MATHEMATICS

M. L. RASULOV

APPLICATION OF THE CONTOUR-INTEGRAL METHOD TO THE SOLUTION OF MIXED PROBLEMS FOR A PARABOLIC SYSTEM

(Presented by Academician I. N. Vekua, March 1, 1967)

In the present note, by the contour-integral method \((^1)\), the mixed problem (1)—(3) for a system of equations of heat and mass transfer \((^2)\) is solved. The existence of a solution of this problem is proved, and its representation in the form of the contour integral (21) is given; this integral converges very well for \(t>0\) in comparison with the Laplace integral, owing to the fact that

\[ \left|\exp\{\lambda^2 t\}\right|\leq \exp\{-\varepsilon |\lambda|^2 t\} \]

along the contour \(S\). This circumstance makes it possible to compute the contour integral and to construct the solution of the problem effectively \((^3)\). Special cases of problem (1)—(3) have been solved by other authors \((^2)\).

1. Consider the mixed problem

\[ \partial v/\partial t=A\Delta v; \tag{1} \]

\[ \lim_{x\to z}\left\{\left(\alpha_0(z)+\alpha_1(z)\frac{\partial}{\partial t}\right) \frac{\partial v(x,t)}{d n_z} + \left(\beta_0(z)+\alpha_1(z)\beta_1(z)\frac{\partial}{\partial t}\right)v(x,t)\right\} =\psi(z), \]

\[ z\in T; \tag{2} \]

\[ v(x,0)=\Phi(x), \tag{3} \]

where:

1) \(A\) is a constant invertible matrix of second order, composed of elements \(a_{ij}\) \((i,j=1,2)\); system (1) is parabolic in the sense of I. G. Petrovsky.

2)

\[ B\left(z,\frac{d}{d n_z},\frac{d}{d t}\right) = \left(\alpha_0(z)+\alpha_1(z)\frac{\partial}{\partial t}\right)\frac{d}{d n_z} + \left(\beta_0(z)+\alpha_1(z)\beta_1(z)\frac{\partial}{\partial t}\right); \]

\(n_z\) is the direction of the inner normal to \(T\) at the point \(z\in T\); \(\alpha_k(z)\), \(\beta_k(z)\) \((k=0,1)\) are matrices of second order, continuous on \(T\); for sufficiently large complex \(\lambda\),

\[ (\alpha_0(z)+\lambda^2\alpha_1(z))^{-1} (\beta_0(z)+\lambda^2\alpha_1(z)\beta_1(z)) \]

is bounded by a constant; \(\psi(z)\) is a continuous vector-function on \(T\); \(T\) is a Lyapunov surface.

3) \(\Phi(x)\) is a continuously differentiable vector-function in the three-dimensional domain \(D\), equal to zero in some boundary strip of the domain.

1. Consider the spectral problem

\[ A\Delta u-\lambda^2 u=\Phi(x); \tag{4} \]

\[ \lim_{x\to z} B\left(z,d/dn_z,\lambda^2\right)u(x,\lambda)=\psi_1(z), \qquad z\in T, \tag{5} \]

in the domain \(D\).

Let \(p,q\) be the roots \(\mu\) of the quadratic equation

\[ \mu^2+(a_{11}+a_{22})\mu+a_{11}a_{22}-a_{12}a_{21}=0. \tag{6} \]

By virtue of condition 1), the real parts of the complex numbers \(p,q\) are negative.

With the aid of the Fourier integral method, a fundamental matrix \(P(x-\xi,\lambda)\) of solutions of the homogeneous system (1) is constructed, with a singularity at

at the point \(x=\xi\). For the elements \(P_{ks}(x-\xi,\lambda)\) of the matrix \(P(x-\xi,\lambda)\) the following formulas hold:

\[ \begin{aligned} P_{11}(x-\xi,\lambda) &=\frac{1}{4\pi(p-q)|x-\xi|} \left\{ \frac{a_{22}+p}{p}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right]\right.\\ &\qquad\qquad\left. -\frac{a_{22}+q}{q}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-q}}\right] \right\},\\[4pt] P_{12}(x-\xi,\lambda) &=-\frac{a_{12}}{4\pi(p-q)|x-\xi|} \left\{ \frac{1}{p}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right]\right.\\ &\qquad\qquad\left. -\frac{1}{q}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-q}}\right] \right\},\\[4pt] P_{21}(x-\xi,\lambda) &=-\frac{a_{11}}{4\pi(p-q)|x-\xi|} \left\{ \frac{1}{p}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right]\right.\\ &\qquad\qquad\left. -\frac{1}{q}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-q}}\right] \right\},\\[4pt] P_{22}(x-\xi,\lambda) &=\frac{1}{4\pi(p-q)|x-\xi|} \left\{ \frac{a_{11}+p}{p}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right]\right.\\ &\qquad\qquad\left. -\frac{a_{11}+q}{q}\exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-q}}\right] \right\}, \end{aligned} \tag{7} \]

