problems are presented in Section 2. There we introduce the concept of consistent Cauchy data, extended Dirichlet-to-Neumann map and the Green function formalism. The inverse transient heat source problem is introduced in Section 3. Basic lemma and theorems about the relative extended Dirichlet-to-Neumann map, existence of solution, reciprocity gap functional in the transient model, uniqueness of solution are demonstrated and discussed. These concepts present original aspects that we are introducing in the present work. We show that the transient problem can be studied with aid of results demonstrated for the Modified Helmholtz Dirichlet Source problem, in Section 4. There the iterative source reconstruction scheme, the uniqueness theorem for these sequences of stationary problems, and the reciprocity gap methodology is presented. We conclude by pointing out the advances introduced by the present work.


## 2. Direct Transient Heat Equation Problem

ByΩ⊂ℜd, d=1,2,3 we denote a bounded space domain with Lipschitz boundary Γ=∂Ω, which means that Ω is locally on one side of its connected boundary. The boundary may be locally parametrized by a Lipschitz continuous function. The more regular boundary will be named as smooth and will be locally parametrized with C∞ functions. In the spatial surface Γ the normal ν is defined almost everywhere and the induced measure on the surface is denoted by dσ. In the time-space ℜd+1, we consider the time interval I:=(0,T), T>0 to form the bounded cylinder Q:=I×Ω, whose lateral time-space surface is Σ:=I×Γ. A section in this cylinder is Ωt:={t}×Ω and the complete cylinder boundary is (2.1)∂Q=Σ¯∪Ω0∪ΩT,
where Ω0 and ΩT are, respectively, the cylinders' bottom and top sections. At cylinder top and bottom there exist the corners Γ0=Ω0¯∩Σ¯⊂ℜd-1 and ΓT=ΩT¯∩Σ¯⊂ℜd-1, respectively.The direct transient heat source initial boundary value problem consists in findingu(t,x) with (t,x)∈Q given a boundary input g(t,x) with (t,x)∈Σ¯, an initial input u0(x) with (t,x)∈Ω0 and a source distribution f(t,x) with (t,x)∈Q that verifies the problem:(2.2)(Pu0,g,f){∂tu-Δu=finQ,u=u0inΩ0,u=gonΣ,
and Dirichlet data compatibility condition, u0=g at the time-space cylinder corner Γ0.It is well known that this direct problem is well posed with a classical unique solution, Isakov introduced by [2], for initial and boundary data and coefficients in spaces of continuous and Holder continuous functions. For Hilbert space framework we need to introduce, following Lions and Magenes [16], anisotropic Sobolev spaces. For r, s>0(2.3)Hr,s(Q):=L2(I;Hr(Ω))∩Hs(I;L2(Ω))
and the associated lateral boundary spaces (2.4)Hr,s(Σ):=L2(I;Hr(Γ))∩Hs(I;L2(Γ)).
Here Hr(Ω) and Hr(Γ), r≥0 are the Hilbert family of Sobolev space in the L2 theory, and the space L2(I;X) denotes the class of functions that are strongly measurable on I=[0,T] with range in X with the following Hilbert norm: (2.5)‖v‖L2(I;X)=(∫I‖v‖X2dt)1/2<∞.
The normal space with null lateral boundary trace will be (2.6)H0,•r,s(Q):=L2(I;H0r(Ω))∩Hs(I;L2(Ω))⊂Hr,s(Q).
A comprehensive presentation of these spaces in the contest of boundary integral operators related with the heat equation and the heat potential can be found in Costabel [17]. The adjoint transient heat problem has a straightforward definition(2.7)(PvT,g,f*){-∂tv-Δv=finQ,v=vTinΩT,v=gonΣ,
and Dirichlet data compatibility condition, vT=g, at the time-space cylinder corner ΓT. The time reversal operator(2.8)κT:Hr,s(Q)⟶Hr,s(Q),v(t,x)⟼κ[v](t,x)=v(T-t,x)
can be used to change changes of variables u*(t,x)=v(T-t,x) and convert the adjoint problem into an equivalent direct problem. The following theorems, lemmas and proposition are important in the well posing of the direct and inverse problems related with the source reconstruction that we are investigating. We adapt the notation in order to make easy the use of these results and address the proofs to the related part in the book Lions and Magenes [16].Theorem 2.1 (first trace theorem).
Foru∈Hr,s(Q) with r>1/2 and s≥0. We can define the continuous and linear mappings: (1)
γ[u]:=u|Σ from Hr,s(Q)→Hr-1/2,s-s/2r(Σ) and(2)
γ1[u]:=∂νu|Σ from Hr,s(Q)→Hr-3/2,s-3s/2r(Σ).
Also, forr≥0 and s>1/2 the continuous mapping
(1)
γ0[u]:=u(0,x) from Hr,s(Q)→Hr-r/2s(Ω0) and(2)
γT[u]:=u(T,x) from Hr,s(Q)→Hr-r/2s(ΩT).Proof.
See Theorem 2.2 on Section 4 of Lions and Magenes [16].Remark 2.2 (nonsurjectivity of traces).
The following mappings are not onto:(1)
γ0,γT:Hr,s(Q)→Hr-r/2s(Ω0) if r>1/2;(2)
γ:Hr,s(Q)→Hr-1/2,s-s/2r(Σ) if s>1/2;(3)
