On inverse problems for pseudoparabolic and parabolic equations of filtration

Abstract

The inverse problems concerning the identification of the coefficient in the second order terms of linear pseudoparabolic equations of filtration in a fissured rock are investigated. The physical and mathematical justification of possible statements of the inverse problem for pseudoparabolic equations is given. New boundary conditions of overdetermination are discussed. Certain parabolic inverse problems relevant to pseudoparabolic ones are considered. The existence and uniqueness of a strong solution to one of the discussed inverse problems for the pseudoparabolic equation is proved in the case of one space variable. It is also established that the inverse problem for the pseudoparabolic equation approximates weakly some inverse problem for the parabolic one.

Keywords:

• filtration
• inverse problem
• pseudoparabolic equation
• parabolic equation
• approximation

AMS Subject Classifications::

• 35K20
• 35K70
• 35Q35
• 35R30
• 76S05

1. Introduction

In 1960, Barenblatt, Zheltov and Kochina 1 proposed the basic concept in the theory of seepage (filtration) of homogeneous liquids in fissured rocks. A fissured rock is considered as a material consisting of pores and permeable blocks which are generally separated from each other by a system of fissures. Compared to the standard arguments of filtration in a porous medium the significant feature given in 1 lies in the fact that (1) two liquid pressures, both in the pores and in the fissures, are introduced at any point in a space and (2) the transfer of liquids between the fissures and the pores is taken into consideration.

Under such an approach the model of the seepage of a liquid in a fissured rock is described by the system of equations 1 (1) where u₁ = u₁(t, x), u₂ = u₂(t, x) are the pressures of the liquid in the fissures and pores, respectively; Δ is the Laplacian operator; d₁ and d₂ are the coefficients of compressibility of the liquid and the blocks; m₀ is the magnitude of the porosity of the blocks at standard pressure; μ is the viscosity of the liquid; ν represents the permeability of fissures. The dimensionless coefficient α characterizes the intensity of the liquid transfer between the blocks and fissures. In general the model can include the nonlinearities arising from fluid type (liquid or gas), concentration (porosity, absorption or saturation) and the exchange rate 2.

Eliminating u₂ from (1.1) we obtain for the pressure of the liquid in the fissures u₁ the so-called fissured medium equation of pseudoparabolic type (2) The parameter κ corresponds to the piezo-conductivity of fissured rock. The coefficient η represents a specific characteristic of fissured rock. The pressure of the liquid in the pores u₂ satisfies an analogous equation. If η tends to zero, it corresponds to a reduction of the block dimensions and an increase of fissuring, and equation (1.2) will obviously tend to coincide with the ordinary filtration equation. Since the natural stratum is involved, the parameters of fissured rock in (1.2) should be determined on the basis of the investigation of their behaviour under the natural nonsteady-state conditions but not on the basis of the tests carried out on rock speciments brought to the surface. This leads to the interest in studying the inverse problems for (1.2) and its analogue.

Pseudoparabolic equations of the form (3) with various differential operators L₁ and L₂ of the even order in spacial variables also arise in the mathematical models of the heat conduction 3, wave processes 4, quasi-stationary processes in semiconductors and magnetics 5 (for more details, see 5,6 and references therein). Similar equations appears in the models for filtration of the two-phase flow in porous media with the dynamic capillary pressure 7–9.

A variety of works have been devoted to the direct problems for (1.3) (apart from the above references, see 10–16). These authors considered initial and initial-boundary-value problems for linear and nonlinear pseudoparabolic equations (1.3). In particular, Gajewski, Groger and Zacharias 12 studied the local solvability of pseudoparabolic operator and operator-differential equations. For operator-differential pseudoparabolic equations, C-and L²-solvability was analysed. Methods of finite-dimensional approximation (in particular, the Galerkin method) were validated. Korpusov and Sveshnikov 5 investigated the blow-up of the solution to certain classes of strongly nonlinear pseudoparabolic equations by the method of energy estimates. In 6,15, Showalter and Ting proved the theorems on the global (in time) solvability of the initial boundary value problem for (1.3) with linear elliptic operators L₁ and L₂ in the strong generalized or weak generalized sense. They also studied the regularity properties and stabilization of the solution to (1.3). The problems for certain linear equations (1.3) with nonlocal boundary conditions is investigated by Kozhanov in 13. Control problems for pseudoparabolic equations were considered in 14,16. For a more comprehensive review we refer the reader to 5.

