V. G. Romanov

ON ONE FORMULATION OF AN INVERSE PROBLEM FOR A GENERALIZED WAVE EQUATION

(Presented by Academician A. N. Tikhonov on 23 XI 1967)

1.

Consider the equation for the function \(u(M,t)\)

\[ n^2(M)\partial^2 u/\partial t^2=\Delta u+a(M)u+f(M,t) \tag{1} \]

under the initial conditions

\[ u(M,0)=\frac{\partial}{\partial t}u(M,0)=0. \tag{2} \]

Here \(M(x,y,z)\) is a point of three-dimensional space; \(\Delta\) is the Laplace operator with respect to the variables \(x,y,z\); \(n(M)\) and \(f(M,t)\) are given functions. Below we give a possible formulation of the problem of reconstructing the coefficient \(a(M)\) from known functionals of the solution of equation (1) under conditions (2). Some results for this problem in the case \(n(M)\equiv 1\) are contained in papers \((^1\text{--}^3)\).

In what follows we shall assume that the function \(f(M,t)\) has the form

\[ f(M,t)=4\pi\delta(t)\delta(M-M_0), \tag{3} \]

where \(M_0(x_0,y_0,z_0)\) is some fixed point, and \(\delta(t)\) is the Dirac delta function. In accordance with this, we denote the solution of the Cauchy problem posed for equation (1) by \(u(M,M_0,t)\). We shall study some features of the connection between the functions \(a(M)\) and \(u(M,M_0,t)\).

Let the function \(n(M)\) be twice continuously differentiable and such that to each pair of points \(M(x,y,z)\), \(P(\xi,\eta,\zeta)\) there corresponds one and only one geodesic \(\Gamma(P,M)\) in the metric defined by the formula

\[ ds=n(x,y,z)\sqrt{dx^2+dy^2+dz^2}, \tag{4} \]

and let, furthermore, \(\tau(P,M)\) be the fundamental function of the central field of characteristics with center at the point \(M\). The physical meaning of \(\tau(P,M)\) is the time in which a disturbance produced at the point \(M\) reaches the point \(P\). Some additional conditions on the function \(n(M)\) will be imposed below. We shall regard the coefficient \(a(M)\) as a continuous function of the point \(M\).

Using S. L. Sobolev’s formula, we find that the solution of equation (1) under conditions (2) satisfies in this case the equation

\[ u(M,M_0,t)=\frac{1}{4\pi} \iiint_{\tau(P,M)\le t} f(P,t-\tau(P,M))\sigma(P,M)\,dv_P+ \]

\[ +\frac{1}{4\pi} \iiint_{\tau(P,M)\le t} \left[\Delta_P\sigma(P,M)+a(P)\sigma(P,M)\right] u(P,M_0,t-\tau(P,M))\,dv_P, \tag{5} \]

in which the subscript \(P\) at the Laplace operator and the volume element \(dv\) means that, when they are evaluated, the variable point is the point \(P(\xi,\eta,\zeta)\), while \(\sigma(P,M)\) is a function determined by the function \(n(M)\), having a singularity at the point \(M\) and satisfying some additional conditions (see \((^4)\)). Since the function \(f(M,t)\) is given by formula (3), the domain of integration in formula (5) degenerates into the domain bounded by the qua-

by the ellipsoid

\[ S_{M,M_0,t}\{\tau(P,M_0)+\tau(P,M)=t\} \]

with foci at the points \(M_0, M\). Indeed, at time \(t\) at the point \(M\), the function \(u(M,M_0,t)\) is affected only by those points \(P\) for which the sum of the travel times of the disturbances from the point \(M_0\) to the point \(P\) and from it to the point \(M\) does not exceed \(t\). Using formulas (3) and (5), for the function

\[ v(M,M_0,t)=u(M,M_0,t)-\sigma(M,M_0)\delta(t-\tau(M,M_0)) \tag{6} \]

we obtain the equation

\[ v(M,M_0,t)=\frac{1}{4\pi}\iiint\limits_{D_{M,M_0,t}} [\Delta_P\sigma(P,M)+a(P)\sigma(P,M)]\times \]

\[ \times[\sigma(P,M_0)\delta(t-\tau(P,M_0)-\tau(P,M))+v(P,M_0,t-\tau(P,M))]\,dv_P, \tag{7} \]

where \(D_{M,M_0,t}\) is the domain bounded by the quasiellipsoid \(S_{M,M_0,t}\).