γ1:Hr,s(Q)→Hr-3/2,s-3s/2r(Σ) if s>3/2.Two important spaces for applications are H2,1(Q) and H1,1/2(Q). For the first, only the normal trace γ1 is surjective. For the second, both the traces on lateral boundary are surjective, that is, γ and γ1. In both cases the traces on initial and final time are not onto.Since we need traces for treating nonhomogeneous problems such as (2.2), Pu0,g,f with its associated inverse source problem, we will introduce the following definitions.Definition 2.3 (Cauchy datum).
By Cauchy datum associated with Problem (2.2) we mean the functions:
(2.9)(u0,g,uT,gν)∈Hr-r/2s(Ω0)×Hr-1/2,s-s/2r(Σ)×Hr-r/2s(ΩT)×Hr-3/2,s-3s/2r(Σ).Definition 2.4 (consistent Cauchy datum).
By consistent Cauchy datum associated with Problem (2.2) we mean the functions:
(2.10)(u0,g,uT,gν)∈(γ0,γ,γT,γ1)[Hr,s],
and by the first trace Theorem 2.1, we have that
(2.11)(γ0,γ,γT,γ1)[Hr,s]⊂Hr-r2s(Ω0)×Hr-12,s-s2r(Σ)×Hr-r2s(ΩT)×Hr-32,s-3s2r(Σ).
Now, since the consistent datum are in the range of the trace operators, the nonhomogeneous problem will be always well posed. From now on, by Cauchy data we will always mean consistent Cauchy data.The well posedness of the problem holds for regular boundaries and may be resumed as follows.Theorem 2.5 (regular datar≥0).
Consider the problemPu0,g,f.
The regular nonhomogeneous problem with consistent data(2.12)(u0,g,f)∈H2(r+1/2)(Ω0)×H2r+3/2,r+3/4(Σ)×H2r,r(Q),r≥0,2r,r∉Z+12
and appropriated compatibility relations in corner Γ0 admit a unique solution u∈H2r+2,r+1(Q). The associated mapping G:(u0,g,f)→u is continuous. Note that for these initial time and Dirichlet data the final time and normal trace are
(2.13)(γT[u],γ1[u])∈H2(r+1/2)(ΩT)×H2r+1/2,r+1/4(Σ).Proof.
See Theorem 6.2 in Section 4 of Lions and Magenes [16].Theorem 2.6 (distributional data-1<r<0).
Consider the problemPu0,g,f.
The distributional nonhomogeneous problem with consistent data(2.14)(u0,g,f)∈H2(r+1/2)(Ω0)×H2r+3/2,r+3/4(Σ)×H2r,r(Q),-1<r<0,
and appropriated compatibility relations in corner Γ0 admit a unique solution u∈H2r+2,r+1(Q). The associated mapping G:(u0,g,f)→u is continuous. Note that for these initial time and Dirichlet data the final time and normal trace are
(2.15)(γT[u],γ1[u])∈H2(r+1/2)(ΩT)×H2r+1/2,r+1/4(Σ).
Here we have defined
(2.16)Hβ(Ω)={Hβ(Ω)ifβ≥0,(H-β(Ω))′ifβ≤0,Hα,α2(Ω)={Hα,α/2(Ω)ifα≥0,(H-α,-α/2(Ω))′ifα≤0.Proof.
See Subsection 15.1 on Section 4 of Lions and Magenes [16].Definition 2.7 (the dual operator domainXr+1(Q)).
Ifr≥0 with 2r and r∉ℤ+1/2,
(2.17)Xr+1(Q)={v∈H2(r+1),r+1(Q),γ[v]=0,γT[v]=0,-∂tv-Δv∈H0,02r,r(Q)},
where H0,02r,r(Q) is the closure of 𝒟(Q) in H2r,r(Q).Lemma 2.8 (adjoint isomorphis of orderr+1).
Xr+1(Q) with the graph norm
(2.18)‖v‖Xr+1(Q)=(‖v‖H2(r+1),r+1(Q)2+‖(-∂t-Δ)v‖H2r,r(Q)2)1/2
is a Hilbert space. For r≥0 and with 2r and r∉ℤ+1/2, the adjoint operator -∂t-Δ is an isomorphism of Xr+1(Q) onto H0,02r,r(Q).Proof.
See Section 7.2 on Section 4 of Lions and Magenes [16].Definition 2.9 (the spaceΞ2(r+1),r+1(Q)).
By definitionXr+1(Q)⊂H2(r+1),r+1(Q). We need a space Ξ2(r+1),r+1(Q) with the following properties: (1)
the space𝒟(Q) is dense in Ξ2(r+1),r+1(Q) for r>-1;(2)
Xr+1(Q)⊂H2(r+1),r+1(Q)⊂Ξ2(r+1),r+1(Q) for r≥0, 2r and r∉ℤ+1/2;(3)
it is least space with these two properties.
This space is independent of properties of the boundary operators and is defined in Section 9.1 of Lions and Magenes [16] by using functions equivalent to the distance to the boundary and interpolation Theorems.Definition 2.10 (the dual spaceΞ-2(r+1),-(r+1)(Q)).
The dual space(2.19)Ξ-2(r+1),-(r+1)(Q)=(Ξ2(r+1),r+1(Q))′
for r≥-1 is a distribution-space-appropriated definition of continuous linear forms on Xr+1(Q) and consequently for source definition.Theorem 2.11 (distributional datar≤-1).
Consider the problemPu0,g,f.
The distributional nonhomogeneous problem with consistent data(2.20)(u0,g,f)∈Ξ2(r+1/2)(Ω0)×H2r+3/2,r+3/4(Σ)×Ξ2r,r(Q),r≤-1,2r,r∉Z+12
and appropriated compatibility relations in corner Γ0 admit a unique solution u∈H2r+2,r+1(Q). The associated mapping G:(u0,g,f)→u is continuous. Note that for these initial time and Dirichlet data the normal trace is
(2.21)(γT[u],γ1[u])∈Ξ2(r+1/2)(ΩT)×H2r+1/2,r+1/4(Σ)Proof.
See Theorem 12.1 of Lions and Magenes [16].The following theorems represent an effort based on domain and range operator modification found on Chapter 4 of the book Lions and Magenes [16], in order to make traces mappings onto. We include them to illustrate the complexity involved. Since our main commitment is with the inverse source problem, we consider it more efficient to simplify the process to assure Cauchy data consistence by restricting its range in the first trace Theorem 2.1.Definition 2.12 (the domain spaceD∂t-Δ-r(Q)).
One has(2.22)D∂t-Δ-r(Q)={u∈H-2r,-r(Q),(∂t-Δu)∈Ξ-2(r+1),-(r+1)(Q)}