if the roots \(p,q\) of the quadratic equation (6) are distinct; here \(|x-\xi|\) denotes the length of the vector \(x-\xi\); \(x=(x_1,x_2,x_3)\), \(\xi=(\xi_1,\xi_2,\xi_3)\).

If, however, the roots \(p,q\) of the quadratic equation (6) coincide, then for the elements \(P_{ks}(x-\xi,\lambda)\) of the matrix \(P(x-\xi,\lambda)\) the formulas are

\[ P_{11}(x-\xi,\lambda) =\frac{1}{8\pi p^{2}} \left\{ \lambda\sqrt{-p}\left(1+\frac{a_{22}}{p}\right) +\frac{2a_{22}}{|x-\xi|} \right\} \exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right], \tag{8} \]

\[ P_{ks}(x-\xi,\lambda) =\frac{a_{ks}}{8\pi p^{2}} \left\{ \frac{\lambda\sqrt{-p}}{|p|} +\frac{2}{|x-\xi|} \right\} \exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right] \]

for \(k=1,\ s=2\) or \(k=2,\ s=1\),

\[ P_{22}(x-\xi,\lambda) =\frac{1}{8\pi p^{2}} \left\{ \lambda\sqrt{-p}\left(1+\frac{a_{11}}{p}\right) -\frac{2a_{11}}{|x-\xi|} \right\} \exp\left[-\lambda\frac{|x-\xi|}{\sqrt{-p}}\right]. \]

1. Denote by \(u_1(x,\lambda)\) the solution of the spectral problem (4)—(5) for the corresponding homogeneous system (4). Seeking \(u_1(x,\lambda)\) in the form of a simple-layer potential

\[ u_1(a,\lambda)=\iint_T P(x-y,\lambda)\mu(y,\lambda)\,dT_y \tag{9} \]

leads to the integral equation

\[ \mu(z,\lambda)+\iint_T K(z,y,\lambda)\mu(y,\lambda)\,dT_y=\psi_1(z,\lambda), \tag{10} \]

where

\[ K(z,y,\lambda)=2A\left[ dP(z-y,\lambda)/dn_z +\left(\alpha_0(z)+\lambda^2\alpha_1(z)\right)^{-1} \left(\beta_0(z)+\lambda^2\alpha_1(z)\beta_1(z)\right) P(z-y,\lambda) \right], \]

\[ \psi_1(z,\lambda)=2A\left(\alpha_0(z)+\lambda^2\alpha_1(z)\right)^{-1}\psi(z). \]

Let \(R_\delta\) be the region of values of \(\lambda\) satisfying the inequalities

\[ \cos\arg\lambda\geq\delta,\qquad |\lambda|\geq R, \]

where \(R\) is sufficiently large and \(\delta\) is a sufficiently small number.

Under conditions 1)—3) of item 1, the kernel \(K(z,y,\lambda)\) of the integral equation (10) has a weak singularity; moreover, for \(\lambda\in R_\delta\) the estimate

\[ |K(z,y,\lambda)|\leq \frac{C}{|z-y|^{2-\alpha}} \exp\{-\varepsilon|\lambda||z-y|\}, \tag{11} \]

holds, where \(\alpha\) is the Lyapunov exponent, and \(C,\varepsilon\) are positive constants.

With the aid of estimate (11) one proves

Theorem 1. Under conditions 1)—3) of Sec. 1, the spectral problem (4)—(5) has a solution \(u_1(x,\lambda)\), analytic in \(\lambda \in R_\delta\), representable in the form of the double-layer potential (9), where \(\mu(y,\lambda)\) is the solution of the integral equation (10), for the resolvent \(R(z,y,\lambda)\) of which an estimate of the form (11) is valid

\[ |R(z,y,\lambda)| \leq \frac{C}{|z-y|^{2-\alpha}} \exp\{-\varepsilon |\lambda|\,|z-y|\}. \tag{12} \]

If \(D_1\) is a domain lying together with its boundary in the domain \(D\), then for all \(x \in \overline{D}_1\) the estimate

\[ \left|\frac{\partial^k u_1(x,\lambda)}{\partial x_i^k}\right| \leq \frac{C}{\sigma^{1+k}}\exp\{-\varepsilon |\lambda|\sigma\} \quad (k=0,1,2), \tag{13} \]

holds, where \(\sigma\) is the distance between the boundaries of the domains \(D\), \(D_1\). For all \(x \in D+T\) the inequality

\[ \left|\frac{d^k}{dn_z^k}u_1(x,\lambda)\right| \leq C \quad (k=0,1) \tag{14} \]

holds.