Consider, along with the quasiellipsoid \(S_{M,M_0,t}\), the family of confocal quasiellipsoids \(S_{M,M_0,T}\) \((T\geq \tau(M,M_0))\). We shall assume that the function \(n(M)\) is such that, for a fixed point \(M\), the point \(P\) can be uniquely determined, at least for \(t\) close to \(\tau(M,M_0)\), by the curvilinear coordinates \(T,\tau,\varphi\), whose meaning is as follows. The coordinate \(T\) determines the quasiellipsoid on which the point \(P\) lies, \(\tau=\tau(P,M_0)\), and \(\varphi\) is the angle of the spherical coordinate system characterizing the direction of the tangent at the point \(M_0\) to the geodesic \(\Gamma(P,M_0)\), if the polar axis of the system is aligned at the point \(M_0\) with the direction of the tangent to the geodesic \(\Gamma(M,M_0)\). In this case formula (7) can be written in the form

\[ v(M,M_0,t)=\frac{1}{4\pi} \iint\limits_{S_{M,M_0,t}} [\Delta_P\sigma(P,M)+a(P)\sigma(P,M)]\sigma(P,M_0)\times \]

\[ \times \left|\frac{\partial(\xi,\eta,\zeta)}{\partial(t,\tau,\varphi)}\right|\,d\tau\,d\varphi +\frac{1}{4\pi}\int\limits_{\tau(M,M_0)}^{t}dT \iint\limits_{S_{M,M_0,T}}[\Delta_P\sigma(P,M)+ \]

\[ +a(P)\sigma(P,M)] \left|\frac{\partial(\xi,\eta,\zeta)}{\partial(T,\tau,\varphi)}\right| v(P,M_0,t-\tau(P,M))\,d\tau\,d\varphi, \tag{8} \]

where \(\partial(\xi,\eta,\zeta)/\partial(T,\tau,\varphi)\) is the Jacobian of the transformation from the Cartesian coordinates of the point \(P\) to the curvilinear ones.

Using the properties of the function \(\sigma(P,M)\) and the assumptions made concerning the function \(n(M)\), one can prove that the functions

\[ R(P,M,M_0,t)=\tau(M,M_0)\sigma(P,M_0)\sigma(P,M) \left|\partial(\xi,\eta,\zeta)/\partial(t,\tau,\varphi)\right|, \]

\[ Q(P,M,M_0,t)=\tau(M,M_0)\sigma(P,M_0)\Delta_P\sigma(P,M) \left|\partial(\xi,\eta,\zeta)/\partial(t,\tau,\varphi)\right| \]

are continuous in the aggregate of their arguments. Letting in equation (8) the argument \(t\) tend to \(\tau(M,M_0)\) and taking into account that the integral over the volume bounded by the quasiellipsoid then tends to zero, we find

\[ v(M,M_0,\tau(M,M_0))= \frac{1}{2\tau(M,M_0)} \int\limits_{\Gamma(M,M_0)} [a(P)R(P,M,M_0,\tau(M,M_0))+ \]

\[ +Q(P,M,M_0,\tau(M,M_0))]\,d\tau, \tag{9} \]

where \(d\tau=n(P)\,ds\), and \(ds\) is the element of length of the arc of the geodesic \(\Gamma(M,M_0)\).

Up to now we have considered equation (1) in unbounded space. We now consider the equation in a domain bounded by a closed surface \(S\), at each point of which there exists a tangent plane, and impose on it, in addition to conditions (2), also the boundary condition

\[ \left.\partial u/\partial n\right|_S=0, \tag{10} \]

where \(\mathbf n\) is the outward normal to \(S\). Let \(M_0\) be a point on the surface \(S\) and

\[ f(M,t)=2\pi\delta(t)\delta(M-M_0). \tag{11} \]

Then for the function \(v(M,M_0,t)\), defined by formula (6), equality (9) also holds.

From formula (9) there follows the following possible formulation of the inverse problem: the value \(v(M,M_0,\tau(M,M_0))\) is known as a function of a pair of points \(M,M_0\) on the surface \(S\); it is required to find the function \(a(M)\).

Since the functions \(R\) and \(Q\) are known, the problem of reconstructing the function \(a(M)\) in the indicated formulation is a problem of integral geometry.

