Third International Conference on Applications of Mathematics to Nonlinear Sciences, Electronic Journal of Differential Equations, Conference 27 (2024), pp. 97–103. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.conf.27.y1 INITIAL TEMPERATURE PROFILE RECOVERY UNDER A MIXED BOUNDARY CONDITION YOUNG HWAN YOU Abstract. The challenge of finding the initial temperature distribution (pro- file) has been addressed for different boundary conditions. Previous studies studied this problem under Dirichlet [2, 6], Neumann [12], and periodic [11]. This article focuses on the problem with mixed boundary conditions. We con- sider a one-dimensional body where temperatures are measured at a specific location x0 and at a finite number of increasing future time points in a bounded interval. 1. Introduction We consider the initial-boundary value problem ∂u ∂t = ∂2u ∂x2 , u(0, t) = 0, ux(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 u(x, t) = ∞∑ k=1 e−( 2k−1 2 )2tf̂k sin (2k − 1 2 x ) . (1.2) where f̂k = 2 π ∫ π 0 f(x) sin ( 2k−1 2 x ) dx. In particular, note that f(x) = u(x, 0) = ∞∑ k=1 f̂k sin (2k − 1 2 x ) . 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 x0 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 2010 Mathematics Subject Classification. 35K05, 35K99, 41A27. Key words and phrases. Initial Temperature profile; mixed boundary condition; temperature measurement; linear time growth; optimal error. ©2024 This work is licensed under a CC BY 4.0 license. Published Month Day, 202X. 97 98 Y. H. YOU EJDE-2024/CONF/27 on f, x0 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 L2([0, π]): Fr = { f ∈ L2([0, π]) : 2r∑ j=1 j2r|f̂j |2 ≤ 1 } (1.3) where r is a positive real number. Furthermore, to avoid ill-posedness appearing due to vanishing temperature of eigenfunctions, they have selected x0 to avoid the nodal sets of eigenfunctions sin kx, k = 1, 2, 3, . . . at x0 and defined exponentially growing future times tk = (2 √ 2)k−1t1, k = 1, 2, 3, . . . with an arbitarly chosen t1 > 0. Later, Aryal and Karki [2] improved DeVore and Zuazua’s result by selecting linearly growing finite future times. tk = (n+ k − 1)t1 (1.4) for k = 2, 3, . . . , n lie within a bounded interval [0, T ] by slightly modifying on the choice of f from the closed subset Fr of L2([0, π]) to a smaller subset Br = { f ∈ L2([0, π]) : ∞∑ k=1 (k + 2)2r|f̂k|2 ≤ 1 } , (1.5) 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 L2([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 L2 closed subspace Cr = { f ∈ L2([0, π]) : ∞∑ k=−∞ (|k|+ e)2r|f̂k|2 ≤ 1 } . (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 Br, the closed subset of L2([0, π]). To avoid nodal sets of the eigenfunctions, sin ( 2k−1 2 x ) , k = 1, 2, 3, . . . , we need to choose an x0 on the one-dimensional rod with sin ( 2k−1 2 x0 ) ̸= 0 for all k = 1, 2, 3, . . . . From Lemma 2.1 in [2], we know that there is an x0 such that | sin kx0| ≥ d for some d > 0 for all k = 1, 2, 3, . . . . It is possible to select 0 ≤ x0 ≤ π/2. Now, define x′ 0 = 2x0, where x0 is the point in [2, Lemma 