Selcuk Journal of
Applied Mathematics
Vol. 4, No. 2, pp. 103–112, 2003Volterra integral equation method for solving
some hyperbolic equation problems
Valery G. Yakhno and Ali I¸sık
Department of Mathematics, Faculty of Arts and Sciences, Dokuz Eylul Univer-sity, Buca, 35160, Izmir, Turkey;
e-mail: valery.yakhno@deu.edu.tr e-mail: a.isik@deu.edu.tr Received: October 30, 2003
Summary. The Cauchy problem for a hyperbolic equation with function coefficients of the first partial derivatives with respect to time and space variables is considered. It is proved by Sobolev’s method that solution of this problem satisfies a 3-D Volterra inte-gral equation. Using this fact the uniqueness theorem for an inverse problem is proved.
Key words: hyperbolic equation of the second order, Cauchy prob-lem, Volterra integral equation, inverse problem.
2000 Mathematics Subject Classification: 35L15, 35R30, 49D05
1. Introduction
The theory of linear hyperbolic equations with function coeffi-cients is very well developed. There are general existence and unique-ness theorems for weak and classical solutions of initial value and initial boundary value problems (see, for example [1–3]). Some par-ticular cases of hyperbolic equations have interesting properties which are useful for numerical methods, inverse problems theory and others. For example, the solution of the Cauchy problem for the wave equa-tion with three space variables and the constant velocity coefficient is
given by Kirchhoff’s explicit formula. If the speed coefficient is a func-tion then there is no explicit formula for the solufunc-tion. However if the speed coefficient depending on three space variables is a smooth func-tion then the solufunc-tion of the Cauchy problem for the wave equafunc-tion satisfies a 3-D Volterra integral equation. This result was obtained by S.Sobolev [4] and is a generalization of Kirchhoff’s formula. This Sobolev result was generalized for some hyperbolic equations [5, 6].
The first part of the present paper is related to generalization of Sobolev’s result for other cases of hyperbolic equations. More pre-cisely we generalized Sobolev Volterra integral equation method of hyperbolic equations with function coefficients of the first partial derivatives with respect to time and space variables. This property of the Cauchy problem solution (that is, to satisfy a 3-D Volterra integral equation), we apply to prove the uniqueness theorem for an inverse problem in the second part of this paper.
2. Initial value problem for hyperbolic equation and property of its solution
Let us consider the following scalar hyperbolic equation
(1) ∂ 2u ∂t2 = c 2Δxu +3 j=1 bj(x)∂u ∂xj + q(x) ∂u ∂t + f (x, t), x∈ R3, t > 0,
subject to the initial data
(2) u(x, 0) = g(x), ∂u ∂t(x, 0) = h(x), where Δx = 3 j=1 ∂2 ∂x2j
is the Laplace operator. We assume that c is a positive constant, bj(x),
q(x)∈ C2(R3), j = 1, 2, 3, g(x)∈ C2(R3)∩ H4(R3), h(x)∈ C1(R3)∩ H3(R3), ∂ ∂t j f (x, t) ∈ C [0, T ]; H4−j(R3) , j = 0, 1, 2; Hm(R3) (m = 1, 2, 3, 4) are the Sobolev spaces. The theory of hyperbolic equa-tions [3] contains existence and uniqueness theorems of a solution
u(x, t) of (1), (2) satisfying ∂ju ∂tj(x, t)∈ C [0, T ]; H4−j ∈ (R3)∩ C2−j(R3) , j = 0, 1, 2, ∂3 ∂t3 ∈ C [0, T ]; H1∈ (R3).
The main goal of this section is to study a property of this solution. This property means that solution of (1), (2) satisfies a 3-D Volterra integral equation. To get this property we use Sobolev’s method [4, 5]. Let u(x, t) be solution of (1), (2). Consider other function u1(x, t) which is given by (3) u1(x, t) = u x, t−|x − x 0| c , where x0= x01, x02, x03∈ R3
is a parameter. Consider the differential operator L which is defined by the formula Lxu1 ≡ ⎛ ⎝Δxu1+c12 3 j=1 bj(x)∂u1 ∂xj ⎞ ⎠ . The following relation holds
(4) σLxu1=−2σ∇xτ∇x∂u∂t1 − c12σf (x, t− τ(x)) − σ∂u1 ∂t Δxτ (x) + c12q(x) + c12 3 j=1bj(x)∂τ(x)∂xj ,
for any function σ(x) from C2(R3) and τ (x) =|x − x0|/c. Here ∇x is the gradient operator.
