Abstract
Full Text
UDC 517.917
MATHEMATICS
S. ELUBAEV
ON AN INVERSE PROBLEM FOR THE TELEGRAPH EQUATION
(Presented by Academician G. I. Marchuk, 24 IV 1969)
This paper is devoted to finding the coefficient for the telegraph equation.
Consider the equation
\[ u_{xy}=au \tag{1} \]
with data on the characteristics
\[ u(x,y,\lambda)\big|_{x=0}=g(y-\lambda), \qquad u(x,y,\lambda)\big|_{y=0}=0, \tag{2} \]
\[ u(x,y,\lambda)\big|_{y=c}=\varphi(x,\lambda), \tag{3} \]
where \(u(x,y,\lambda)\) is a family of solutions of (1), depending on the parameter \(\lambda\) \((0<\lambda<c)\); \(a=a(x,y)\) is the coefficient of equation (1); \(g(y-\lambda)\) is the Dirac function; \(\varphi(x,\lambda)\) is a function having a continuous mixed second derivative with respect to \(x\) and \(\lambda\) in the rectangle \(Q=[0,h;0,c]\). The rectangle \(Q\) is bounded by the coordinate axes and the straight lines \(x=h,\ y=c\); \(h\) and \(c\) are positive constants.
Let \(a(x,y)\) be an unknown continuous function in the rectangle \(Q\), and let
\[ \sup_{0\le x\le h,\ 0\le y\le c} |a(x,y)|=M, \]
where \(M\) is some positive number.
Let \(c\) and \(h\) satisfy the condition
\[ M^* \operatorname{ch} S < 1, \tag{4} \]
where \(S\) and \(M^*\) are certain positive numbers, which will be discussed below.
Multidimensional inverse problems in a linearized formulation were considered in work (1).
In this paper the problem is considered in the following formulation: for a given function \(\varphi(x,\lambda)\) in the rectangle \(Q\), it is required to find the coefficient \(a(x,y)\) of equation (1) in this rectangle.
The solution of equation (1) satisfying the boundary conditions (2) is equivalent to the solution of the integral equation:
\[ u(x,y,\lambda)=g(y-\lambda)+\int_{0}^{y}\int_{0}^{x} d(\xi,\eta)\,u(\xi,\eta,\lambda)\,d\xi\,d\eta . \tag{5} \]
Equation (5) is a Volterra integral equation of the second kind with respect to \(u(x,y,\lambda)\).
From (5), applying the method of successive approximations, we obtain
\[ u(x,y,\lambda)=g(y-\lambda)+ \sum_{n=1}^{\infty}\int_{0}^{y}\int_{0}^{x} g(t-\lambda)\,a_n(x,y,\tau,t)\,d\tau\,dt, \tag{6} \]
where
\[ a_n(x,y,\tau,t)=\int_t^y\int_\tau^x a(\xi,\eta)a_{n-1}(\xi,\eta,\tau,t)\,d\xi\,d\eta \]
is the \(n\)-th iterated kernel for \(a(x,y)\). In view of the boundedness of \(a(x,y)\) in the rectangle \(Q\), the series in (6) will converge absolutely and uniformly in this rectangle. Therefore, interchanging the order of summation and integration in (6), we obtain
\[ u(x,y,\lambda)=g(y-\lambda)+\int_0^y\int_0^x g(t-\lambda)R(x,y,\tau,t)\,d\tau\,dt, \tag{7} \]
where
\[ R(x,y,\tau,t)=\sum_{n=1}^{\infty} a_n(x,y,\tau,t),\qquad a_1=a(\tau,t). \]
From (7), by virtue of condition (3), we obtain
\[ \varphi(x,\lambda)=g(c-\lambda)+\int_0^c\int_0^x g(t-\lambda)R(x,c,\tau,t)\,d\tau\,dt. \tag{8} \]
Differentiating relation (8) with respect to \(x\), and then with respect to \(\lambda\), we obtain
\[ -\varphi''_{x\lambda}(x,\lambda)=a(x,\lambda)+\int_\lambda^c\int_0^x a(x,\eta)R(x,\eta,\tau,\lambda)\,d\tau\,d\eta. \tag{9} \]
Relation (9) can be written in the form
\[ a(x,\lambda)=Aa, \tag{10} \]
where
\[ Aa=-\varphi''_{x\lambda}(x,\lambda)-\int_\lambda^c\int_0^x a(x,\eta)R(x,\eta,\tau,\lambda)\,d\tau\,d\eta \tag{11} \]
is a nonlinear operator.
We now show that this operator is a contraction operator in the space \(C[0,h;0,c]\). Take two arbitrary elements \(a(x,y)\) and \(b(x,y)\) from the space \(C[0,h;0,c]\),
\[ \rho(a,b)=\max_{x,y\in C[0,h;0,c]} |a-b|. \]
Let for \(b(x,y)\) in the rectangle \(Q\) the condition
\[ \sup_{0\le x\le h,\;0\le y\le c}|b(x,y)|=M_1 \]
be satisfied.
Denote by \(M^*\) the larger of the numbers \(M\) and \(M_1\). Using relation (11), we write \(\rho(Aa,Ab)\):
\[ \rho(Aa,Ab)=\|Aa-Ab\|= \]
\[ =\left\|\int_\lambda^c\int_0^x \left[b(x,\eta)R^*(x,\eta,\tau,\lambda)-a(x,\eta)R(x,\eta,\tau,\lambda)\right]\,d\tau\,d\eta\right\|, \tag{12} \]
where
\[ R^*(x,\eta,\tau,\lambda)=\sum_{n=1}^{\infty} b_n(x,\eta,\tau,\lambda),\qquad b_1=b(\tau,\lambda). \]
From (12) we obtain
\[ \rho(Aa, Ab) \leqslant \rho(a,b) M^{*}hcS, \tag{13} \]
where \(S\) is the sum of the series
\[ \sum_{n=1}^{\infty} \frac{(n+1)\,|M^{*}hc|^{\,n-1}}{(n-1)!\,(n-1)!}. \]
From (13), in view of condition (4), we see that the operator \(A\) is a contraction operator. Therefore, on the basis of the contraction mapping principle, from the functional equation (10) we determine the unique coefficient \(a(x,y)\) in the rectangle \(Q\).
Kzyl-Orda State
Pedagogical Institute
named after N. V. Gogol
Received
11 III 1969
CITED LITERATURE
- M. M. Lavrent’ev, V. G. Romanov, DAN, 171, No. 6 (1966).
- S. Elubaev, Vestn. AN KazSSR, 1969.