2.1]. Then sin ( 2k−1 2 x′ 0 ) = sin ( (2k − 1)x0 ) ̸= 0 for all k = 1, 2, 3, . . . . Therefore, for eigenfunctions, sin ( 2k−1 2 x ) , k = 1, 2, 3, . . . , we EJDE-202X/CONF/27 INITIAL TEMPERATURE PROFILE RECOVERY 99 can choose x0 as x′ 0. Furthermore, we know that∣∣ sin (2k − 1 2 x0 )∣∣ ≥ d for some d > 0 (1.7) Finally, for any forward time sequence tk as in (1.4), the corresponding discrete temperature measurements u(x0, tk), k = 1, 2, . . . are sufficient to determine the initial profile f uniquely. To see this, consider a holomorphic function F (z) = ∞∑ j=1 z(2j−1)2 f̂j sin (2j − 1 2 x ) , where z ∈ D = {z ∈ C : |z| < 1}. Choose zk = e−tk/4 ∈ D where k = 1, 2, 3, . . . . Since zk converges in D, we can uniquely determine F and hence f̂j . Therefore, u(x0, tj) uniquely determine f . 2. 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 L2([0, π]) into a point in Rn, and a decoder is a continuous function that maps each point y ∈ Rn into an element of L2([0, π]). In our setting, an encoder is a continuous function en mapping f ∈ B into en(f) = (u1, u2, . . . , un) ∈ Rn where uk = u(x0, tk), k = 1, 2, . . . , n, is n temperature measurements. On the other hand, a decoder is a continuous function Mn mapping (u1, u2, . . . , un) ∈ Rn into an approximation f̄n of f . The optimal error of approximation to f is defined as δen,Mn (Br, L 2) = sup f∈Br ∥f − f̄n∥L2 (2.1) where f̄n = Mn(en(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 en : f 7→ (u1, u2, . . . , un) and a continuous decoder Mn : (u1, u2, . . . , un) 7→ f̄n as described above, we have δen,Mn (Br, L 2) ≥ Cn−r (2.2) where C is a constant depending on r only. 100 Y. H. YOU EJDE-2024/CONF/27 2.2. Main theorem. Our main goal is to recover an initial profile f with a de- sired rate of accuracy order n−r once we observe n temperature measurements u(x0, tj), j = 1, 2, . . . , n. More precisely, we have the following theorem. Theorem 2.2. Fix x0 in [0, π] as described in (1.7). Given n ∈ N and a bounded interval [0, T ], we choose t1 ∈ (0, T ] such that the next n − 1 future times tj := (n + j − 1)t1, j = 2, . . . n are also in the interval [0, T ]. Let f ∈ Br. Then for the known n temperature measurements u(x0, tj), j = 1, 2, . . . , n, there exists f̄n, an approximation to f , in L2([0, π]) such that ∥f − f̄n∥L2 ≤ Cn−r, (2.3) where C is a constant that depends on r and t1. Setting ck = f̂k sin ( 2k−1 2 x0 ) from (1.2), we first estimate ck and construct an approximation of ck. 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 cj = f̂j sin ( 2k−1 2 x0 ) are bounded. More precisely, |cj | ≤ (j + 2)−r, j = 1, 2, . . . . Now, we define an approximation c̄k of ck. Later, we will use c̄k when construct- ing an approximation f̄n of f . Note that u(x0, tk) = ∞∑ j=1 cje −( 2j−1 2 )2t = k−1∑ j=1 cje −( 2j−1 2 )2t + cke −( 2k−1 2 )2t + ∞∑ j=k+1 cje −( 2j−1 2 )2t, k = 1, 2, 3, . . . , n. From this, we obtain ck = u(x0, tk)e ( 2k−1 2 )2t − k−1∑ j=1 cje [( 2k−1 2 )2−( 2j−1 2 )2]tk − ∞∑ j=k+1 cje [( 2k−1 2 )2−( 2j−1 2 )2]tk = u(x0, tk)e ( 2k−1 2 )2t − k−1∑ j=1 cje (k+j−1)(k−j)tk − ∞∑ j=k+1 cje (k+j−1)(k−j)tk Now we define an approximation c̄k to ck as c̄k := u(x0, tk)e ( 2k−1 2 )2t − k−1∑ j=1 (j + 2)−re(k+j−1)(k−j)tk − n∑ j=k+1 (j + 2)−re(k+j−1)(k−j)tk (2.4) To calculate an error bound between ck and c̄k, we will need a couple of estimates. Lemma 2.4. Let p∗ = T [( T 2t1 + 3 2 ) 2 − 2] and r ≥ p∗. Then, for the n future times tk, k = 1, 2, . . . , n as in Theorem 2.2 and j = 1, . . . , k − 1, we obtain (k2 − j2 + k + j)tk ≤ r. EJDE-202X/CONF/27 INITIAL TEMPERATURE PROFILE RECOVERY 101 Proof. Note that tk = (n+ k − 1)t1 ≤ T for k = 1, . . . , n. Therefore, tn = (2n− 1)t1 ≤ T ⇒ n+ 1 ≤ ( T 2t1 + 3 2 ) (2.5) Then (k2 − j2 + k + j)tk ≤ (k2 + 2k − 1)tk ≤ ((k + 1)2 − 2)tk ≤ ((n+ 1)2 − 2)T ≤ p∗ by (2.5) ≤ r. □ Proposition 2.5. For the n future times tk, k = 1, 2, 3, . . . as in Theorem 2.2, we consider a real number p∗ in Lemma 2.4 and r ≥ p∗. Then |ck − c̄k| ≤ C(t1)ke −2ktk , k = 1, 2, 3, . . . , n. Proof. From the definition of ck and c̄k, we obtain |ck − c̄k| = ∣∣∣ k−1∑ j=1 ((j + 2)−r − cj)e (k+j−1)(k−j)tk − ∞∑ j=k+1 cje (k+j−1)(k−j)tk + n∑ j=k+1 (j + 2)−re(k+j−1)(k−j)tk ∣∣∣ ≤ 2 k−1∑ j=1 (j + 2)−re(k+j−1)(k−j)tk + 2 ∞∑ j=k+1 (j + 2)−re(k+j−1)(k−j)tk ≤ 2 k−1∑ j=1 (j + 2)−re(k+j−1)(k−j)tk + 2(k + 3)−r ∞∑ j=k+1 e(k+j−1)(k−j)tk From the second sum on the right side of the above inequality, we have (k + 3)−r ∞∑ j=k+1 e(k+j−1)(k−j)tk ≤ (k + 3)−r ∞∑ j=k+1 e−(k+j−1)tk = (k + 3)−r ∞∑ j=0 e−(2k+j)tk = (k + 3)−re−2ktk ∞∑ j=0 e−jtk ≤ (k + 3)−re−2ktks(t1) (2.6) where s(t1) := ∞∑ j=0 e−jt1 102 Y. H. YOU EJDE-2024/CONF/27 From Lemma 2.4 and (2.6), we obtain |ck − c̄k| ≤ 2 k−1∑ j=1 (j + 2)−re(k+j−1)(k−j)tk + 2(k + 3)−r ∞∑ j=k+1 e(k+j−1)(k−j)tk ≤ e−2ktk [ 2 k−1∑ j=1 (j + 2)−re((k+j−1)(k−j)+2k)tk + 2(k + 3)−rs(t1) ] ≤ e−2ktk [ 2 k−1∑ j=1 e−re(k 2−j2+k+j)tk + 2(k + 3)−rs(t1) ] ≤ e−2ktk [ 2(k − 1) + 2 (k + 3)r s(t1) ] ≤ ke−2ktk [2 + 2s(t1)] = C(t1)ke −2ktk , (2.7) □ Proof of Theorem 2.2. We define an approximation of f ∈ Br as f̄n(x) := n∑ k=1 ˆ̄fk sin (2k − 1 2 x ) where ˆ̄fk = c̄k/sin ( 2k−1 2 x0 ) . By defining ˆ̄fk = 0 for all k ≥ n+ 1, we obtain ∥f − f̄n∥2 = ∥∥ ∞∑ k=1 (f̂k − ˆ̄fk) sin (2k − 1 2 x )∥∥∥2 L2([0,π]) ≤ C1 ∞∑ k=1 ∣∣f̂k − ˆ̄fk ∣∣2 for some constant C1 ≤ C1 [ n∑ k=1 ∣∣f̂k − ˆ̄fk ∣∣2 + ∞∑ k=n+1 ∣∣f̂k∣∣2] ≤ C1 [ n∑ k=1 ∣∣f̂k − ˆ̄fk ∣∣2 + n−2r ∞∑ k=n+1 (k + 2)2r ∣∣f̂k∣∣2] ≤ C1 [ n∑ k=1 ∣∣∣ ck sin kx0 − c̄k sin kx0 ∣∣∣2 + n−2r ] ≤ C2(t1) n∑ k=1 k2e−4ktk + C1n −2r by (1.7) and Proposition 2.5 ≤ C2(t1) n∑ k=1 k2 e4k(n+k−1)t1 + C1n −2r ≤ C2(t1) e4nt1 n∑ k=1 k2 + C1n −2r ≤ C2(t1) e4nt1 n(n+ 1)(2n+ 1) 6 + C1n −2r EJDE-202X/CONF/27 INITIAL TEMPERATURE PROFILE RECOVERY 103 = C2(t1) n3 6e4nt1 ( 1 + 1 n )( 2 + 1 n ) + C1n −2r For each pair of the parameters r and t1, the sequence{m2r+3 6e4mt1 ( 1 + 1 m )( 2 + 1 m )} is convergent. So, there exists a constant c(r, t1) such that n2r+3 6e4nt1 ( 1 + 1 n )( 2 + 1 n ) ≤ c(r, t1). (2.8) Thus from (2.8), we have ∥f − f̄n∥2L2 ≤ C(r, t1)n −2r. Therefore, ∥f − f̄n∥L2 ≤ Cn−r, where C is a constant depending on t1 and r. 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