Let σ = σ(x, x0) be solution of the following problem
(5) 2∇xτ (x)∇xσ + σ ⎛ ⎝xτ (x)− 1 c2q(x)− 1 c2 3 j=1 bj(x) ∂τ ∂xj ⎞ ⎠ = 0, (6) σ(x, x0) = O 1 |x − x0| as x−→ x0.
Then the equality (4) may be written as follows (7) σLxu1 = divx −∂u1 ∂t 2σ∇xτ − 1 c2σf (x, t− τ(x)).
Remark 1. We note that we can find a solution of (5), (6) using the method of characteristics. This solution is given by the explicit formula (8) σ(x, x0) = 1 |x − x0|exp |x − x0| 2c 1 0 q(x 0+ (x− x0)z)dz × exp ⎛ ⎝ 1 2c2 1 0 3 j=1 bj(x0+ (x− x0)z)(xj − x0j)dz ⎞ ⎠ .
Using the formula (8) we obtain the following properties of the func-tion σ(x, x0):
1. lim x→x0σ(x, x
0)|x − x0| = 1;
2. σ(x, x0) is twice continuously differentiable function if x= x0; 3. |L∗xσ(x, x0)| ≤ O |x − x0|−2, as x→ x0; 4. lim r→+0 |x−x0|=r ∂σ(x, x0) ∂n dS =−4π. Here L∗xσ(x, x0) = Δxσ(x, x0)− 1 c2 3 j=1 ∂ ∂xj bj(x)σ(x, x0) is the adjoint operator to Lx.
Let u1(x, t), σ(x, x0) be functions defined by (3), (8). Using (7) and Green’s formula
|x−x0|≤ct σ(x, x0)Lxu1(x, t)− u1(x, t)L∗xσ(x, x0) dx = |x−x0|=ct σ(x, x0)∂u1∂n(x,t) − u1(x, t)∂σ(x,x∂n 0) + c12 3 j=1bj(x)njσ(x, x0)u1(x, t) dSx, we find
(9) |x−x0|≤ctdivx −∂u1(x, t) ∂t 2σ(x, x 0)∇x|x − x0| c dx − 1 c2 |x−x0|≤ctσ(x, x0)f x, t− |x−xc 0| )dx − |x−x0|≤ctu1(x, t)L∗xσ(x, x0)dx = |x−x0|=ct σ(x, x0)∂u1∂n(x,t) − u1(x, t)∂σ(x,x∂n 0) + c123j=1bj(x)njσ(x, x0)u1(x, t) dSx.
Applying Ostrogradskii’s formula the equation (9) may be written as
(10) 1 c2 |x−x0|≤ctσ(x, x 0)fx, t−|x − x0| c dx + |x−x0|≤ctu1(x, t)L∗xσ(x, x0)dx + |x−x0|=ct σ(x, x0)∂u1∂n(x,t)− u1(x, t)∂σ(x,x∂n 0) dSx + c12 |x−x0|=ct 3 j=1bj(x)njσ(x, x0)u1(x, t)dSx + 1c |x−x0|=ct2σ∂u1∂t(x,t)dS = 0.
Using the properties 1)-4) of σ(x, x0), (see Remark 1) and the rela-tions u1 x,|x − x 0| c = u(x, 0) = g(x), ∂u1(x, t) ∂t t=|x−x0|c−1 = ∂ ∂tu(x, 0) = h(x)
the formula (10) can be written as
(11) u(x, t) = F (x, t) + 4π1 |x−ξ|≤ctu ξ, t−|ξ−x|c L∗ξσ(ξ, x)dξ,
where F (x, t) = 4π1 |x−ξ|=ct σ(ξ, x)∂g(ξ)∂n − g(ξ)∂σ(ξ,x)∂n + 1 c2 3 j=1bj(ξ)nj σ(ξ, x)g(ξ) +σ(ξ,x)c h(ξ) dSξ + 4πc12 |x−ξ|≤ctσ(ξ, x)f ξ, t− |ξ−x|c dξ.
