UDC 517.946.9: 534.11

MATHEMATICAL CONSIDERATION OF CERTAIN PROBLEMS ON LONGITUDINAL VIBRATIONS OF COMPOSITE BEAMS CONSISTING OF ELASTIC AND VISCOELASTIC PARTS

S. I. GAIDUK

A number of works, for example [1—4], have been devoted to the consideration of problems on longitudinal vibrations of simple viscoelastic beams described by the equation
\[ \frac{\partial^2 u}{\partial t^2} = \alpha \frac{\partial^2 u}{\partial x^2} + \beta \frac{\partial^3 u}{\partial x^2 \partial t}, \]
where
\[ \alpha=\frac{E}{\rho}; \qquad \beta=\frac{\mu}{\rho}; \]
\(E\) is the modulus of elasticity; \(\mu\) is the coefficient of viscosity; \(\rho\) is the density of the beam material. Similar problems for composite beams consisting of separate segments with different physical properties, apparently, have not been considered up to the present time.

In the present work a rigorous solution is given for two problems on longitudinal vibrations of finite composite viscoelastic beams; moreover, as in [4], the case is considered in which one end of the beam is fixed and the other is free. To obtain the solution of the problems and their mathematical justification, the residue method and the contour-integral method developed in [5] are applied.

§ 1. PROBLEM ON LONGITUDINAL VIBRATIONS OF A COMPOSITE BEAM CONSISTING OF TWO VISCOELASTIC SEGMENTS

1. Formulation of the problem. It is required to find solutions of the equations

\[ \frac{\partial^2 u_1}{\partial t^2} = a_1 \frac{\partial^2 u_1}{\partial x^2} + b_1 \frac{\partial^3 u_1}{\partial x^2 \partial t} \qquad (a<x<b,\quad 0<t<T) \tag{1.1.1} \]

and

\[ \frac{\partial^2 u_2}{\partial t^2} = a_2 \frac{\partial^2 u_2}{\partial x^2} + b_2 \frac{\partial^3 u_2}{\partial x^2 \partial t} \qquad (b<x<c,\quad 0<t<T) \tag{1.1.2} \]

under the initial conditions

\[ u_1\big|_{t=0}=\varphi_1(x), \qquad \left.\frac{\partial u_1}{\partial t}\right|_{t=0} = \psi_1(x) \qquad (a<x<b), \]

\[ u_2\big|_{t=0}=\varphi_2(x), \qquad \left.\frac{\partial u_2}{\partial t}\right|_{t=0} = \psi_2(x) \qquad (b<x<c), \tag{1.1.3} \]

and the boundary conditions

\[ u_1\big|_{x=a}=0, \qquad \left. a_2 \frac{\partial u_2}{\partial x} + b_2 \frac{\partial^2 u_2}{\partial x \partial t} \right|_{x=c} =0 \qquad (0<t<T). \tag{1.1.4} \]

and the coupling conditions*)

\[ \begin{gathered} \alpha_0 u_1\big|_{x=b}=\beta_0 u_2\big|_{x=b}\quad (0<t<T),\\ \alpha_1 \frac{\partial u_1}{\partial x}\bigg|_{x=b} =\beta_1 \frac{\partial u_2}{\partial x}\bigg|_{x=b}\quad (0<t<T) \end{gathered} \tag{1.1.5} \]

under the assumption that the following conditions are satisfied:

a) the function \(\varphi_1(x)\) on the interval \([a,b]\) and the function \(\varphi_2(x)\) on the interval \([b,c]\) have three continuous first derivatives and piecewise-continuous fourth derivatives;

b) the function \(\psi_1(x)\) on the interval \([a,b]\) and the function \(\psi_2(x)\) on the interval \([b,c]\) have continuous first derivatives and piecewise-continuous second derivatives;

c) \(\varphi_1(a)=0,\ \varphi_1''(a)=0,\ \psi_1(a)=0,\ \psi_2'(c)=0\), and
\[ \alpha_0\varphi_1(b)=\beta_0\varphi_2(b),\quad \alpha_1\varphi_1'(b)=\beta_1\varphi_2'(b),\quad \alpha_0\psi_1(b)=\beta_0\psi_2(b),\quad a_1\alpha_0\varphi_1''(b)=a_2\beta_0\varphi_2''(b). \]

The constants \(a_j,\ b_j\ (j=1,2)\) and \(\alpha_k,\ \beta_k\ (k=0,1)\) are assumed to be nonzero and positive.