with r≥0, 2r and r∉ℤ+1/2. Provided the graph norm
(2.23)‖v‖D∂t-Δ-r(Q)=(‖v‖H-2r,-r(Q)2+‖(∂t-Δ)v‖Ξ-2(r+1),-(r+1)(Q)2)1/2,
it is a Hilbert space.Theorem 2.13.
The mappings(2.24)γ:D∂t-Δ-r(Q)⟶H-2r-1/2,-r-1/4(Σ),γ1:D∂t-Δ-r(Q)⟶H-2r-3/2,-r-3/4(Σ)
for every r≥0 with 2r and r∉ℤ+1/2 are continuous and linear. Further these mappings are onto if their domains and range are restricted to
(2.25)γ:V(Q)⟶H0,•2r+1/2,r+1/4(Σ),γ1:V(Q)⟶H0,•2r+3/2,r+3/4(Σ),
where 𝒱={v∈H0,•2(r+1),r+1(Q),(-∂t-Δ)v∈H2(r-1),r-1(Q)}.Theorem 2.14.
The mappings(2.26)γ0:D∂t-Δ-r(Q)⟶H-2r+1/2(Ω0),γT:D∂t-Δ-r(Q)⟶H-2r+1/2(ΩT)
for every r≥0 with 2r and r∉ℤ+1/2 are continuous and linear. Further these mappings are onto if their domains and range are restricted to
(2.27)γ0:V1(Q)⟶H02r+1/2(Ω0),γT:V1(Q)⟶H02r+3/2(ΩT),
where 𝒱1={v∈H2(r+1),r+1(Q),(-∂t-Δ)v∈H0,02(r-1),r-1(Q)}.Proof.
See Theorem 10.4 and Lemma 10.2 in Section 4 of Lions and Magenes [16].Remark 2.15.
The data in the lateral and bottomd-dimensional surfaces of time-space cylinder can be considered as Dirichlet prescribed boundary data in the extended boundary Ω0∪Σ¯ of the transient heat equation problem. This set is adjoint by the extended boundary ΩT∩Σ¯ for the adjoint problem. Since the transient heat problem has one derivative in time and two derivatives in space, the problem and its adjoint are posed with the following set of Cauchy data: prescribed only Dirichlet on the bottom Ω0 and on the top ΩT of the cylinder and both Dirichlet and Neumann data at the lateral cylinder surface Σ. If there exists data compatibility at corners Γ0 and ΓT, it will give to the transient heat problem a character similar to the Poisson Laplace equation.Lemma 2.16 (solution operator).
The solution operator is a continuous but not injective linear operator associated with the Direct Problem (2.2) Pu0,g,f(1)
for regular data withr≥0 and if 2r and r∉ℤ+1/2(2.28)S:H2(r+1/2)(Ω)×H2r+3/2,r+3/4(Σ)×H2r,r(Q)⟶H2r+2,r+1(Q),(2)
for distributional data with-1<r<0(2.29)S:H2(r+1/2)(Ω)×H2r+3/2,r+3/4(Σ)×H2r,r(Q)⟶H2r+2,r+1(Q),(3)
for distributional data withr≤-1 and if 2r and r∉ℤ+1/2(2.30)S:Ξ2(r+1/2)(Ω)×H2r+3/2,r+3/4(Σ)×Ξ2r,r(Q)⟶H2r+2,r+1(Q),
defined by
(2.31)S(u0,g,f)=u
when u∈H2r+2,r+1(Q) is solution of Problem (2.2) with initial data (u0,g)=(u|Ω0,u|Σ).Proof.
It is a combination of results found in Theorem 6.2, Subsection 15.1 and Theorem 12.1 on Chapter 4 of Lions and Magenes [16]. Note that operator S is continuous, but not necessarily one to one.Definition 2.17 (extended dirichlet-to-neumann map).
We call The extended Dirichlet-to-Neumann map for the Problem (2.2) (1)
for regular data withr≥0 and 2r and r∉ℤ+1/2:
(2.32)ΛΩ,Σf:H2r+1(Ω0)×H2r+3/2,r+3/4(Σ)⟶H2r+1(ΩT)×H2r+1/2,r+1/4(Σ),(2)
for distributional data with-1<r<0(2.33)ΛΩ,Σf:H2r+1(Ω0)×H2r+3/2,r+3/4(Σ)⟶H2r+1(ΩT)×H2r+1/2,r+1/4(Σ),(3)
for distributional data withr≤-1 and 2r and r∉ℤ+1/2:
(2.34)ΛΩ,Σf:Ξ2r+1(Ω0)×H2r+3/2,r+3/4(Σ)⟶Ξ2r+1(ΩT)×H2r+1/2,r+1/4(Σ),
defined by
(2.35)ΛΩ,Σf[(u0,g)]=(u|ΩT,∂νu|∂Ω)
when u∈H2r+2,r+1(Q) is solution of Problem (2.2) with initial data (u0,g)=(u|Ω0,u|Σ).Note  that this operator can be viewed as a combination of the standard Dirichlet to Neumann map in the spatial boundary with the input-to-output map in the time boundary, that is, in initial and final interval times, found in control theory.Definition 2.18 (dirichlet green function).
By the Dirichlet Green's functionG(t,x,τ,ζ) for the Problem (2.2) we mean its solution with source δ(x-ζ,t-τ), (t,x,τ,ζ)∈Q×Q, and homogeneous Dirichlet data on the extended boundary ΩT×Σ, that is, G(t,x,τ,ζ)=0 for (t,x) on ΩT×Σ.Remark 2.19.
For regular data the Green's function exist, Costabel [17], and we can show that
(2.36)u(t,x)=∫Ω0u0(ζ)G(t,x,0,ζ)dζ+∫Σg(τ,ζ)∂G(t,x,τ,ζ)∂ν(τ,ζ)dσ(τ,ζ)+∫Qf(τ,ζ)G(t,x,τ,ζ)dζdτ
for (t,x)∈Q¯ is an explicit solution S to Problem (2.2). By using problem's (2.2) linearity we formally decompose the solution in three parts
(2.37)u=S[u0,g,f]:=S[u0,0,0]+S[0,g,0]+S[0,0,f],S[u0,g,f]:H2(r+1/2)(Ω)×H2r+3/2,r+3/4(Σ)×H2r,r(Q)⟶H2r+2,r+1(Q),
where
(2.38)S[u0,0,0]:H2(r+1/2)(Ω)⟶H2r+2,r+1(Q)
is the homogeneous Dirichlet zero source initial value auxiliary problem solution and
(2.39)S[0,g,0]:H2r+3/2,r+3/4(Σ)⟶H2r+2,r+1(Q)
is the zero source zero initial value auxiliary Dirichlet problem solution and
(2.40)S[0,0,f]:H2r,r(Q)⟶H2r+2,r+1(Q)
is the zero data auxiliary Dirichlet auxiliary problem.Lemma 2.20 (composition of trace and solution).
The extended Dirichlet-to-Neumann map is a composition of the final time trace and the lateral boundary normal trace with the Solution operator:(2.41)ΛΩ,Σf[u0,g]=(γT,γ1)S[u0,g,f]=(γT∘S,γ1∘S)[u0,g,f]=(uT,gν).Proof.
Consequence of the definition.