Remark 2. The formula (10) for a particular case bj ≡ 0, j = 1, 2, 3; q(x) = 0 will be written as Kirchhoff’s formula for the solution of the Cauchy problem for the wave equation.
Remark 3. Let us consider the function in the form of Neumann’s series u(x, t) = ∞ n=0 un(x, t), where u0(x, t) = F (x, t), n≥ 1, un(x, t) = 1 4π |ξ−x|≤ctun−1 ξ, t− |ξ − x| c L∗ξσ(ξ, x)dξ.
We can show that
1) the series∞n=0un(x, t) is uniformly convergent to a continuous function u(x, t) on (T ),
2) the function u(x, t) is a unique solution of the integral equation (10) for (x, t)∈ (T ),
where
(T ) = {(x, t) : 0 ≤ t ≤ T − |x|
c }, T > 0.
3. Uniqueness theorem of an inverse problem
In this section the property of the Cauchy problem solution, which was studied in the section 2, is applied to prove the uniqueness theo-rem of an inverse problem for the integral equation (1). We suppose here that q is a even twice continuously differentiable function de-pending on x3, c is a constant, bj(x), j = 1, 2, g(x), h(x), f (x, t) are even and b3 is odd functions satisfying conditions of the section 2.
The main object of the study here is the following inverse problem. Inverse problem. Let T be a positive number, X = [−T/c, T/c],
f (x, t), g(x), h(x) be given functions, q(x3) ∈ C2(X) be unknown even function. Find q(x3) if the solution of (1), (2) complies with the data
(12) u(0, t) = G(t),
where G(t) is a function known for t∈ [0, T ].
Remark 4. The similar inverse problems for Klein-Gordon-Fock equation were studied by V.G.Romanov [6], Rakesh [7].
The main result of this section is the uniqueness theorem.
Theorem. Let h(0, 0, x3) = 0 for x3 ∈ X and qi(x3), i = 1, 2 be solutions of the inverse problem corresponding the same data
G(t), t∈ [0, T ]. Then q1(x3)≡ q2(x3) for x3 ∈ X.
Proof. Let ui(x, t), i = 1, 2 be two solutions of the Cauchy prob-lem (1), (2) corresponding to q = qi(x3), i = 1, 2, respectively.
Sub-tracting equations (1)-(3) for q = q2(x3) from equations (1)-(3) for
q = q1(x3) we find (13) ∂ 2u˜ ∂t2 = c 2˜u +3 j=1 bj(x)∂ ˜u ∂xj + q1(x3) ∂ ˜u ∂t + ˜q(x3) ∂u2(x, t) ∂t , x = (x1, x2, x3)∈ R3, t > 0, (14) u(x, 0) = 0,˜ ∂ ˜u(x, 0) ∂t = 0, x∈ R 3, (15) u(0, t) = 0,˜ where ˜ q(x3) = q1(x3)− q2(x3), u(x, t) = u˜ 1(x, t)− u2(x, t).
The Cauchy problem (13), (14) is similar to (1), (2). Using reason-ing of the section 2, we find that the solution ˜u(x, t) of the problem
(13), (14) satisfies the following integral equation which is similar to the equation (11). (16) ˜ u(x, t) = 4πc12 |x−ξ|≤ctσ(ξ, x)˜q(ξ3)∂u2 ∂t ξ, t−|ξ−x|c dξ + 4π1 |x−ξ|≤ctu˜ ξ, t−|ξ−x|c L∗ξσ(ξ, x)dξ,
where σ(x, x0) is defined by the formula (8) in which q = q1(x3). Using spherical coordinates
ξ = x + rα, α = (cos ϕ sin θ, sin ϕ sin θ, cos θ),
0≤ ϕ < 2π, 0 < θ < π, dw = sin θdθdϕ, the second integral of (16) can be written in the form
(17) 1 4π ct 0 2π 0 π 0 u˜ ξ, t− |ξ − x| c L∗ξσ(ξ, x) ξ=x+rα r2dwdr.