2. Residue representation of the solution of problem (1.1.1)—(1.1.5). We associate with the main problem (1.1.1)—(1.1.5) the following auxiliary, so-called spectral, problem:

\[ (a_j+b_j\lambda^2)\frac{d^2 y_j}{dx^2}-\lambda^4 y_j=-f_j(x,\lambda), \tag{1.2.1} \]

\[ (a<x<b\ \text{for } j=1,\quad b<x<c\ \text{for } j=2), \]

\[ y_1\big|_{x=a}=0,\qquad \frac{dy_2}{dx}\bigg|_{x=c}=0, \tag{1.2.2} \]

\[ \alpha_k\frac{d^k y_1}{dx^k}\bigg|_{x=b} = \beta_k\frac{d^k y_2}{dx^k}\bigg|_{x=b}\quad (k=0,1), \tag{1.2.3} \]

where

\[ f_j(x,\lambda)=\lambda^2\varphi_j(x)+\psi_j(x)-b_j\varphi_j''(x)\quad (j=1,2), \]

and here the operation of differentiation with respect to \(t\) in problem (1.1.1)—(1.1.5) is put in correspondence, in problem (1.2.1)—(1.2.3), with the complex parameter \(\lambda^2\).

The solution of problem (1.2.1)—(1.2.3) can be represented with the aid of the matrix Green function \(G(x,\xi,\lambda)\) in the form [6]

\[ y_1(x,\lambda)=\int_a^b G_1(x,\xi,\lambda)F_1(\xi,\lambda)\,d\xi+ \]

\[ +\int_b^c G_{12}(x,\xi,\lambda)F_2(\xi,\lambda)\,d\xi, \tag{1.2.4} \]

*) The constants \(\alpha_k,\ \beta_k\ (k=0,1)\) are chosen depending on the type of elastic connection joining the ends of the sections of the composite rod, whose mass is neglected. If the joined ends have the same cross section and are at all times located together, and the modulus of elasticity of the material of the connection is equal to \(\gamma\), then in the coupling conditions one must put \(\alpha_0=\beta_0=1\) and \(\alpha_1=\beta_1=\gamma\); then they take the form

\[ \frac{\partial^k u_1}{\partial x^k}\bigg|_{x=b} = \frac{\partial^k u_2}{\partial x^k}\bigg|_{x=b} \quad (k=0,1). \]

\[ y_2(x,\lambda)=\int_a^b G_{21}(x,\xi,\lambda)F_1(\xi,\lambda)\,d\xi+ \int_b^c G_2(x,\xi,\lambda)F_2(\xi,\lambda)\,d\xi, \tag{1.2.5} \]

where

\[ F_j(x,\lambda)=\frac{f_j(x,\lambda)}{a_j+b_j\lambda^2}, \tag{1.2.6} \]

\[ G_1(x,\xi,\lambda)= \begin{cases} G_{11}(x,\xi,\lambda), & \text{for } \xi \leqslant x,\\ G_{11}^{*}(x,\xi,\lambda), & \text{for } \xi \geqslant x, \end{cases} \tag{1.2.7} \]

\[ G_2(x,\xi,\lambda)= \begin{cases} G_{22}(x,\xi,\lambda), & \text{for } \xi \leqslant x,\\ G_{22}^{*}(x,\xi,\lambda), & \text{for } \xi \geqslant x, \end{cases} \tag{1.2.8} \]

\[ G_{11}(x,\xi,\lambda)= \frac{1}{\Delta(\lambda)} \left[ \alpha_1\beta_0\,\operatorname{ch} z_2(c-b)\operatorname{ch} z_1(b-x)+ \right. \]

\[ \left. +\frac{z_2}{z_1}\alpha_0\beta_1\,\operatorname{sh} z_2(c-b)\operatorname{sh} z_1(b-x) \right]\operatorname{sh} z_1(\xi-a), \tag{1.2.9} \]

\[ G_{11}^{*}(x,\xi,\lambda)=G_{11}(\xi,x,\lambda), \]