The investigation of inverse problems for pseudoparabolic equations goes back to the 1980s. The first result obtained by Rundell in 17 is concerned with the inverse problems of the identification of an unknown source f in the (1.3) with linear operators L₁ and L₂, L₁ = L₂. Rundell proved the global existence and uniqueness theorems in the case that f depends only on x or t. Another kind of inverse problems is considered in 18,19. These works are devoted to problems of reconstructing the kernels in integral term of (1.3) with the integro-differential operator L₂. As for the determination of unknown coefficients in (1.3) we mention the result of Mamayusupov 20. He proved the uniqueness theorem and found an algorithm for solving the inverse problem with respect to u(t, x), functions b(y), c(y) and a constant a for the equation provided that u(t, x, 0), u_y(t, x, 0) and u(0, x, y) are given. Here δ(t, x, y) is the Dirac delta function. To the authors' knowledge, inverse problems of the identification of unknown variable coefficients in the terms of the second and third order of (1.3) have not been posed and studied yet.

This article is concerned with the inverse problem of finding the unknown coefficient k = k(t) in equation (4) where η and f are given. In the physical context of k(t) this means to find the unknown average hydraulic characteristics d₁ and d₂.

Our study is divided into three parts. The first part contains the physical and mathematical justification of new statements of the inverse problem for (1.4). We also consider certain parabolic inverse problems relevant to pseudoparabolic ones. This will be discussed in Section 2.

In Section 3 we shall establish the existence and uniqueness of a solution to the inverse problem for (1.4) in the case of one spacial variable (see Problem 1 below).

As mentioned above, when passing to the limit η → 0 equation (1.4) formally tends to coincide with the standard linear equation of filtration in a porous medium (5) It is well known that the direct initial boundary value problem for pseudoparabolic equation in a bounded domain Ω ⊂ Rⁿ approximates the corresponding problem for parabolic equation 15. In particular, under certain assumptions the solution u^η of Equation (1.4) with the initial data u^η(0, x) = u₀(x) tends to the solution u of (1.5) with the same initial condition in the norm of L²(Ω) for all t ≥ 0 as η → 0. However, for the inverse problems, so far as the authors know, these remain unsolved. We discuss it in Section 4. Such an investigation is also of interest in studying the inverse problems for parabolic equations whose principal terms contain unknown coefficients.

2. On posing inverse problems

The physical processes modelled by (1.4) occur in bounded domains. Therefore the initial and boundary conditions must be imposed for (1.4). To find out mathematical formulation of these we start from the model (1.1).

Let a problem be considered in a domain of the stratum Ω ⊂ R³ with boundary ∂Ω, t ∈ (0, T) and T is a positive real number. As discussed above, u₁ and u₂ both satisfy (1.4). The initial data for u₁, u₂ are 10 respectively. Since the first equation of (1.1) is elliptic, the boundary conditions for u₁ can take the form or where n is the outward unit normal to ∂Ω. u₀ is given on Ω and A, B, g are given functions on (0, T) × ∂Ω. From here and the second equation (1.1) we obtain the boundary conditions for u₂: or

From the above we conclude that, in general, the initial data for (1.4) are given as (6) where U₀(x) is a known function. If k(t) ≠ 0, then among the possible types of the boundary conditions the most important can be written as the condition of the Dirichlet type (7) or (8) the Neumann type (9) and the general mixed type (10) Here g₁ and g₂ are given functions on (0, T) × ∂Ω. In the case of u₁ the Dirichlet data (2.3) with g₁(t, x) = g(t, x) and g₂(t, x) = ηg_t(t, x) comes from (2.2). The formulae (2.4), (2.5) with the same functions g₁ and g₂ are deduced from the appropriate Neuman and mixed boundary conditions for u₁. The boundary data for u₂ are of the form (2.3–2.5) with g₁(t, x) = g(t, x) and g₂(t, x) ≡ 0. Thus, three direct initial boundary value problems can be posed for (1.4) when k(t) is known.

In physical context the Neumann condition (2.4) describes the given flow. The mixed condition (2.5) deals with the heat or mass transfer through the boundary ∂Ω.