Let \(Q(x,\xi,\lambda)\) be the regular part of the Green matrix \(G(x,\xi,\lambda)\) of problem (4)—(5):

\[ G(x,\xi,\lambda)=P(x-\xi,\lambda)-Q(x,\xi,\lambda). \tag{15} \]

Seeking \(Q(x,\xi,\lambda)\) in the form of a single-layer potential

\[ Q(x,\xi,\lambda)=\iint_T P(x-y,\lambda)\mu(y,\xi,\lambda)\,dT_y, \tag{16} \]

we arrive at the integral equation:

\[ \mu(z,\xi,\mu)+\iint_T K(z,y,\lambda)\mu(y,\xi,\lambda)\,dT_y = f(z,\xi,\lambda), \tag{17} \]

where

\[ f(z,\xi,\lambda)= 2A\left[ \frac{d}{dn_z} + \left(\alpha_0(z)+\lambda^2\alpha_1(z)\right)^{-1} \left(\beta_0(z)+\lambda^2\alpha_1(z)\beta_1(z)\right) \right]P(z-\xi,\lambda). \]

As is seen, the integral equations (10) and (17) differ from each other only in the free terms \(\psi_1(z,\lambda)\), \(f(z,\xi,\lambda)\). Consequently,

\[ \mu(z,\xi,\lambda)= f(z,\xi,\lambda) - \iint_T R(z,y,\lambda)f(y,\xi,\lambda)\,dT_y. \tag{18} \]

From (7), (8), (11), (12), (16), and (18) it follows that

Theorem 2. Under conditions 1)—3) of Sec. 1, for all \(\lambda \in R_\delta\) there exists a solution analytic in \(\lambda\)

\[ u_2(x,\lambda,\Phi)= -\iiint_D G(x,\xi,\lambda)\Phi(\xi)\,dD_\xi; \tag{19} \]

the regular part \(Q(x,\xi,\lambda)\) of the Green matrix is determined by formula (16). For every pair of points \(x,\xi\) from the domain \(D_1\), lying together with its boundary in the domain \(D\), the estimate

\[ \left| \frac{\partial^k Q(x,\xi,\lambda)}{\partial x_i^k} \right| \leq \frac{C}{\sigma^{k+3}} \exp\{-\varepsilon |\lambda|\,|x-\xi|\} \quad (k=0,1,2). \tag{20} \]

holds. For all \(x \in D+T\) lying on the normal \(n_z\) \((z \in T)\), and \(\xi \in D_1\), the inequality

\[ |dQ(x,\xi,\lambda)/dn_z| \leq C/\sigma^2 \]

is satisfied.

1. Let \(S\) be an infinite open contour of the \(\lambda\)-plane, coinciding with the boundary of the domain \(R_\delta\) outside a circle of sufficiently large radius centered at the origin. Following the proof scheme of Theorems 38, 39 of the author’s book \({}^{(2)}\), it is not difficult to prove

Theorem 3. Under conditions 1)—3) of item 1, the mixed problem (1)—(3) has a solution \(v(x,t)\), representable by the formula

\[ v(x,t)=\frac{1}{\pi\sqrt{-1}}\int_S e^{\lambda^2 t} \left\{ \frac{u_1(x,\lambda)}{\lambda} -\lambda \iiint_D G(x,\xi,\lambda)\Phi(\xi)\,dD_\xi \right\}\,d\lambda . \tag{21} \]

Azerbaijan State University
named after S. M. Kirov

Received
2 IX 1966

CITED LITERATURE

\({}^{1}\) M. L. Rasulov, The Contour Integral Method, Nauka, 1964. \({}^{2}\) A. V. Lykov, Yu. A. Mikhailov, Theory of Energy and Matter Transfer, Minsk, 1959. \({}^{3}\) M. L. Rasulov, DAN, 128, No. 3 (1959).