\[ G_{12}(x,\xi,\lambda)= \frac{\beta_0\beta_1}{\Delta(\lambda)} \operatorname{sh} z_1(x-a)\operatorname{ch} z_2(c-\xi), \tag{1.2.10} \]

\[ G_{22}(x,\xi,\lambda)= \frac{1}{\Delta(\lambda)} \left[ \alpha_0\beta_1\,\operatorname{sh} z_1(b-a)\operatorname{ch} z_2(\xi-b)+ \right. \]

\[ \left. +\frac{z_1}{z_2}\alpha_1\beta_0\,\operatorname{ch} z_1(b-a)\operatorname{sh} z_2(\xi-b) \right]\operatorname{ch} z_2(c-x), \tag{1.2.11} \]

\[ G_{22}^{*}(x,\xi,\lambda)=G_{22}(\xi,x,\lambda), \]

\[ G_{21}(x,\xi,\lambda)= \frac{\alpha_0\alpha_1}{\Delta(\lambda)} \operatorname{ch} z_2(c-x)\operatorname{sh} z_1(\xi-a), \tag{1.2.12} \]

\[ \Delta(\lambda)= \alpha_1\beta_0 z_1\operatorname{ch} z_1(b-a)\operatorname{ch} z_2(c-b)+ \]

\[ +\alpha_0\beta_1 z_2\operatorname{sh} z_1(b-a)\operatorname{sh} z_2(c-b), \tag{1.2.13} \]

\[ z_j=\frac{\lambda^2}{\sqrt{a_j+b_j\lambda^2}}\qquad (j=1,2). \tag{1.2.14} \]

The following is valid

Theorem 1. Under conditions a)—c) of item 1, the expansion formulas hold

\[ \frac{1}{2\pi i}\sum_{\nu}\int_{C_{\nu}} y_j(x,\lambda)\lambda^{2s+1}\,d\lambda = \begin{cases} \varphi_j(x), & \text{for } s=0,\\ \psi_j(x), & \text{for } s=1 \end{cases} \tag{1.2.15} \]

\[ (j=1,2), \]

moreover, the convergence of the series is uniform with respect to \(x\in[a,b]\) for \(j=1\) and \(x\in[b,c]\) for \(j=2\), where \(C_\nu\) is a simple closed contour containing

inside only one special point \(\lambda_\nu\) of the subintegral function, and where the special points are numbered in the order of increase of their moduli.

Proof. Since for large \(|\lambda|\) and for any constant \(\alpha\) the equality

\[ e^{-\frac{\alpha\lambda^2}{\sqrt{a_j+b_j\lambda^2}}} = e^{-\frac{\alpha\lambda}{\sqrt{b_j}}} \left\{1+O\left(\frac{1}{\lambda}\right)\right\}, \tag{1.2.16} \]

is valid, it can be shown that for the indicated \(|\lambda|\)

\[ \Delta(\lambda)=\lambda e^{-\mu(m_1+m_2)\lambda} \left\{[M_1]+[M_2]e^{2\mu m_1\lambda} +[M_2]e^{2\mu m_2\lambda}+[M_1]e^{2\mu(m_1+m_2)\lambda}\right\}, \tag{1.2.17} \]

where \(\mu=1\) for the left \(\lambda\)-half-plane and \(\mu=-1\) for the right \(\lambda\)-half-plane,

\[ m_1=\frac{b-a}{\sqrt{b_1}},\qquad m_2=\frac{c-b}{\sqrt{b_2}},\qquad [M_k]=M_k+O\left(\frac{1}{\lambda}\right)\quad (k=1,2), \]

\[ M_1=\frac{1}{4}\left(\frac{\alpha_1\beta_0}{\sqrt{b_1}}+\frac{\alpha_0\beta_1}{\sqrt{b_2}}\right),\qquad M_2=\frac{1}{4}\left(\frac{\alpha_1\beta_0}{\sqrt{b_1}}-\frac{\alpha_0\beta_1}{\sqrt{b_2}}\right). \]