The inverse problem of identification of the unknown coefficient k(t) with each one of the above boundary conditions is underdetermined, so that in order to recover k(t) we are forced to impose an additional condition. The identification of k(t) here can be based on the boundary data of the pointwise or integral type. This leads to the pointwise or integral condition of overdetermination, which is in general written as (11) or (12) Here σ₁, σ₂ are real numbers, ω₁(t, s), ω₂(t, s) are given functions, x₀ ∈ ∂Ω and Γ ⊆ ∂Ω. In the case of the Dirichlet boundary problem (1.4), (2.1), (2.2) after substituting (2.3) into (2.6) and (2.7) the conditions of overdetermination take the form (13) or (14) respectively. Here

If ω₂(t, s) ≡ 1 and μ₂ ≡ 0, then the integral condition means a given flux of a liquid through the surface Γ, for instance, the total discharge of a liquid through the surface of the ground. Similar nonlocal conditions were applied to control problems in 14,16 and to elliptic inverse problem in 21.

In the case of the problems (1.4), (2.1), (2.4) and (1.4), (2.1), (2.5), as the condition of overdetermination, (2.6) or (2.7) are to be taken of the form (15) (16) respectively. In particular, for the problem (1.4), (2.1), (2.4) We will not consider these cases in this article.

The inverse problems for (1.4) are closely allied with the appropriate problems for (1.5). If U ≡ u₀ − ηΔu₀ in (2.1) where and u₀|_∂Ω = g(0, x), then Equation (1.4) formally tends to coincide with (1.5) when η → 0, the initial condition takes the form (17) (2.4), (2.5) pass into the standard Nemmann and mixed boundary data and (2.6), (2.7) transform into the conditions (18) and (19) respectively.

Inverse problem of restoring k(t) in (1.5) under the conditions (2.2), (2.12) and (2.13) with σ₁ = 0 were studied by Cannon and Rundell 22 (see also the references therein). The inverse problems of finding k(t) in (1.5) under the integral condition (2.14) has not been considered yet.

3. Existence and uniqueness

In this section we consider the inverse problem of finding k(t) in (1.2) under the integral conditions (2.9) in the case of one spacial variable on Ω = (0, l). Hereafter, we shall use the symbols (·, ·) and ‖·‖ to denote the inner product and the norm of L²(0, l), respectively, and ‖·‖_i (i = 1, 2) for the norm of . We also keep the notation (20) for h(t) ∈ C([0, T] ).

Problem 1

Find the pair of functions (u(t, x), k(t)) satisfying equation (21) the initial condition (22) the boundary conditions (23) and (24) provided that f(t, x), u₀(x), β₁(t), β₂(t), ϕ₁(t), ϕ₂(t) and η are given.

From the results in 12, Chapter 5] and 6 it can be easily shown that if , f ∈ C([0, T] ; L²(Q_T)), β₁, β₂ ∈ C¹ ([0, T] ) and k(t) ∈ C([0, T] ) are known, then the direct initial boundary value problem (3.2–3.4) has a solution and the solution is unique.

By the strong solution of Problem 1 we mean a pair (u(t, x), k(t)) such that

	(a) ; (b) k(t) ∈ C([0, T] );
	(c) Equations (3.2–3.5) are satisfied.

To prove the existence and uniqueness of a strong solution to Problem 1 we need the following result.

Lemma 3.1

Let v(t, x) be a solution belonging to of problem where η is a positive constant and q(t), g(t, x), v₀(x), β₁(t), β₂(t) are given functions with properties q ∈ C([0, T] ), β₁, β₂ ∈ C¹([0, T] ), , g ∈ C([0, T] ; L²(0, l)), v₀(0) = β₁(0), v₀(l) = β₂(0). Suppose that β₁(t), β₂(t), v₀(x) are non-negative, g(t, x) is non-negative almost everywhere in , q(t) is positive on [0, T] and the inequality v₀(x) − G(0, x) ≥ 0 holds for some function G(t, x) satisfying Then v(t, x) ≥ 0 in [0, T]  × [0, l] .

This lemma is just an extension of the result due to Rundell and Stecher 23.

Theorem 3.2

Let η be a positive constant. Assume that

i.	f(t, x) ∈ C([0, T] ; L2(0, l)), β1(t), β2(t), ϕ1(t) ∈ C1([0, T] ), ϕ2(t) ∈ C([0, T] ),
ii.	f(t, x) is non-negative almost everywhere in , u0(x), ϕ1(t), ϕ2(t) are non-negative, u0(0) = β1(0), u0(l) = β2(0), (25)
iii.	there exist positive constants ϕ0, γ and non-negative constants α0, α1, such that αi ≤ 1 (i = 0, 1), α0 + α1 ≠ 2, (26) (27) where (28) (29) (30) (31) Then Problem 1 has a unique solution . Moreover, coefficient k(t) satisfies the inequality (32) with some positive constants k0, k1.