## 3. Inverse Transient Heat Equation Source Problem

The inverse source problem that we address consists in the recovery of the sourcef, knowing the extended Dirichlet-to-Neumann map ΛΩ,Σf. When r=0, the data are regular, the Green's function exists and f∈L2(Q). Let us investigate this situation, and then, we will show that the unique information available in this inverse problem is given only by one measurement, say, the bottom and top Dirichlet data and lateral cylinder Cauchy boundary data. The inverse problem IP(u0,g),(uT,gν)f is to find f∈L2(Q) such that(3.1)(IP(u0,g),(uT,gν)f)(uT,gν)=ΛΩ,Σf(u0,g)
for all given data pair (u0,g)×(uT,gν) corresponding to different solutions to the direct problem. By taking the normal trace at lateral cylinder boundary of the solution (2.36), we obtain that (3.2)γ1[u]=∂u∂ν(t,x)|Σ=Λ•,Σf[(u0,g)]=γ1[S[u0,0,0]]+γ1[S[0,g,0]]+γ1[S[0,0,f]]
or (3.3)γ1[u](t,x)=∫Ω0u0(ζ)∂G(t,x,0,ζ)∂ν(t,x)|Σdζ-∫Σg(τ,ζ)∂2G(t,x,τ,ζ)∂ν(t,x)∂ν(τ,ζ)|Σdσ(τ,ζ)+∫Qf(τ,ζ)∂G(t,x,τ,ζ)∂ν(t,x)|Σdζdτ
is an explicit expression of the lateral part of the extended Dirichlet-to-Neumann map Λ•,Σf. mapping for f∈L2(Q)(3.4)Λ•,Σf:H1(Ω)×H3/2,3/4(Σ)⟶H1/2,1/4(Σ).
Note that it appears decomposed in its partial traces(3.5)Λ•,Σf[(u0,g)]:=Λ•,Σ0[(u0,0)]+Λ•,Σ0[(0,g)]+Λ•,Σf[(0,0)],
where (3.6)Λ•,Σ0[(⋅,0)]:H1(Ω0)⟶H1/2,1/4(Σ)
is the zero source auxiliary initial value zero Dirichlet problem solution lateral normal trace and (3.7)Λ•,Σ0[(0,⋅)]:H3/2,3/4(Σ)⟶H1/2,1/4(Σ)
is the zero source nonhomogeneous Dirichlet zero initial auxiliary problem solution lateral normal trace and
(3.8)Λ•,Σf[(0,0)]∈H1/2,1/4(Σ)
is the homogeneous Dirichlet zero initial source auxiliary problem solution lateral normal trace.By taking the final time trace at cylinder top boundary of the solution (2.36) we obtain that (3.9)γT[u]=u(T,⋅)=ΛΩ,•f[(u0,g)]=γT[S[u0,0,f]]+γT[S[0,g,0]]+γT[G[0,0,f]]
or (3.10)γT[u](t,x)=∫Ω0u0(ζ)G(T,x,0,ζ)dζ-∫Σg(τ,ζ)∂G(T,x,τ,ζ)∂ν(τ,ζ)dσ(τ,ζ)+∫Qf(τ,ζ)G(T,x,τ,ζ)dζdτ
is an explicit expression of the final part of the input-to-output map (3.11)ΛΩ,•f:H1(Ω0)×H3/2,3/4(Σ)⟶H1(ΩT).
Note that its appears decomposed in its partial traces(3.12)ΛΩ,•f[(u0,g)]:=ΛΩ,•0[(u0,0)]+ΛΩ,•0[(0,g)]+ΛΩ,•f[(0,0)],
where (3.13)ΛΩ,•0[(⋅,0)]:H1(Ω0)⟶H1(ΩT)
is the zero source auxiliary initial value zero Dirichlet problem solution final trace,(3.14)Λ•,Σ0[(0,⋅)]:H3/2,3/4(Σ)⟶H1(ΩT)
is the zero source nonhomogeneous Dirichlet zero initial auxiliary problem solution final trace and (3.15)ΛΩ,•f[(0,0)]∈H1(ΩT)
is the homogeneous Dirichlet zero initial source auxiliary problem solution final trace.Definition 3.1 (relative extended dirichlet-to-neumann map).
Consider two problemsPu0,g,f and Pu0,g,0, one with source f and two other with zero source, but both with the same consistent initial time and Dirichlet data. By the Relative extended Dirichlet-to-Neumann map for f∈L2(Q) we mean the application:
(3.16)ΛΩ,Σf-ΛΩ,Σ0:H1(Ω0)×H3/2,3/4(Σ)⟶H1(ΩT)×H1/2,1/4(Σ).Note that the consistence of data (u0,g) is necessary to existence of solution to the problems Pu0,g,f and Pu0,g,0.Lemma 3.2.
Letuj, j=1,2,3,… be different solutions of Problem (2.2) with the same source f∈L2(Q) and different initial time and Dirichlet data (u0j,gj), j=1,2,3,…, respectively. Then(i)
the relative extended Dirichlet-to-Newman operatorΛΩ,Σf-ΛΩ,Σ0 is an operator whose functional value depends only on the source function f∈L2(Q), but is independent of the initial time and Dirichlet data (u0,g),(ii)
tor all solution of consistent data problemsPf,u0j,gj, j=1,2,3,…, with the same source, the source satisfies the systems of integral equations
(3.17)∫Qf(τ,ζ)(G(T,x,τ,ζ),∂G(t,x,τ,ζ)∂ν(t,x))dζdτ=(ΛΩ,Σf-ΛΩ,Σ0)[u0j,gj]=ΛΩ,Σf[0,0]
which depend only on the Relative extended Dirichlet-to-Neumann map.Proof.
Both results (i) and (ii) are trivial consequences of (3.3) and (3.10).Remark 3.3.
Note that in this case the unique information available for source reconstruction is given by only one measurement, that is, that final-Neumann boundary measurement(3.18)(uT,∂ν(t,x)u)=ΛΩ,Σf[u0,g]=ΛΩ,Σf[0,0]∈H1(ΩT)×H1/2,1/4(Σ)
corresponding to some specific initial-Dirichlet data (u0,g), which may be assumed as zero without loss of generality.


### 3.1. Existence of Regular Solution

The fact that only one Cauchy data can be used in a nonredundant source reconstruction suggests a method for solved inverse source problem that is based on the formulation of a high-order well posed Dirichlet equation problem. As observed by Friedman [18] and explored in this work, the parabolic transient heat equation behavior is similar to that of elliptic stationary equation. They coexist in a larger class of partial differential equations. In that class we found also the fourth-order in space and second order in time equation that is obtained if we apply simultaneously the transient heat operator (2.2) and its adjoint (2.7) to an auxiliary field related with the fields u as in problem (3.20). We have thus existence of solution in an especial class analogous to the class introduced in Alves et al. [10] for the Helmholtz operator problem,(3.19)H∂t-Δr,s={f∈Hr,s(Q)∣(-∂t-Δ)f=0}.Proposition 3.4.
Suppose that the available consistent Cauchy data to be used in the reconstruction is(u0,g)×(uT,gν)∈(H1(Ω0),H3/2,3/4(Σ))×(H1(ΩT),H1/2,1/4(Σ)), then there exist a solution to inverse source problem, problem (3.1).Proof.
In order to determine the Relative extended Dirichlet-to-Neumann Map, we first use the data to solve the zero source auxiliary problemPu0,g,0, obtaining ΛΩ,Σ0(u0,g)=(uT0,gν0). Since this datum is consistent, this solution exists. Then ΛΩ,Σ0(0,0)=ΛΩ,Σf(u0,g)-ΛΩ,Σ0(u0,g). We can now consider the following fourth-order problem formulated, as it has been suggested in the introduction of this subsection:
(3.20)(P0,u0,uT,g,gν4){(-∂t-Δ)(∂t-Δ)v=(-∂t2+Δ2)v=0inQ,v=0inΩ0,v=ΛΩ,•0(0,0)inΩT,v=0onΣ,∂νv=Λ•,Σ0(0,0)onΣ.
The fourth-order operator (-∂t2+Δ2) can be separated with an unbounded sequence of parameters λ into two elliptic operators, one being biharmonic type in space (-λ+Δ2), and the other second order in time (-∂t2+λ). Since the initial condition is zero, by finite energy method, the regularity of these operators is H1(I,L2(Ω)) and L2(I;H2(Ω)), respectively, and it is well posed and has a unique solution v=S(u0,g,uT,gν)∈H2,1(Q). The inverse source solution will be f=(-∂t-Δ)v.Remark 3.5.
The fourth-order problem used to show existence has the same equation used for modeling the direct problem involving elastic vibration of beams or plates,-∂2u-c2Δ2u=F, but with an unphysical imaginary parameter c=1i. Obviously, as a direct problem equation (3.20) is unphysical, but as a fourth-order model for the inverse source reconstruction with source in the class (3.19), it is just a method for source reconstruction based on higher order equation. Analytical solutions for this equation in one-dimension and two-dimensions polar geometry can be determined by noting the following complete and dense sets, respectively: (1)
{Xn(x)=cosh(κnx)-cos(κnx)-((cosh(κn)-cos(κn))/(sinh(κn)-sin(κn)))(sinh(κnx)-sin(κnx)})⊂L2(0,1) where κn, n=1,2,… are positive roots of the equation cosh(κ)cos(κ)=1;(2)
{ϕmn(r,θ)=[Im(κmnJm(κmnr)-Jm(κmnIm(κmnr)]exp(1imθ)}⊂L2(B(0,1)) where κmn, m,n=1,2,… are the positive roots of the systems of equations(3.21)Jm(κ)I′(κ)-Im(κ)J′(κ)=0,m=1,2,….
Note that m,n→∞, κn→∞ and κm,n→∞. In the case of the hyperbolic behavior of the wave-type equation that appears in beams and plates, the data in time are initial and the initial derivative traces and can be adjusted in the non-homogeneous case (this is expected since the boundary control is exact, Alves et al. [19]). In our fourth-order problem that combines the heat equation and its adjoint, we have a two-point boundary at times t=0 and t=T that cannot be simultaneously controlled (this parabolic type problem is only null controllable, Tucsnak and George Weiss [20]) and it is not possible to adjust at the same time the two time boundaries. Since by Lemma 3.2. we can always avoid nonhomogeneous initial condition, the problem is well posed. So, we have here a heat equation control analogous to the harmonic control in the Laplacian Dirichlet operator source reconstruction from boundary Neumann data problem.