For an exponential polynomial with asymptotically constant coefficients of the form

\[ H(\lambda)=[M_1]+[M_2]e^{2\mu m_1\lambda} +[M_2]e^{2\mu m_2\lambda}+[M_1]e^{2\mu(m_1+m_2)\lambda}, \]

where

\[ M_1\ne 0, \]

the following is known [7, 5]: 1) the polynomial under consideration has a countable set of zeros located in a strip \(D_h\) of bounded width \(h\), with center at the origin of the \(\lambda\)-plane, whose boundaries are parallel to the imaginary axis; 2) if from the strip \(D_h\) one removes the interiors of circles of sufficiently small radius \(\delta\) with centers at the zeros \(\lambda_k\) \((k=1,2,3\ldots)\) of the polynomial \(H(\lambda)\) (in what follows we shall denote them by \(O_\delta^{(k)}\) \((k=1,2,3,\ldots)\)), then in the remaining part of the strip \(D_h\) the inequality

\[ |H(\lambda)|\ge M_\delta, \tag{1.2.18} \]

holds, where \(M_\delta>0\) and depends on the choice of \(\delta\); 3) if the zeros \(\lambda_k\) \((k=1,2,3\ldots)\) of the polynomial \(H(\lambda)\) are numbered in the order of increase of their moduli, then for them the asymptotic representation

\[ |\lambda_k|=\frac{k\pi}{m_1+m_2} \left\{1+O\left(\frac{1}{k}\right)\right\} \tag{1.2.19} \]

holds.

Integrating by parts the terms standing on the right-hand sides of equalities (1.2.4) and (1.2.5), taking into account conditions a)—b) of item 1, the above-mentioned equalities, after a number of transformations, can be represented in the form

\[ y_j(x,\lambda)=\frac{\varphi_j(x)}{\lambda^2} +\frac{\psi_j(x)}{\lambda^4} +\frac{a_j\varphi_j''(x)}{\lambda^6} +\frac{1}{\lambda^5}W_1^{(j)}(x,\lambda)+ \]

\[ +\frac{1}{\lambda^7}W_2^{(j)}(x,\lambda)\quad (j=1,2), \tag{1.2.20} \]

where the functions \(W_p^{(j)}(x,\lambda)\) \((j,p=1,2)\) are bounded in modulus outside the circles \(Q_\delta^{(k)}\) for sufficiently large \(|\lambda|\).

We now take in the \(\lambda\)-plane a sequence of expanding concentric circles \(O_n\) \((n=1,2,3,\ldots)\), with common center at the origin and with radii \(r_n\to\infty\) as \(n\to\infty\), separated from the circles \(Q_\delta^{(k)}\) by a certain distance greater than \(\delta\) (in view of (1.2.19), this can always be done).

Substituting (1.2.20) into

\[ \lim_{n\to\infty}\frac{1}{2\pi i}\int_{O_n} y_j(x,\lambda)\lambda^{2s+1}\,d\lambda \quad (j=1,2,\ s=0,1) \]

and calculating the limits, we obtain, as in [4], that

\[ \lim_{n\to\infty}\frac{1}{2\pi i}\int_{O_n} y_j(x,\lambda)\lambda^{2s+1}\,d\lambda = \begin{cases} \varphi_j(x), & \text{for } s=0,\\ \psi_j(x), & \text{for } s=1 \end{cases} \quad (j=1,2). \]

Since, according to the well-known residue theorem,

\[ \begin{aligned} \lim_{n\to\infty}\frac{1}{2\pi i}\int_{O_n} y_j(x,\lambda)\lambda^{2s+1}\,d\lambda &= \frac{1}{2\pi i}\sum_{\nu}\int_{C_\nu} y_j(x,\lambda)\lambda^{2s+1}\,d\lambda, \end{aligned} \tag{1.2.21} \]

where the sum over \(\nu\) on the right-hand side of the last equality extends over all singular points of the integrand, equality (1.2.21) also proves Theorem 1 formulated above.