Proof

Main idea of the proof lies in the reduction of Problem 1 to an equivalent inverse problem with a nonlinear operator equation for the coefficient k(t) 24. By virtue of (3.5), (3.9) we have (33)

Multiplying (3.2) by b(x), integrating over (0, l) and substituting (3.14) into the resulting equation yields This equation can be written, in view of the relations b−ηb_xx ≡ 0, b(0) = 1, b(l) = 0, (1, b) = η^1/2tanh(lη^−1/2/2), and as (34) It is easily seen that Problem 1 and problems (3.2–3.4), (3.15) are equivalent. Therefore we shall prove the same assertion for (3.2–3.4), (3.15) as that in Theorem 3.2.

For the proof of existence of solution (u(t, x), k(t)) we use the iteration scheme : (35) (36)

First, let us prove inductively that for each i = 1, 2, 3, … problem (3.16), (3.17) has a unique solution satisfying (37) (38) with Indeed, for i = 1 the results in 12 enable us to establish that the problem (3.16) has a unique solution . Then Lemma 3.1 and the hypotheses (3.6–3.8) and (3.12) of Theorem 3.2 give the inequality from which the inequality (39) follows. Here we use the inequality ‖b‖² ≤ (η^1/2/2)coth(l/η^1/2). The smoothness of the input data, (3.7), (3.17) and (3.20) lead to the estimate with k₀, k₁ mentioned above. Next assume that for i ≥ 1 and kⁱ ∈ C([0, T] ) satisfy (3.18) and (3.19), respectively. Then the arguments similar to those in the proofs of (3.18) and (3.19) for i = 1 yield that the problem (3.16), (3.17) with i + 1 also has a unique solution , and uⁱ⁺¹ and kⁱ⁺¹ satisfy (3.18) and (3.19), respectively. Thus, by induction there exists a unique solution to the problem (3.16), (3.17), and uⁱ and kⁱ satisfy (3.18) and (3.19), respectively, for every i = 1, 2, 3, … .

Second, in order to prove the convergence of {(uⁱ, kⁱ)} we prepare the estimates in higher norms. Multiplying (3.16)₁ by and integrating over (0, l), we have by integration by parts and Schwarz' inequality Applying Gronwall's lemma to this inequality leads to (40) where C₁ is a positive constant depending on , η, ‖u₀‖₂, and . Furthermore, multiplying (3.16)₁ by , integrating over (0, l) we get by integration by parts and the inequalities (3.19), (3.21) and hence (41) From (3.16)₁ with the help of (3.19), (3.21), (3.22) the estimate (42) is easily derived. Here positive constants C₂, C₃ depend on η, l, T, k₁, ‖u₀‖₂, C₁ and .

Let Bt virtue of (3.13), (3.18) and (3.19) we have (43) Obviously, ũⁱ is the solution of problem (44) Multiplying (3.25)₁ by and integrating over (0, l), we obtain in aid of the integration by parts and Schwarz' inequality By Gronwall's lemma, this together with (3.21) implies that (45) where C₄ is a positive constant depending on η, C₁. Now, let us introduce the following equivalent norm in C[0, T]  for a positive constant m determined later. Then, (3.24) and (3.26) yield (46) with a positive constant C₅ dependent on l, η, γ, k₁, C₄ and T. Inequality (3.27) shows that there exists a limit k(t) of the sequence {kⁱ(t)} when . This in turn provides the convergence of {uⁱ} to a function , since (3.26) and holds with a constant C₆ depending on C₄, k₁, η, T. Letting i → ∞ in (3.16), (3.17) we see that the pair (u(t, x), k(t)) is the solution of problem (3.2–3.4), (3.15). Besides, u(t, x) and k(t) satisfy the estimates (3.18), (3.19), (3.21–3.23).

Uniqueness of the solution (u(t, x), k(t)) is obvious.▪

The hypotheses (3.7), (3.8) and (3.10–3.12) can be satisfied by some large set of given data. For instance, if u₀ ≥ 0 and u_0xx ≤ 0 on (0, l), then (3.8) is fulfilled by the maximum principle for elliptic equations. If, in addition, f is a nonnegative convex function (in particular, f ≡ 0), ϕ₁ ≥ 0, β₁ > β₂ ≥ 0, , and for t ∈ [0, T] , then (3.10–3.12) holds, for instance, with α₀, α₁ ≥ 0, α₀ + α₁ ≤ 1. By (3.9), so that (3.7) is valid when ϕ₂ > 0 and η is small enough.

