Abstract
A non-linear wave equation with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional (space) integral measurement. The global in time existence, uniqueness as well as the regularity of a solution is addressed. A new numerical algorithm based on Rothe’s method is designed and error estimates are derived.
1. Introduction
We consider a bounded domain with sufficiently smooth boundary . The symbol stands for the outer normal vector associated with . In this paper, we are interested in determining of an unknown couple obeying the following non-linear hyperbolic problem of second order(1) (1)
where and are given real functions. By we denote the usual convolution in time, namely . The missing time-convolution kernel will be recovered from the following integral-type measurement(2) (2)
The integral-type over-determination in inverse problems (IPs) combined with evolutionary partial differential equations (PDEs) has been studied in several papers, e.g. [Citation1–Citation3] and the references therein.
A well-known example of (a homogeneous version of) the general wave equation is the telegraph equation. It describes the voltage inside a piece of telegraph/transmission wire, whose electrical properties per unit length are: resistance , inductance , capacitance and conductance of leakage current :
where , and When considering transport for long distances, the memory effects can be modelled by a additional convolution term.
Identification of missing memory kernels in evolutionary PDEs represents an interesting topic in IPs. Intensive study of such problems was implemented in 1990’s cf. [Citation4–Citation16]. The paper [Citation12] deals with a linear hyperbolic equation without a damping term. Here, the time convolution operator is applied to the Laplacian of solution. The missing data are recovered from a point measurement. There is no constructive algorithm for finding a solution for this IP. The manuscript [Citation11] studies identification of convolution kernels in abstract linear hyperbolic integro-differential problems. The author shows local in time solvability of this IP. No constructive algorithm for recovery of missing convolution kernel is present. The study performed in [Citation14] is devoted to a one-dimensional linear hyperbolic integro-differential problem. The author uses finite differences for numerical scheme with dependent time and space mesh-steps. The error estimates are derived under high regularity of solution. Bukhgeĭm and Dyatlov [Citation4] presents a nice study of properties of Dirichlet-to-Neumann maps for memory reconstruction for linear settings. The paper [Citation16] studies IPs for identifying memory kernels in linear heat conduction and viscoelasticity by the method of least squares with Tikhonov regularization. In [Citation8] a global in time existence and uniqueness result for an IP arising in the theory of heat conduction for materials with memory has been studied. The Ref. [Citation6] derives some local and global in time existence results for the recovery of memory kernels, but there is no description of constructive algorithms how to find a solution. The papers [Citation9,Citation10] deals with the identification of missing memory kernel in a parabolic problem.
The ultimate highlight of this paper is to design a constructive algorithm for finding the solution and to recover the unknown kernel in non-linear hyperbolic problems. This is not achieved by minimization of a cost functional (which is typical for IPs [Citation17–Citation19]), but on the time discretization based on method of lines.[Citation20,Citation21] This technique was already applied for parabolic settings in [Citation9]. First, we start with derivation of a suitable variational formulation. Section 2 is devoted to a time discretization, where (based on backward Euler scheme) the continuous problem is approximated by a sequence of steady state settings at each point of a time partitioning. Section 3 deals with existence and uniqueness of a solution. Stability analysis of approximates is performed in appropriate function spaces and convergence (based on compactness argument) is established in Theorem 3.1. Uniqueness is addressed in Theorem 3.2. Theorem 3.3 shows the error estimates for time discretization.
Denote by the standard inner product of and its induced norm. When working at the boundary we use a similar notation, namely , and . By we denote the set of abstract functions equipped with the usual norm and is furnished with the norm with , cf. [Citation22]. The symbol stands for the dual space to . The symbols and will denote generic positive constants (they may change from line to line) depending only on a priori known quantities, where is small and is large. Further, we set
Denoting the primitive function of by we may rewrite the convolution kernel by integration by parts formula in an another way as follows
Hence the governing PDE takes the form(3) (3)
in . We take a test function , and derive after integration over and making use of the Green’s first identity(P) (P)
Setting in (EquationP(P) (P) ) we obtain together with the measurement that(MP) (MP)
The relations (EquationP(P) (P) ) and (EquationMP(MP) (MP) ) represent the variational formulation of (Equation1(1) (1) ) and (Equation2(2) (2) ).
2. Time discretization
Rothe’s method [Citation20,Citation21] represents a constructive method suitable for solving evolution problems. Using a simple discretization in time, a time-dependent problem is approximated by a sequence of elliptic problems which have to be solved successively with increasing time step. This standard technique is in our case complicated by the unknown convolution kernel . There exists a simple way to overcome this difficulty.
