INITIAL TEMPERATURE PROFILE RECOVERY UNDER A MIXED BOUNDARY CONDITION

. The challenge of finding the initial temperature distribution (profile) has been addressed for different boundary conditions. Previous studies studied this problem under Dirichlet [2, 6], Neumann [12]


Introduction
We consider the initial-boundary value problem ∂u ∂t = ∂ 2 u ∂x 2 , u(0, t) = 0, u x (π, t) = 0, u(x, 0) = f (x), (1.1)This equation describes the behavior of the temperature distribution, denoted by u(x, t), within a thin, uniform, one-dimensional rod of length π under the mixed boundary condition.These initial boundary conditions are also called as Dirichlet-Neumann boundary conditions.For a known integrable function f over the interval [0, π], it is well known that problem (1.1) has a solution with the Fourier sine series representation where fk = 2 π π 0 f (x) sin 2k−1 2 x dx.In particular, note that But, what if we do not know the initial temperature profile f (x)?Suppose we only have temperature measurements (u(x, t)) at a specific location x 0 on the rod and at a finite future times.Can we still recover f with a certain desired rate of accuracy?This type of problem is highly ill-posed without further assumptions on f, x 0 and future time selections.This type of recovering the initial profile from discrete sample has been widely studied under the context of fractional Laplacian or under the different boundary conditions or other contexts in recent times (see [1,2,6,8,9,10,11,12,13]) and has many applications in many areas such as mathematical biology or physics (see [3,7,14,15]).
To avoid ill-posedness and to recover an initial temperature profile under the Dirichlet boundary conditions, DeVore and Zuazua [6] have assumed that the initial data f lies in the following closed subset of L 2 ([0, π]): where r is a positive real number.Furthermore, to avoid ill-posedness appearing due to vanishing temperature of eigenfunctions, they have selected x 0 to avoid the nodal sets of eigenfunctions sin kx, k = 1, 2, 3, . . .at x 0 and defined exponentially growing future times Later, Aryal and Karki [2] improved DeVore and Zuazua's result by selecting linearly growing finite future times.
for k = 2, 3, . . ., n lie within a bounded interval [0, T ] by slightly modifying on the choice of f from the closed subset F r of L 2 ([0, π]) to a smaller subset Their time selection is far more practical in real-life applications than DeVore and Zuazua's exponentially growing time selection.
For the Neumann boundary conditions, the author (with Karki and Shawn) [12] used Fourier cosine represention and a modified L 2 ([0, π]) closed subspace to recover the initial profile.
For the periodic boundary conditions, the author (with Karki and Allison) [11] adopted complex analytic method by defining the L 2 closed subspace (1.6) In particular, the complex analytic approach in [11] turns out to be a unified solution approach for Dirichlet and Neumann boundary conditions.
In this article, we establish a result corresponding to the initial-boundary value problem described in (1.1).For a future linear time selection, we use the time selection (1.4).We also assume that f is in B r , the closed subset of L 2 ([0, π]).To avoid nodal sets of the eigenfunctions, sin 2k−1 2 x , k = 1, 2, 3, . . ., we need to choose an x 0 on the one-dimensional rod with sin 2k−1 2 x 0 ̸ = 0 for all k = 1, 2, 3, . . . .From Lemma 2.1 in [2], we know that there is an x 0 ̸ = 0 for all k = 1, 2, 3, . . . .Therefore, for eigenfunctions, sin 2k−1 2 x , k = 1, 2, 3, . . ., we can choose x 0 as x ′ 0 .Furthermore, we know that Finally, for any forward time sequence t k as in (1.4), the corresponding discrete temperature measurements u(x 0 , t k ), k = 1, 2, . . .are sufficient to determine the initial profile f uniquely.To see this, consider a holomorphic function where we can uniquely determine F and hence fj .Therefore, u(x 0 , t j ) uniquely determine f .

Optimal approximation error
In this section, first, we briefly recall the theory of manifold width as discussed in [4,5] to develop a measurement algorithm and recall a lower bound for the optimal error of approximation to an initial profile.Then, it is natural to ask whether there is an upper bound with a certain desired accuracy.In our main result, we tackle the upper bound problem.
2.1.Lower bound on optimal error.From [1, 2, 6, 13], we briefly recall a measurement algorithm and then discuss a lower bound on the optimal error of the approximation.For reader's reference, we like to briefly discuss the approach here.The development measurement algorithm uses the theory of manifold width in [4,5] and is indeed an encoder coupled with a decoder.An encoder is a continuous function that maps each element of a compact subspace B of L 2 ([0, π]) into a point in R n , and a decoder is a continuous function that maps each point y ∈ R n into an element of L 2 ([0, π]).In our setting, an encoder is a continuous function e n mapping f ∈ B into e n (f ) = (u 1 , u 2 , . . ., u n ) ∈ R n where u k = u(x 0 , t k ), k = 1, 2, . . ., n, is n temperature measurements.On the other hand, a decoder is a continuous function M n mapping (u 1 , u 2 , . . ., u n ) ∈ R n into an approximation fn of f .The optimal error of approximation to f is defined as where fn = M n (e n (f )).
In [2] we obtain a lower bound for this optimal error (also see [6]) as below.
Theorem 2.1.For the measurement algorithm defined with an encoder e n : f → (u 1 , u 2 , . . ., u n ) and a continuous decoder M n : (u 1 , u 2 , . . ., u n ) → fn as described above, we have where C is a constant depending on r only.

Main theorem.
Our main goal is to recover an initial profile f with a desired rate of accuracy order n −r once we observe n temperature measurements u(x 0 , t j ), j = 1, 2, . . ., n.More precisely, we have the following theorem.
Setting c k = fk sin 2k−1 2 x 0 from (1.2), we first estimate c k and construct an approximation of c k .Then we give the proof of the theorem at the end of this section.
From [2, Lemma 2.3], we have the following Lemma with the same proof.
Lemma 2.3.The coefficients c j = fj sin 2k−1 2 x 0 are bounded.More precisely, |c j | ≤ (j + 2) −r , j = 1, 2, . . . .Now, we define an approximation ck of c k .Later, we will use ck when constructing an approximation fn of f .Note that From this, we obtain Now we define an approximation ck to c k as To calculate an error bound between c k and ck , we will need a couple of estimates.