### 3.2. The Regular Data Class

Definition 3.6 (regular class).
We call a class of distributional consistent dataℛ regular if its relative extended Dirichlet-to-Neumann map is regular
(3.22)R={(u0,g),(uT,gν)∈(H2r+1(Ω0)×H2r+3/2,r+3/4(Σ))×(H2r+1(ΩT)×H2r+1/2,r+1/4(Σ))∣-1<r<0;(ΛΩ,Σf-ΛΩ,Σ0)[u0,g]∈(γT,γ1)[H2,1(Q)]⊂H1(ΩT)×H1/2,1/4(Σ)}
or
(3.23)R={(u0,g),(uT,gν)∈(Ξ2r+1(Ω0)×H2r+3/2,r+3/4(Σ))×(Ξ2r+1(ΩT)×H2r+1/2,r+1/4(Σ))∣r<-1;(ΛΩ,Σf-ΛΩ,Σ0)[u0,g]∈(γT,γ1)[H2,1(Q)]⊂H1(ΩT)×H1/2,1/4(Σ)}.Theorem 3.7.
A inverse problemIP(u0,g),(uT,gν)f admits a solution f∈L2(Q) only if (u0,g),(uT,gν)∈ℛ.Proof.
Necessity: suppose the IP(u0,g),(uT,gν)f has a solution f∈L2(Q). Let us consider the following auxiliary problems: P0,0,f with solution w0=S[0,0,f] and Pu0,g,0 with solution w1=S[u0,g,f]. Then, by the additivity principle for linear operators
(3.24)ΛΩ,Σf[u0,g]=ΛΩ,Σf[0,0]+ΛΩ,Σ0[u0,g]⟹(ΛΩ,Σf-ΛΩ,Σ0)[u0,g]=ΛΩ,Σf[0,0].
Since f∈L2(Q) implies by regularity that w0∈H2,1(Q), by the first trace Theorem 2.1. the data are in ℛ.
Sufficiency: suppose the data are in the regular class. Consider the fourth-order problem (3.20). As we have shown in Section 3.4, it is well posed with a solution v∈H2,1(Q) and the source f=(-∂-Δ)v∈L2(Q) is solution to the inverse problem with data in the class ℛ.


### 3.3. The Reciprocity Gap Functional

The second Green formula(3.25)∫Q((∂tu-Δu)v-u(-∂tv-Δv))dxdt=∫Σ(γ[u]γ1[v]-γ1[u]γ[v])dσ(t,x)-∫ΩTγT[u]γT[v]dx+∫Ω0γ0[u]γ0[v]dx
applied to problems Pu0,g,f with normal trace at the cylinder lateral boundary Σ, γ1[u]=gν, initial value at Ω0, γ0[u]=u0 and the adjoint problem PuT,γ[v],0* with uT time T value and zero source yield the following expression for reciprocity gap functional in the transient heat equation context:(3.26)∫Qfvdxdt=∫Σ(gγ1[v]-gνγ[v])dσ(t,x)-∫ΩTuTγT[v]dx+∫Ω0u0γ0[v]dx
or, by using the extended Dirichlet-to-Neumann notation,(3.27)∫Qfvdxdt=∫Σ(gγ1[v]-Λ•,Σf[u0,g]γ[v])dσ(t,x)-∫ΩTΛΩ,•f[u0,g]γT[v]dx+∫Ω0u0γ0[v]dx.
By subtracting the extended Dirichlet-to-Neumann map for the zero source problem Pu0,g,0 with the same data, we obtain the following weak form for the systems (3.17):(3.28)∫Qfvdxdt=-∫ΩTΛΩ,•f[0,0]γT[v]dx-∫ΣΛ•,Σf[0,0]γ[v]dσ(t,x).
We call (3.28) as the transient heat reciprocity gap equation. This is a weak variational equation that must be tested for all functions in(3.29)H-∂t-Δ2,1(Q)={v∈H2,1(Q)∣-∂tv-Δv=0}.


### 3.4. Nonobservability

Let(3.30)R0:={(u0,g),(uT,gν)∈RsuchthatΛΩ,Σf[0,0]=0}.
Let us consider the following set of sources:(3.31)F0:={f∈L2(Q);fisasolutionofIP(u0,g),(uT,gν)f;(u0,g),(uT,gν)∈R0}.
Note that by the Transient Reciprocity Gap equation (3.28)(3.32)f∈F0⟹∫Qfvdxdt=0∀v∈H-∂t-Δ2,1(Q)
and ℱ0⊂(H-∂t-Δ2,1(Q))⊥ is a set of non observable sources.Remark 3.8 (ill-posed problem).
Since the extended Dirichlet-to-Neumann map is a composition of the solution operatorS and the trace operator, this non uniqueness result is expected. Also, trace operator may not be onto, as we have pointed in Remark 2.2, and problems with existence may occur if the data are not consistent. Finally, we note that the integral systems (3.17) and the variational (3.28) are compact operators, so instability problems are expected in numerical solutions. In that bad situation, a generalized concept of inverse need to be adopted, Engl et al. [21].


### 3.5. Uniqueness of Solution in Classical Spaces

The source reconstruction of consistent boundary data is not possible, if we do not have additional information about the source. As an example, we may reconstruct a time independent source in the transient model or even a source in the source class (3.19). In this work, we are interested in the reconstruction of a source which is given by a characteristic function defined with a star-shaped support. This is a generalization of the classical results for reconstruction of star-shape sources in the Laplace Poisson Dirichlet Problem, Roberty and Alves [9], and in the Helmholtz Poisson Dirichlet Problem, Roberty and Rainha [15]. Isakov [2] treats a problem for reconstruction of an unknown domain moving in time with the prescribed exterior thermal potential. He proves the following theorem.Theorem 3.9.
Considerω with ω¯⊂Ω. Let Qj⊂Q, for j=1,2, with Qj¯⊂ω×[0,T], with spatial boundaries Σj:=∂xQj of class C1+λ and satisfying:
(3.33)thesetsQj∩{t=τ},(Q∖(Q1¯∪Q2¯))∩{t=τ},(Q∖Qj¯∩{t=τ})
are connected for 0<τ<T. Let ωj=(Qj¯∩{t=0})¯o and
(3.34)f∈Cλ,λ/2(Q¯),f>0onQ¯.
If solutions uj to the problems:
(3.35)∂tuj-Δuj=fχ(Qj)onQ,uj=0inΩ0
coincides on Q∖ω×(0,T), then Q1=Q2.In his proof, he first demonstrates in two auxiliary lemmas that:(1)
in the initial time the two source support,Q1∪{t=0}=Q2∪{t=0}, must coincide and that(2)
there is a timet0>0 such that the relative potential associated with the sources
(3.36)u=U(⋅,fχ(Q2∩ω1×(-T,0]))-U(⋅,fχ(Q1∩ω1×(-T,0]))
satisfies uf(χ(q2)-χ(Q1))≤0 on ℜn×(0,t0)and he uses these results to prove that the hypotheses that the source may be unequal are contradictory.In this way, we expect that when the data are sufficiently regular, the moving characteristic source can be reconstructed. A common method for numerical solution of the transient heat equation is based on the construction of time marching schemes by the solution of a sequence of modified Helmholtz equations. In these schemes, at each time increment the source must be reconstructed. For the class of problems we also have demonstrated a uniqueness theorem in Roberty and Rainha [15]. We will return to this theorem after the presentation of this scheme.