Condition (3.10) may seem peculiar, however the following example shows that it cannot be removed in general.

Example

Let u(t, x) satisfy (47) where β and ϕ₂ are positive constants. The input data of problem (3.28) fit with all conditions of Theorem 3.2 except (3.10) when we choose ϕ₂ > 0, 0 < η ≤ 1/4 (48) and (49) Indeed, if β obeys (3.30), then there exists a σ > 0 such that for any α₁, 0 ≤ α₁ ≤ 1.

Let This function satisfies (50)

From (3.31)_1,2,3 it follows that the solution w is given in the form (51) with for any integrable function k(t). Substituting (3.32) into (3.31)₄ and defining we obtain the ordinary differential equation (52) Since y(0) = 1, the solution to (3.33) is given by (53) Note that h′(1) = 0 because of (3.29). Moreover, if (54) then h″(1) < 0, so that h(y) attains a maximum at this point. Therefore, for β satisfying (3.30), (3.35) there exists a positive number κ such that which is impossible because of (3.34) and the positivity of ϕ₂. Consequently, (3.28) has no solution in the framework of an integrable function k(t).

4. Approximation of parabolic inverse problem

As mentioned in Section 2, when passing to the limit η → 0, Equation (3.2) formally tends to coincise with the linear parabolic equation and Problem 1 transforms to the following parabolic inverse problem.

Problem 2

Find the pair of functions (u(t, x), k(t)) satisfying equation (55) the initial and boundary conditions (3.3), (3.4) and (56) provided that f(t, x), u₀(x), β₁(t), β₁(t), ϕ₁(t) and ϕ₂(t) and are given.

Hereafter, by the solution of Problem 2 we mean a pair (u(t, x), k(t)) such that

a.	;
b.	k(t) ∈ L∞(0, T), k(t) > 0;
c.	Equations (3.3), (3.4), (4.1) and (4.2) are satisfied.

Problem 2 is the limit case (η → 0) for Problem 1, i.e. the solution (u^η, k^η) of Problem 1 tends to (u, k) as η → 0. This implies that Problem 2 is solved by relying on the results on Problem 1.

Theorem 4.1

Let η₀ be a positive constant and η ∈ (0, η₀]. Suppose that the conditions (i), (ii) of Theorem 3.2 hold. Let also (iii′) f_t ∈ L²(Q_T), (3.8) and (3.12) are fulfilled with α₀ = α₁ = 0, (57) (58) (59) Then as η → 0 and the limit (u, k) is a solution to Problem 2. Moreover, (60) where r(η) is a bounded continuous function of η on (0, η₀], and (61)

Proof

Without loss of generality we can assume that η₀ is chosen in such a way that (62) holds for any η ∈ (0, η₀]. Therefore one can easily check that under the hypotheses (4.3–4.5) all the assumptions of Theorem 3.2 are fulfilled with α₀ = α₁ = 0. This implies that Problem 1 has a unique solution and estimates (3.18), (3.19), (3.21–3.23) hold for any η, 0 < η ≤ η₀. Thus our next step is to get a uniform lower bound of k^η and then the uniform estimates of and .

Let us consider the function (63)

The pair (w^η(t, x), k^η(t)) is the solution of problem (64) For every k^η(t) one can represent w^η as where (65) and w_0n and f_1n(θ) are Fourier coefficients of w₀ and f₁, respectively. Let By (3.19), for any η > 0. Then we get (66)

Since and w₀(0) = w₀(l) = 0, it is easy to derive from Schwarz' inequality that (67) The positive constant K depends on l . To estimate the second term in the right-hand side of (4.12) we use Hölder inequality and the smoothness of f, so that we have where K₂ is a positive constant depending on l, T. Hence (68) Thus, (4.12–4.14) give (69) From (3.1), (3.7), (3.15), (4.8), (4.15) and the definition of it follows that hence (70) The positive constant K₄ depends on K₁, , , , l. Since , there exists a number such that

Now it remains to estimate . Let Then it is easily seen that g(y) has a unique positive real root, which can be taken as and (71)

From (4.17) we conclude that if then . Otherwise, . Consequently, we get (72) that is, (4.6) is proved. It is clear that r(η) satisfies (4.7). The uniform estimates of , and are derived from (3.1), (4.9), (4.10)_1,2,3, (4.18) in the same way as (3.21–3.23): (73) (74)

Thus, from (3.13), (3.18), (4.19), (4.20) it follows that there exists a subsequence of and a pair of functions (u, k) such that (75) as η′ → 0. By virtue of the embedding theorem, (4.21) implies that (76) as η′ → 0.