For ease of explanation we consider an equidistant time partitioning of the time frame with a step for any . We use the notation and for any function we write
Further, we set to omit double indices. We will consider a decoupled system with unknowns for . At time we infer from (Equation3(3) (3) ) the backward Euler scheme(4) (4)
where . Like (EquationP(P) (P) ) and (EquationMP(MP) (MP) ) one obtains for that(DPi) (DPi)
along with and(DMPi) (DMPi)
Note that for a given we solve first (EquationDMP(DMPi) (DMPi) ) and then (EquationDP(DPi) (DPi) ). Further, we increase to .
Lemma 2.1:
Let be bounded, i.e. . Moreover assume that , , , . Then there exist and such that for any and each we have
(i) | there exist and obeying (EquationDMP(DMPi) (DMPi) ) and (EquationDP(DPi) (DPi) ) | ||||
(ii) | . |
Proof:
(i) Set . Then for any we may write by triangle inequality that
We apply the following recursive deduction for .
(1) | Let and be given. Then (EquationDMP(DMPi) (DMPi) ) implies the existence of such that(5) (5) | ||||
(2) | The existence and uniqueness of follows from (EquationDP(DPi) (DPi) ) by the Lax-Milgram lemma. |
which is valid for any . An application of the discrete Grönwall lemma [Citation24] gives the uniform bound of .
Lemma 2.2:
Let the conditions of Lemma 2.1 be satisfied. Moreover assume that . Then there exists such that for any
Proof:
If we set in (EquationDP(DPi) (DPi) ) and sum up for we obtain(6) (6)
Let be any sequence of real numbers, then the following obvious identity holds true
Thus
and For any real sequences and the following (summation by parts) identity takes place(7) (7)
For the third term of (Equation6(6) (6) ) we get by (Equation7(7) (7) )
by Cauchy’s inequality, the trace theorem and Young’s inequality. The fourth guy in (Equation6(6) (6) ) is easily bounded by
as is bounded, see Lemma 2.1. The last term in the left-hand side of (Equation6(6) (6) ) is
again as is bounded, see Lemma 2.1. The right-hand side of (Equation6(6) (6) ) can be estimated as follows
Collecting the estimates together and fixing a sufficiently small we obtain
Applying the discrete Grönwall lemma we conclude the proof.
Lemma 2.3:
Let be bounded, i.e. , and globally Lipschitz continuous. Suppose that (EquationMP(MP) (MP) ) is valid at the time (compatibility condition). Moreover assume that , , , and . Then there exists such that
(i) | |||||
(ii) | |||||
(iii) | . |
Proof:
(i) Multiplying (Equation4(4) (4) ) by and summing up for we get
We apply the Green’s theorem to find out It holds
and
Employing the Cauchy and Young inequalities we easily see that We recall the Nečas inequality [Citation25](8) (8)
Summation by parts formula, Cauchy and Young inequalities, trace theorem together with (Equation8(8) (8) ) allow us to deduce that
Summarizing all estimates and fixing a suitable small we arrive at Employing Grönwall’s lemma we conclude the proof.
(ii) The assertion follows from (i) and (Equation4(4) (4) ) as follows(iii) Subtract (EquationMP(MP) (MP) ) at the time from (EquationDMP(DMPi) (DMPi) ) for to observe(9) (9)
Applying the -operator to (EquationDMP(DMPi) (DMPi) ) we get for (10) (10)
Taking into account Lemmas 2.1, 2.2 and Lipschitz continuity of we easily see that
3. Existence of a solution, uniqueness and error estimates
Now, let us introduce the following piecewise linear functions in time
and step functions
Similarly we define , , , , and . These prolongations are also called Rothe’s (piecewise linear and continuous, or piecewise constant) functions. Now, we can rewrite (EquationDP(DPi) (DPi) ) and (EquationDMP(DMPi) (DMPi) ) on the whole time frame asFootnote1(DP) (DP)
and(DMP) (DMP)
Now, we are in a position to prove the existence of a weak solution to (EquationP(P) (P) ) and (EquationMP(MP) (MP) ).
Theorem 3.1:
Suppose the conditions of Lemma 2.3 are fulfilled. Then there exists a solution to (EquationP(P) (P) ) and (EquationMP(MP) (MP) ), where with , and with .