## 3.1. Existence of Regular Solution

The fact that only one Cauchy data can be used in a nonredundant source reconstruction suggests a method for solved inverse source problem that is based on the formulation of a high-order well posed Dirichlet equation problem. As observed by Friedman [18] and explored in this work, the parabolic transient heat equation behavior is similar to that of elliptic stationary equation. They coexist in a larger class of partial differential equations. In that class we found also the fourth-order in space and second order in time equation that is obtained if we apply simultaneously the transient heat operator (2.2) and its adjoint (2.7) to an auxiliary field related with the fields u as in problem (3.20). We have thus existence of solution in an especial class analogous to the class introduced in Alves et al. [10] for the Helmholtz operator problem,(3.19)H∂t-Δr,s={f∈Hr,s(Q)∣(-∂t-Δ)f=0}.Proposition 3.4.
Suppose that the available consistent Cauchy data to be used in the reconstruction is(u0,g)×(uT,gν)∈(H1(Ω0),H3/2,3/4(Σ))×(H1(ΩT),H1/2,1/4(Σ)), then there exist a solution to inverse source problem, problem (3.1).Proof.
In order to determine the Relative extended Dirichlet-to-Neumann Map, we first use the data to solve the zero source auxiliary problemPu0,g,0, obtaining ΛΩ,Σ0(u0,g)=(uT0,gν0). Since this datum is consistent, this solution exists. Then ΛΩ,Σ0(0,0)=ΛΩ,Σf(u0,g)-ΛΩ,Σ0(u0,g). We can now consider the following fourth-order problem formulated, as it has been suggested in the introduction of this subsection:
(3.20)(P0,u0,uT,g,gν4){(-∂t-Δ)(∂t-Δ)v=(-∂t2+Δ2)v=0inQ,v=0inΩ0,v=ΛΩ,•0(0,0)inΩT,v=0onΣ,∂νv=Λ•,Σ0(0,0)onΣ.
The fourth-order operator (-∂t2+Δ2) can be separated with an unbounded sequence of parameters λ into two elliptic operators, one being biharmonic type in space (-λ+Δ2), and the other second order in time (-∂t2+λ). Since the initial condition is zero, by finite energy method, the regularity of these operators is H1(I,L2(Ω)) and L2(I;H2(Ω)), respectively, and it is well posed and has a unique solution v=S(u0,g,uT,gν)∈H2,1(Q). The inverse source solution will be f=(-∂t-Δ)v.Remark 3.5.
The fourth-order problem used to show existence has the same equation used for modeling the direct problem involving elastic vibration of beams or plates,-∂2u-c2Δ2u=F, but with an unphysical imaginary parameter c=1i. Obviously, as a direct problem equation (3.20) is unphysical, but as a fourth-order model for the inverse source reconstruction with source in the class (3.19), it is just a method for source reconstruction based on higher order equation. Analytical solutions for this equation in one-dimension and two-dimensions polar geometry can be determined by noting the following complete and dense sets, respectively: (1)
{Xn(x)=cosh(κnx)-cos(κnx)-((cosh(κn)-cos(κn))/(sinh(κn)-sin(κn)))(sinh(κnx)-sin(κnx)})⊂L2(0,1) where κn, n=1,2,… are positive roots of the equation cosh(κ)cos(κ)=1;(2)
{ϕmn(r,θ)=[Im(κmnJm(κmnr)-Jm(κmnIm(κmnr)]exp(1imθ)}⊂L2(B(0,1)) where κmn, m,n=1,2,… are the positive roots of the systems of equations(3.21)Jm(κ)I′(κ)-Im(κ)J′(κ)=0,m=1,2,….
Note that m,n→∞, κn→∞ and κm,n→∞. In the case of the hyperbolic behavior of the wave-type equation that appears in beams and plates, the data in time are initial and the initial derivative traces and can be adjusted in the non-homogeneous case (this is expected since the boundary control is exact, Alves et al. [19]). In our fourth-order problem that combines the heat equation and its adjoint, we have a two-point boundary at times t=0 and t=T that cannot be simultaneously controlled (this parabolic type problem is only null controllable, Tucsnak and George Weiss [20]) and it is not possible to adjust at the same time the two time boundaries. Since by Lemma 3.2. we can always avoid nonhomogeneous initial condition, the problem is well posed. So, we have here a heat equation control analogous to the harmonic control in the Laplacian Dirichlet operator source reconstruction from boundary Neumann data problem.


## 3.2. The Regular Data Class

Definition 3.6 (regular class).
We call a class of distributional consistent dataℛ regular if its relative extended Dirichlet-to-Neumann map is regular
(3.22)R={(u0,g),(uT,gν)∈(H2r+1(Ω0)×H2r+3/2,r+3/4(Σ))×(H2r+1(ΩT)×H2r+1/2,r+1/4(Σ))∣-1<r<0;(ΛΩ,Σf-ΛΩ,Σ0)[u0,g]∈(γT,γ1)[H2,1(Q)]⊂H1(ΩT)×H1/2,1/4(Σ)}
or
(3.23)R={(u0,g),(uT,gν)∈(Ξ2r+1(Ω0)×H2r+3/2,r+3/4(Σ))×(Ξ2r+1(ΩT)×H2r+1/2,r+1/4(Σ))∣r<-1;(ΛΩ,Σf-ΛΩ,Σ0)[u0,g]∈(γT,γ1)[H2,1(Q)]⊂H1(ΩT)×H1/2,1/4(Σ)}.Theorem 3.7.
A inverse problemIP(u0,g),(uT,gν)f admits a solution f∈L2(Q) only if (u0,g),(uT,gν)∈ℛ.Proof.
Necessity: suppose the IP(u0,g),(uT,gν)f has a solution f∈L2(Q). Let us consider the following auxiliary problems: P0,0,f with solution w0=S[0,0,f] and Pu0,g,0 with solution w1=S[u0,g,f]. Then, by the additivity principle for linear operators
(3.24)ΛΩ,Σf[u0,g]=ΛΩ,Σf[0,0]+ΛΩ,Σ0[u0,g]⟹(ΛΩ,Σf-ΛΩ,Σ0)[u0,g]=ΛΩ,Σf[0,0].
Since f∈L2(Q) implies by regularity that w0∈H2,1(Q), by the first trace Theorem 2.1. the data are in ℛ.
Sufficiency: suppose the data are in the regular class. Consider the fourth-order problem (3.20). As we have shown in Section 3.4, it is well posed with a solution v∈H2,1(Q) and the source f=(-∂-Δ)v∈L2(Q) is solution to the inverse problem with data in the class ℛ.