In the case \(n(M)=\mathrm{const}\), all restrictions imposed on the function \(n(M)\) are satisfied. The geodesic lines \(\Gamma(M,M_0)\) in this case are straight lines joining the points \(M,M_0\), and we have a well-studied \((^5)\) problem of integral geometry. From the results contained in the book \((^5)\) there follows the theorem:

Theorem 1. For \(n(M)=\mathrm{const}\), the coefficient \(a(M)\) of equation (1) in the class of continuous functions is uniquely reconstructed from the function \(v(M,M_0,\tau(M,M_0))\).

In the case of an arbitrary function \(n(M)\), uniqueness of reconstruction of the function \(a(M)\), of course, will not hold. However, in a number of cases uniqueness of reconstruction also holds for a variable function \(n(M)\). On the basis of the results of article \((^6)\), the following theorem can be proved:

Theorem 2. Let the function \(n(M)\) satisfy all the conditions indicated in the text and, in addition, the following conditions:

1) there exists such a finite point \(M^*\) that \(n(M)\) is a function only of the distance to the point \(M^*\) (or \(n(M)\) depends only on one coordinate, which corresponds to an infinitely remote point \(M^*\));

2) the vertices of the curves \(\Gamma(M,M_0)\) (i.e., the points on the geodesics \(\Gamma(M,M_0)\) least distant from the point \(M^*\)) densely (in the sense of \((^6)\)) fill the whole domain bounded by the surface \(S\).

Then the reconstruction of the continuous function \(a(M)\) from the function \(v(M,M_0,\tau(M,M_0))\) is unique.

We note that, since the curves \(\Gamma(M,M_0)\) corresponding to Theorems 1, 2 are plane curves, to reconstruct the function \(a(M)\) it is in fact necessary to know the function \(v(M,M_0,\tau(M,M_0))\) as a function of 3 variables, and not 4, as might seem. Indeed, it is enough to place the points \(M,M_0\) on the contours of sections of the surface \(S\) by a one-parameter family of planes.

II. Analogously to the preceding discussion, one can pose the problem of reconstructing the function \(a(M)\) for the equation

\[ \partial^2 u/\partial t^2=Lu+a(M)u+f(M,t), \tag{12} \]

where \(M(x_1,x_2,x_3)\) is a point of a three-dimensional domain bounded by a smooth contour \(S\); \(L\) is a given elliptic operator of the form

\[ Lu\equiv \sum_{i=1}^{3}\left(a_i(M)\frac{\partial^2 u}{\partial x_i^2}+b_i(M)\frac{\partial u}{\partial x_i}\right)+h(M)u, \tag{13} \]

and \(f(M,t)\) is defined by formula (11). Then, under conditions (2), (10), for the function \(v(M,M_0,t)\), defined by formula (6), formula (9) can be obtained. The function \(\sigma(P,M)\) here, of course, has a different meaning: it is expressed in terms of the coefficients \(a_i(M)\), \(b_i(M)\) \((i=1,2,3)\) of the operator \(L\).

For \(a_i(M)=c_i\) \((i=1,2,3)\), where \(c_i\) are constants, and under some smoothness conditions on the coefficients \(b_i(M)\) \((i=1,2,3)\), a theorem analogous to Theorem 1 holds.

In the case where \(a_i(M), b_i(M)\) \((i=1,2,3)\) depend only on the distance to some point \(M^*\) and the second of the conditions of Theorem 2 is fulfilled, so-

there is also uniqueness of the reconstruction of the coefficient \(a(M)\) in the class of continuous functions.

Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
21 VI 1967

REFERENCES

\(^{1}\) Yu. M. Berezanskii, Tr. Mosk. matem. obshch., 7 (1958).
\(^{2}\) M. M. Lavrent’ev, V. G. Romanov, DAN, 171, No. 6, 1279 (1966).
\(^{3}\) V. G. Romanov, DAN, 173, No. 4 (1967).
\(^{4}\) S. L. Sobolev, Tr. Seismological Institute, No. 6, 1 (1930).
\(^{5}\) I. M. Gel’fand, M. I. Graev, N. Ya. Vilenkin, Integral Geometry and Related Questions of Representation Theory, Moscow, 1962.
\(^{6}\) V. G. Romanov, Sibirsk. matem. zhurn., 8, No. 5, 1206 (1967).