Proof:
The functions and are uniformly bounded in . By Arzelà Ascoli theorem we get for a subsequence of (which we denote by the same symbol again to skip double indices) that in and Hence, is a.e. in bounded. Moreover we have(11) (11)
A priori estimates say that . Due to the compact embedding we may invoke [Citation20, Lemma 1.3.13] to claim the existence of with and a subsequence of (denoted by the same symbol again) such that
The limit function is moreover Lipschitz continuous, which follows from
when passing to the limit for . Similarly we have(12) (12)
The stability result and the compact embedding give (by [Citation20, Lemma 1.3.13]) the existence of with and a subsequence of (denoted by the same symbol again) such that
The limit function is again Lipschitz continuous, which follows from
when passing to the limit for . Analogously, we get(13) (13)
Moreover, taking into account , letting go to infinity and differentiating the result with respect to the time variable we see that .
Finally, the regularity of , together with the convergences above allow us to pass to the limit for in (EquationDP(DP) (DP) ) and (EquationDMP(DMP) (DMP) ) to arrive at (EquationP(P) (P) ) and (EquationMP(MP) (MP) ). Thus the couple is a solution to (EquationP(P) (P) ) and (EquationMP(MP) (MP) ).
Now, we are in a position to state unicity of solution. Suppose and solve (EquationP(P) (P) )–(EquationMP(MP) (MP) ). We set and . Then by subtracting the corresponding variational formulations from each other we obtain
Theorem 3.2:
Let be bounded, i.e. , and globally Lipschitz continuous. Moreover assume that , , , and . Then the problem (EquationP(P) (P) )–(EquationMP(MP) (MP) ) has at most one solution satisfying with , and with .
Proof:
The Lipschitz continuity of and (14b) imply
which by Grönwall’s lemma gives(15) (15)
We put in (14a) and integrate in time Using Cauchy’s inequality, we obtain successively the bounds
as is Lipschitz,
using . Taking into account and (Equation15(15) (15) )
Finally, we deduce that by (Equation15(15) (15) )
Grouping all estimates together we see that
Grönwall’s lemma implies that and from (Equation15(15) (15) ) we conclude . By differentiation we have .
The convergences of Rothe’s functions towards the weak solution (EquationP(P) (P) )–(EquationMP(MP) (MP) ) (as stated in the proof of Theorem 3.1) have been shown for a subsequence. Note, that taking into account Theorem 3.2 we see that the whole Rothe’s functions converge against the solution. Error estimates are addressed in the next theorem.
Theorem 3.3:
Suppose the conditions of Lemma 2.3 are fulfilled. Then
Proof:
Subtract (EquationMP(MP) (MP) ) from (EquationDMP(DMP) (DMP) ) to get(16) (16)
Clearly
by the mean value theorem, and . Triangle inequality, (Equation12(12) (12) ), (Equation13(13) (13) ), Lipschitz continuity of imply Using and (Equation11(11) (11) ) we easily get
This together with (Equation16(16) (16) ) and (Equation13(13) (13) ) say Hence, by Grönwall’s argument we obtain(17) (17)
Now, subtract (EquationDP(DP) (DP) ) from (EquationP(P) (P) ) to get
Based on the triangle inequality, (Equation11(11) (11) )–(Equation13(13) (13) ) and
we are allowed to write(18) (18)
Now, we set and integrate in time(19) (19)
The following estimate is the classical bottleneck for hyperbolic problems, i.e. the bound
A simple deduction yields
Similarly we deduce
as . Further as . Using the Lipschitz continuity of we derive in a standard way Summarizing all estimates we arrive at
which holds true in . By Grönwall’s argument we conclude
Clearly
and by (Equation17(17) (17) ) we conclude
4. Numerical experiment
In this section, we support theoretical results from previous sections on a concrete example, testing convergence of the computational scheme and error estimates from Theorem 3.3. Let’s take and . Further we set
One can easily check that the exact solution to (Equation1(1) (1) ) and (Equation2(2) (2) ) is and . We divide the space interval in 200 equidistant subintervals. We choose the time step as , where . We denote the left-hand side of inequality from Theorem 3.3 as :
The errors for mentioned are depicted in Figure , where the errors are plotted as a function of . The linear regression line through all the data points is given by , which indicates the convergence rate . This error estimate is better that the theoretical result from Theorem 3.3. The absolute error for ,i.e. , is shown for in Figure .
5. Conclusion
A semilinear hyperbolic integro-differential problem of second order with an unknown convolution kernel is considered. The well-posedness of a weak solution for the IBVP is proved. The missing integral kernel is recovered from an additional space-integral measurement. A numerical algorithm based on Rothe’s method is established, convergence of approximations towards the exact solution is demonstrated and the error estimates are derived. The IP was reformulated to a direct system of two equations. Please note that the suggested algorithm involves time derivatives of measurements, thus the IP is moderately ill-posed. There arises a natural question how to deal with noisy data? In such a case we suggest to regularize the measurements first and then to apply the suggested scheme.
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1 when .
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