## 3.3. The Reciprocity Gap Functional

The second Green formula(3.25)∫Q((∂tu-Δu)v-u(-∂tv-Δv))dxdt=∫Σ(γ[u]γ1[v]-γ1[u]γ[v])dσ(t,x)-∫ΩTγT[u]γT[v]dx+∫Ω0γ0[u]γ0[v]dx
applied to problems Pu0,g,f with normal trace at the cylinder lateral boundary Σ, γ1[u]=gν, initial value at Ω0, γ0[u]=u0 and the adjoint problem PuT,γ[v],0* with uT time T value and zero source yield the following expression for reciprocity gap functional in the transient heat equation context:(3.26)∫Qfvdxdt=∫Σ(gγ1[v]-gνγ[v])dσ(t,x)-∫ΩTuTγT[v]dx+∫Ω0u0γ0[v]dx
or, by using the extended Dirichlet-to-Neumann notation,(3.27)∫Qfvdxdt=∫Σ(gγ1[v]-Λ•,Σf[u0,g]γ[v])dσ(t,x)-∫ΩTΛΩ,•f[u0,g]γT[v]dx+∫Ω0u0γ0[v]dx.
By subtracting the extended Dirichlet-to-Neumann map for the zero source problem Pu0,g,0 with the same data, we obtain the following weak form for the systems (3.17):(3.28)∫Qfvdxdt=-∫ΩTΛΩ,•f[0,0]γT[v]dx-∫ΣΛ•,Σf[0,0]γ[v]dσ(t,x).
We call (3.28) as the transient heat reciprocity gap equation. This is a weak variational equation that must be tested for all functions in(3.29)H-∂t-Δ2,1(Q)={v∈H2,1(Q)∣-∂tv-Δv=0}.


## 3.4. Nonobservability

Let(3.30)R0:={(u0,g),(uT,gν)∈RsuchthatΛΩ,Σf[0,0]=0}.
Let us consider the following set of sources:(3.31)F0:={f∈L2(Q);fisasolutionofIP(u0,g),(uT,gν)f;(u0,g),(uT,gν)∈R0}.
Note that by the Transient Reciprocity Gap equation (3.28)(3.32)f∈F0⟹∫Qfvdxdt=0∀v∈H-∂t-Δ2,1(Q)
and ℱ0⊂(H-∂t-Δ2,1(Q))⊥ is a set of non observable sources.Remark 3.8 (ill-posed problem).
Since the extended Dirichlet-to-Neumann map is a composition of the solution operatorS and the trace operator, this non uniqueness result is expected. Also, trace operator may not be onto, as we have pointed in Remark 2.2, and problems with existence may occur if the data are not consistent. Finally, we note that the integral systems (3.17) and the variational (3.28) are compact operators, so instability problems are expected in numerical solutions. In that bad situation, a generalized concept of inverse need to be adopted, Engl et al. [21].


## 3.5. Uniqueness of Solution in Classical Spaces

The source reconstruction of consistent boundary data is not possible, if we do not have additional information about the source. As an example, we may reconstruct a time independent source in the transient model or even a source in the source class (3.19). In this work, we are interested in the reconstruction of a source which is given by a characteristic function defined with a star-shaped support. This is a generalization of the classical results for reconstruction of star-shape sources in the Laplace Poisson Dirichlet Problem, Roberty and Alves [9], and in the Helmholtz Poisson Dirichlet Problem, Roberty and Rainha [15]. Isakov [2] treats a problem for reconstruction of an unknown domain moving in time with the prescribed exterior thermal potential. He proves the following theorem.Theorem 3.9.
Considerω with ω¯⊂Ω. Let Qj⊂Q, for j=1,2, with Qj¯⊂ω×[0,T], with spatial boundaries Σj:=∂xQj of class C1+λ and satisfying:
(3.33)thesetsQj∩{t=τ},(Q∖(Q1¯∪Q2¯))∩{t=τ},(Q∖Qj¯∩{t=τ})
are connected for 0<τ<T. Let ωj=(Qj¯∩{t=0})¯o and
(3.34)f∈Cλ,λ/2(Q¯),f>0onQ¯.
If solutions uj to the problems:
(3.35)∂tuj-Δuj=fχ(Qj)onQ,uj=0inΩ0
coincides on Q∖ω×(0,T), then Q1=Q2.In his proof, he first demonstrates in two auxiliary lemmas that:(1)
in the initial time the two source support,Q1∪{t=0}=Q2∪{t=0}, must coincide and that(2)
there is a timet0>0 such that the relative potential associated with the sources
(3.36)u=U(⋅,fχ(Q2∩ω1×(-T,0]))-U(⋅,fχ(Q1∩ω1×(-T,0]))
satisfies uf(χ(q2)-χ(Q1))≤0 on ℜn×(0,t0)and he uses these results to prove that the hypotheses that the source may be unequal are contradictory.In this way, we expect that when the data are sufficiently regular, the moving characteristic source can be reconstructed. A common method for numerical solution of the transient heat equation is based on the construction of time marching schemes by the solution of a sequence of modified Helmholtz equations. In these schemes, at each time increment the source must be reconstructed. For the class of problems we also have demonstrated a uniqueness theorem in Roberty and Rainha [15]. We will return to this theorem after the presentation of this scheme.