We are now in a position to show that (u, k) is a solution of Problem 2. Indeed, satisfies the identity (77) for every . In view of (4.21), (4.22), we can pass to the limit in (4.23). Since (because of (4.20)), we have (78) Moreover, from (3.13), (3.18), (4.18–4.20), one can derive the estimates (79) (80) (81) (82) From (4.24–4.28) it follows that (u(t, x), k(t)) satisfies the parabolic equation (4.1) for almost all (t, x) ∈ Q_T. Furthermore, by (4.21)_1,2, u(t, x) obeys (3.3) and (3.4). It remains to verify (4.2). Let v(t, x) = g(t, x)h(t), where g(t,x) and h(t) are arbitrary functions of classes and L²(0, T), respectively, satisfying g(t, 0) = β₁(t) and g(t, l) = 0. Then it is clear that the identity (83) holds because of (3.5). Passing to the limit η′ → 0 in (4.29) similar to the above implies that (84) By virtue of (4.1), the integration by parts in the second term of the left-hand side of (4.30) gives for any h(t) ∈ L²(0, T). This yields that (u(t, x), k(t)) satisfies (4.2) for almost all t ∈ (0, T).▪

Theorem 4.2

Theorem 4.1 remains true if .

Proof

Since this theorem is proved in just the same way as Theorem 4.1, we focus on the estimate (4.6). Let us turn to (4.12). If , then The positive constant c depends on l. Hence from which follows. Here the positive constant c₁ depends on c, , and in turn depends on l, T. Now we only need to repeat the arguments leading to (4.18). It gives the estimate Thus, (4.6) is proved.▪

Under the hypotheses of Theorem 4.1 the solution to Problem 2 is unique in the class

Indeed, let (u₁, k₁(t)) and (u₂, k₂(t)) be two solutions of Problem 2 (u₁, k₁(t)), (u₂, k₂(t)) ∈ V × L^∞(0, T). Then (w(t, x), p(t)) = (u₁ − u₂, k₁(t) − k₂(t)) is a solution of the problem (85) Multiplying (4.31)₁ by −w_xx and integrating over (0, l), we can easily obtain by the integraton by parts (86) From (3.29), (4.31)₃ and the multiplicative inequality 12 it follows that (87) Since , (4.32) and (4.33) give, according to Gronwall's lemma, which implies from (4.31)_2,3 that w = 0 for almost all (t, x) ∈ Q_T and p = 0 for almost all t ∈ (0, T).

5. Summary

We discussed certain new statements of the inverse problems for pseudoparabolic equations by the example of the linear fissured rock equation. The conditions of overdetermination similar to (2.8–2.11) are available for the linear pseudoparabolic equations (1.3) with the operators L₁ and L₂ of more general form and also for the nonlinear equations with various functions γ₁ and γ₂ arising in generalized models of the liquid flow in porous media 2,7.

The results of Section 3 assure not only the existence and uniqueness of the solution to Problem 1, but the stability as well. Under the hypotheses of Theorem 3.2 the stability of the solution is provided by that the operator in right-hand side of (3.15) is a contruction in the norm . Moreover, this makes possible a numerical reconstruction of the solution to Problem 2 by the iteration scheme (3.16), (3.17).

Theorems 4.1 and 4.2 give the sufficient conditions of approximation of Problem 2 by Problem 1. This implies the existence of the solution to Problem 2. As indicated above, this solution is unique. The hypotheses of the theorems fitted well with the results of Cannon and Rundell 22. Moreover, the assumptions of Theorems 4.1 and 4.2 on ϕ₂ are weaker than in 22. However, in general, Problem 1 cannot be taken as the regularization of Problem 2 unlike the direct initial boundary problems (see the example of Section 3).

This article provides guidelines for further considerations. We considered only the one-dimensional inverse problems for (1.1) and (1.4) with the Dirichlet boundary data and condition of overdetermination (2.9). The techniques used in this article can be applied to more dimensions with suitable extensions of the boundary data and to the case of pointwise measurements and the case of Neumann or mixed boundary conditions. The method is also available to recover the coefficient η. These problems are the subject of further studies.