The first integral of (16) may be written in terms of coordinates (r, ϕ, ξ3). These coordinates we get if θ in the spherical coordinates is changed by ξ3 according to the formula ξ3 = x3+ r cos θ. As a result we have
¯
ξ = ¯x +r2− (ξ3− x3)2ν, ¯ξ = (ξ1, ξ2), ¯x = (x1, x2),
ν = (cos ϕ, sin ϕ), 0≤ ϕ < 2π, dξ = −rdϕdξ3dr,
and the first integral of (16) is presented in the form
(18) 1 4πc2 ct 0 2π 0 x3+r x3−r r σ(ξ, x)˜q(ξ3) ×∂u2 ∂t ξ, t−r c ¯ξ=¯x+√ r2−(ξ3−x3)2ν dξ3dϕdr.
Applying the operator
∂ ∂t
2
to (16) and using presentations (17), (18) for integral of (16) we find (19) ∂ 2u˜ ∂t2(x, t) = c 2 tσ(ξ, x)h(ξ) ¯ξ=¯x,r=ct ˜ q(ξ3) ξ3=x3+ct ξ3=x3−ct + 1 4π 2π 0 x3+ct x3−ct ∂ ∂t (tσ(ξ, x)h(ξ)) ¯ξ=¯x+√ (ct)2−(ξ3−x3)2ν, r=ct + tσ(ξ, x)∂ 2u2 ∂t2 (ξ, 0) ¯ξ=¯x+√ (ct)2−(ξ3−x3)2 ν, r=ct ˜ q(ξ3)dξ3dϕ
+ 1 4πc2 ct 0 2π 0 x3+r x3−r rσ(ξ, x) ×∂3u2 ∂t3 ξ, t− r c ¯ξ=¯x+√ r2−(ξ3−x3)2 ν ˜ q(ξ3)dξ3dϕdr + 1 4π ct 0 2π 0 π 0 ∂2u˜ ∂t2 ξ, t−r c L∗ξσ(ξ, x) ξ=x+rα r2dwdr.
Substituting x = 0 into (19) and using formulas (12), (8), the evenness of h, f, g, q1, q2, ˜q, bj, j = 1, 2 and oddness of b3with respect
to x3 we obtain (20) q(ct) =˜ 1 c tσ(ξ, 0)h(ξ) ¯ξ=0, ξ 3=ct −1 G(t) − 1 2π 2π 0 ct 0 ∂ ∂t tσ(ξ, 0)h(ξ) ¯ξ=√ (ct)2−ξ2 3 ν, r=ct +tσ(ξ, 0)∂ 2u 2 ∂t2 (ξ, 0) ¯ξ=√ (ct)2−ξ2 3ν, r=ct ˜ q(ξ3)dξ3dϕ − 1 4πc2 ct 0 2π 0 r −rrσ(ξ, 0) ∂3u2 ∂t3 ξ, t−r c ¯ξ=√ r2−ξ2 3 ν ˜ q(ξ3)dξ3dϕdr − 1 4π ct 0 2π 0 π 0 ∂2u˜ ∂t2 ξ, t−r c L∗ξσ(ξ, 0) ξ=rα r2dwdr .
Equations (19), (20) may be written as a system of two integral equations relative to two functions W (x, t), ˜q(ct), where
W (x, t) = ∂ 2u˜ ∂t2(x, t)− c 2[tσ(ξ, x)h(ξ)] ¯ξ=¯x, r=ctq(ξ˜ 3) ξ3=x3+ct ξ3=x3−ct .
This system will be homogeneous Volterra type with the polar kernel. According to the theory of Volterra integral equations this system has zero solution only. Therefore ˜q(x3) ≡ 0 for x3 ∈ [0, X], and the theorem is proved.
Remark 5. The property of the Cauchy problem solution (that is to satisfy a 3-D Volterra integral equation) can be generalized for general hyperbolic equations of the second order with the function
coefficients depending on space variables. The uniqueness theorems for inverse problems of these general hyperbolic equations may be proved using this property also.
Acknowledgment. The authors would like to thank the Faculty of Arts and Sciences of Dokuz Eylul Universty in Izmir for a harmo-nious environment.
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