Theorem 2. Under the conditions a)—c) of Sec. 1, if problem (1.1.1)—(1.1.5) has a solution \(u_j(x,t)\) \((j=1,2)\), continuous for \(x\in[a,b]\) \((j=1)\), \(x\in[b,c]\) \((j=2)\), \(t\in[0,T]\), and having continuous derivatives entering into the problem \(\left[\dfrac{\partial u_j}{\partial t}\right.\) for \(a<x<b\) \((j=1)\), \(b<x<c\) \((j=2)\), \(0\le t<T\);

\[ \frac{\partial u_j}{\partial x} \quad\text{for } a<x\le b\ (j=1),\quad b\le x\le c\ (j=2),\quad 0<t<T;\quad \frac{\partial^2 u_2}{\partial x\partial t} \quad\text{for } b<x\le c,\quad 0<t<T; \]

the remaining derivatives for \(a<x<b\) \((j=1)\), \(b<x<c\) \((j=2)\), \(0<t<T\big]\), then it can be represented in the form of the complete integral residue

\[ u_j(x,t)=\frac{1}{2\pi i}\sum_{\nu}\int_{C_\nu} y_j(x,\lambda)\lambda e^{\lambda^2 t}\,d\lambda \quad (j=1,2). \tag{1.2.22} \]

The validity of the theorem is verified by substituting formula (1.2.22) into equations (1.1.1), (1.1.2) and conditions (1.1.3)—(1.1.5).

3. Proof of existence of a solution. Let \(D_h\) be the strip described in Sec. 2, and let \(R(h,\beta)\) be the set of values \(\lambda\) lying outside the strip \(D_h\) and satisfying the conditions:

\[ -\frac{\pi}{2}+\beta\le \arg\lambda \le \frac{\pi}{2}-\beta,\qquad -\frac{\pi}{4}<\beta<\frac{\pi}{4},\qquad \beta=\text{const}. \]

Denote by \(L\) the open contour lying in the region \(R(h,\beta)\) and having as its asymptotes the straight lines

\[ \arg \lambda=-\frac{\pi}{2}+\beta \quad\text{and}\quad \arg \lambda=\frac{\pi}{2}-\beta . \]

Let \(L^*\) be the contour symmetric to the contour \(L\) with respect to the imaginary axis of the \(\lambda\)-plane, and let \(O_n\) \((n=1,2,\ldots)\) be the sequence of circles described in item 2. Denote by \(L_1^{(n)}\) and \(L_2^{(n)}\) the parts of the contours \(L\) and \(L^*\), respectively, lying inside \(O_n\), and by \(M_1^{(n)}\) and \(M_2^{(n)}\) the arcs of the circles \(O_n\) located between \(L\) and \(L^*\) and lying in the upper and lower \(\lambda\)-half-planes, respectively (Fig. 1).

Let \(\Gamma_n\) be the closed contour consisting of the arcs \(L_1^{(n)}\), \(L_2^{(n)}\), \(M_1^{(n)}\), \(M_2^{(n)}\) (Fig. 1).

Theorem 3. If conditions a)—c) of item 1 are fulfilled, problem (1.1.1)—(1.1.5) has a solution possessing the properties indicated in Theorem 2 and representable by the formula

\[ u_j(x,t)=\frac{1}{\pi i}\int_L y_j(x,\lambda)\lambda e^{\lambda^2 t}\,d\lambda \]

\[ (j=1,2), \tag{1.3.1} \]

where the contour of integration \(L\) was described above.

Fig. 1

Proof. To prove that the function defined by formula (1.3.1) satisfies equations (1.1.1)—(1.1.2), the boundary conditions (1.1.4), and the conjugation conditions (1.1.5), it is sufficient to prove the uniform convergence of the integrals:

\[ \int_L \lambda^{2s+1}\frac{d^m y_j}{dx^m}e^{\lambda^2 t}\,d\lambda \tag{1.3.2} \]

\[ (s=0,\ m=0,1,2;\quad s=1,\ m=2;\quad s=2,\ m=0\ \text{for } j=1 \]

\[ \text{and}\quad s=0,\ m=0,1,2;\quad s=1,\ m=1,2;\quad s=2,\ m=0\ \text{for } j=2) \]

with respect to \(t\in(0,T)\) and \(x\), varying over the intervals indicated for each of the integrals (1.3.2) in Theorem 2. This will make it possible to differentiate the required number of times with respect to \(x\) and \(t\) and to pass to the limit as \(x\to a\), \(x\to b\), and \(x\to c\) under the integral sign in formula (1.3.1).