## 4. Theθ-Scheme and the Modified Helmholtz Model for the Transient Heat Problem

One type of source that can be uniquely reconstructed from Neumann boundary measurements in a model based on Poisson equation with the Laplace operatorΔ is star-shaped characteristic sources. This uniqueness resulted may be easily extended to a modified Helmholtz equation based model, Roberty and Rainha [15], and thus we present now an algorithm for moving transient source reconstruction in the heat equation based on this result. Let the source be given by(4.1)f(t,x)=χω(t)(x)inQ,
where ω(t), t∈[0,T] is a representation of the star-shape source boundary. For one-dimensional problems it is a set with two points. For two or three-dimensional problems it is a moving Lipschitz parametric curve or surface in which the parameter has been omitted. With this in mind, we may rewrite the transient problem as(4.2)(Pχω){∂tu-Δu=χω(t)(x)inQ,-Δu0=χω(0)(x)inΩ0,u=0onΣ;
with transient Neumann history γ1[u]=∂νu=gν in the lateral cylinder boundary.The initialu0 can be determined as solution of the Poisson problem -Δu0=χω(0), if the initial shape ω(0) is known. If it is not known, then we can use the Cauchy data, g(0) and gν(0), to solve the static inverse problem, by using the methodology in Roberty and Alves [9]. Consider a partition of the time interval [0,T] into N subintervals of length τ>0. Let {t0,t1,t2,…,tn,tn+1,…,tN} be the knots of this partition, with t0=0andtN=T. For tn<t<tn+1, n=0,1,N-1we use the θ-scheme approach for the discretization of (4.2). Define, for a function h(t,x), a linear θ weighted approximation δθ(h)(x) by(4.3)δθ(h)(x)=θh(tn+1,x)+(1-θ)h(tn,x)
We start by approximating the time derivative ∂tu in (4.2), by a first order forward difference(4.4)∂u∂t(t,x)≅u(tn+1,x)-u(tn,x)τ,x∈Ω.
The diffusion and the characteristic source respectively by(4.5)Δu(t,x)≅δθ(Δu)(x),x∈Ω,χω(t)(x)≅δθ(χω)(x),x∈Ω.By denotingun+1(x) with x∈Ω, the approximate solution at the time step tn+1, the transient system (4.2) is approximated by the following sequence of stationary problems:(4.6)(Hχωn+1){-Δun+1+λun+1=fn+χω(tn+1)inΩ,un+1=0onΓ,withgν(tn+1):=∂νun+1onΓ;
for n=0,1,2,…,N. Here λ=1/τθ and (4.7)fn=un+τ(1-θ)Δun+τθχω(tn)(x)τθ.
Note that Δun+χω(tn)=∂un/∂t and that the initial Poisson problem determining the u0 and χω(0) is(4.8)(Hg,χω0){-Δu0=χω(0)(x)inΩ,u0=0onΓ,withgν(0):=∂νu0onΓ.The sequence of modified Helmholtz source inverse problem (4.6) starting with stationary problem (4.8) may be used to model a scheme for the reconstruction of star-shaped sources χω(tn)(x), for a time knot sequence, showing its movement and deformation in the external domain Ω. For this, we only need to know the transient Neumann data with zero Dirichlet datum on the external boundary Γ=∂Ω. Since we do not have experimental data, we will solve the direct problem with a different method, adding noise, and do an experimental data synthesis.


### 4.1. Iterative Source Reconstruction Scheme

The source at timetn may be further calculated as(4.9)fn=λθun-1-θθfn-1,forn=1,2,…
with f-1=0 and f0=λu0. Since the discretized direct problem (4.6) and (4.8) are linear, it may be decomposed into two subproblems separating the known part of the source from the part to be reconstructed, that is, fn and χω(tn+1). Let yn+1, n=-1,0,1,… be a solution of(4.10)(Hfnn+1){-Δyn+1+λyn+1=fninΩ,yn+1=0onΓ,withgyν(tn+1):=∂νyn+1onΓ;
and let wn+1, n=-1,0,1,… be solution of(4.11)(Hχωn+1){-Δwn+1+λwn+1=χω(tn+1)(x)inΩ,wn+1=0onΓ,withgwν(tn+1):=∂νwn+1onΓ.Then, by the superposition principle, the solution of (4.6) is un+1=wn+1+yn+1 and the Neumann data will be the sum of the decomposed parts gν(tn+1)=gwν(tn+1)+gyν(tn+1). The Y-problems (4.10) form a discrete sequence of problems with continuous source fn that may be solved before the time increment at tn begins. Its normal derivatives may be calculated and(4.12)gyν(tn+1):=∂νyn+1onΓ,
subtracted from the synthetic transient Neumann data at knot tn+1(4.13)gwν(tn+1)=gν(tn+1)-gyν(tn+1),
to produce the data for the modified Helmholtz (4.11) that will be used in the reconstruction of the source χω(tn+1) at time tn+1. Note that by using the Reciprocity gap functional the characteristic star-shaped source may be reconstructed without solving the direct problem (4.11). By using the second Green's formula, this inverse problem is modeled with a nonlinear Fredholm integral equation of first kind.In this modified Helmholtz equation formulation to the transient problem the uniqueness can be proved for each time interval.Theorem 4.1.
Consider the direct problem (4.11) and its associated inverse problem with two sources χω1(tn+1) and χω2(tn+1). Let ω1(tn+1), ω2(tn+1)⊂Ω domains with C2 boundary and ω1(tn+1)∖ω2(tn+1), ω2(tn+1)∖ω1(tn+1), ω1(tn+1)∩ω2(tn+1) be connected. If the Cauchy data for the two problems are the same, then ω1(tn+1)=ω2(tn+1).Proof.
The proof can be found in Roberty and Rainha [15]. It is based on contradictory consequences of the maximum principle and the initial hypotheses that the sources are unequal.


## 4.1. Iterative Source Reconstruction Scheme

The source at timetn may be further calculated as(4.9)fn=λθun-1-θθfn-1,forn=1,2,…
with f-1=0 and f0=λu0. Since the discretized direct problem (4.6) and (4.8) are linear, it may be decomposed into two subproblems separating the known part of the source from the part to be reconstructed, that is, fn and χω(tn+1). Let yn+1, n=-1,0,1,… be a solution of(4.10)(Hfnn+1){-Δyn+1+λyn+1=fninΩ,yn+1=0onΓ,withgyν(tn+1):=∂νyn+1onΓ;
and let wn+1, n=-1,0,1,… be solution of(4.11)(Hχωn+1){-Δwn+1+λwn+1=χω(tn+1)(x)inΩ,wn+1=0onΓ,withgwν(tn+1):=∂νwn+1onΓ.Then, by the superposition principle, the solution of (4.6) is un+1=wn+1+yn+1 and the Neumann data will be the sum of the decomposed parts gν(tn+1)=gwν(tn+1)+gyν(tn+1). The Y-problems (4.10) form a discrete sequence of problems with continuous source fn that may be solved before the time increment at tn begins. Its normal derivatives may be calculated and(4.12)gyν(tn+1):=∂νyn+1onΓ,
subtracted from the synthetic transient Neumann data at knot tn+1(4.13)gwν(tn+1)=gν(tn+1)-gyν(tn+1),
to produce the data for the modified Helmholtz (4.11) that will be used in the reconstruction of the source χω(tn+1) at time tn+1. Note that by using the Reciprocity gap functional the characteristic star-shaped source may be reconstructed without solving the direct problem (4.11). By using the second Green's formula, this inverse problem is modeled with a nonlinear Fredholm integral equation of first kind.In this modified Helmholtz equation formulation to the transient problem the uniqueness can be proved for each time interval.Theorem 4.1.
Consider the direct problem (4.11) and its associated inverse problem with two sources χω1(tn+1) and χω2(tn+1). Let ω1(tn+1), ω2(tn+1)⊂Ω domains with C2 boundary and ω1(tn+1)∖ω2(tn+1), ω2(tn+1)∖ω1(tn+1), ω1(tn+1)∩ω2(tn+1) be connected. If the Cauchy data for the two problems are the same, then ω1(tn+1)=ω2(tn+1).Proof.
The proof can be found in Roberty and Rainha [15]. It is based on contradictory consequences of the maximum principle and the initial hypotheses that the sources are unequal.


## 5. Conclusions

We have presented a methodology for star-shape source reconstruction in the transient heat problem by using one set of Cauchy data history. With the adoption of an anisotropic Sobolev Hilbert mathematical framework, we can treat the problem with a methodology analogous to that used to study stationary elliptic problems. Therefore, by introducing a finite differences timeθ-scheme, we developed an algorithm based on a modified Helmholtz system, for which we have already studied the mathematically inverse source reconstruction problem.



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*Source: 987545-2011-01-16.xml*