The uniform convergence of the integrals (1.3.2) is proved in the same way as in [4], with the aid of the estimate

\[ \left|\frac{d^m y_j}{dx^m}\right|\leq A|\lambda|^{m-1} \quad (j=1,2;\ m=0,1,2), \tag{1.3.3} \]

valid outside the circles \(O_\delta^{(k)}\), which can be obtained by differentiating the equalities (1.2.4), (1.2.5) twice with respect to \(x\), using inequality (1.2.18).

To prove the fulfillment of the initial conditions, it is necessary to prove the uniform convergence of the integrals (1.3.2) for \(s=0,\ m=0\) and \(s=1,\ m=0\) with respect to \(t\in(0,T)\). The proof is carried out with the aid of the inequalities (1.2.20), in the same way as in [4].

4. Uniqueness of the solution and its continuous dependence on the initial conditions.

Theorem 4. If the conditions a)—c) of Sec. 1 are satisfied, the solution of the problem (1.1.1)—(1.1.5) is unique in the class of functions representable in the form (1.2.22), and depends continuously on the initial conditions.

Proof. Let the problem (1.1.1)—(1.1.5) have two sufficiently smooth solutions \(u_j^{(1)}(x,t)\) and \(u_j^{(2)}(x,t)\). Then their difference is a sufficiently smooth solution of the problem (1.1.1)—(1.1.5) with zero initial conditions. This solution, according to Theorem 2, is representable in the form of the complete integral residue (1.2.22), where \(\varphi_j(x)=0\) and \(\psi_j(x)=0\) \((j=1,2)\). Consequently, it is identically equal to zero. Thus,

\[ u_j^{(1)}(x,t)=u_j^{(2)}(x,t). \]

Substituting into (1.2.4), (1.2.5), instead of the functions \(F_j(\xi,\lambda)\), their values (1.2.6), and integrating twice by parts the integrals under whose signs the functions \(\varphi_j(\xi)\) enter, after a number of transformations we obtain

\[ \begin{aligned} y_j(x,\lambda)=\frac{\varphi_j(x)}{\lambda} +\frac{1}{\lambda^3}\Bigg[ &\int_a^b \omega_1^{(j)}(x,\xi,\lambda)\varphi_1''(\xi)\,d\xi +\int_a^b \omega_2^{(j)}(x,\xi,\lambda)\psi_1(\xi)\,d\xi \\ &+\int_b^c \omega_3^{(j)}(x,\xi,\lambda)\varphi_2''(\xi)\,d\xi +\int_b^c \omega_4^{(j)}(x,\xi,\lambda)\psi_2(\xi)\,d\xi \Bigg]\quad (j=1,2), \end{aligned} \tag{1.4.1} \]

where the functions \(\omega_p^{(j)}(x,\xi,\lambda)\) \((j=1,2;\ p=1,2,3,4)\) are bounded outside the circles \(Q_\delta^{(k)}\) for sufficiently large \(|\lambda|\).

Substituting, instead of the functions \(y_j(x,\lambda)\) in the right-hand side of the equality,

\[ u_j(x,t)=-\frac{1}{\pi i}\int_L y_j(x,\lambda)\lambda e^{\lambda^2t}\,d\lambda = -\frac{1}{2\pi i}\lim_{n\to\infty}\int_{\Gamma_n} y_j(x,\lambda)\lambda e^{\lambda^2t}\,d\lambda \]

their values (1.4.1) and changing the order of integration, which is possible by virtue of the uniform convergence of the integrals

\[ \Phi_p^{(j)}(x,\xi,t)= \]

\[ =\frac{1}{2\pi i}\lim_{n\to\infty}\int_{\Gamma_n} \frac{\omega_p^{(j)}(x,\xi,\lambda)}{\lambda^2}\,e^{\lambda^2t}\,d\lambda \quad (p=1,2,3,4) \]

with respect to \(x\in [a,b]\) \((j=1)\), \(x\in [b,c]\) \((j=2)\), \(t\in[0,T]\), we obtain

\[ u_j(x,t)=\varphi_j(x)+\int_a^b \Phi_1^{(j)}(x,\xi,t)\varphi_1''(\xi)\,d\xi+ \]

\[ +\int_a^b \Phi_2^{(j)}(x,\xi,t)\psi_1(\xi)\,d\xi+ \int_b^c \Phi_3^{(j)}(x,\xi,t)\varphi_2''(\xi)\,d\xi+ \]

\[ +\int_b^c \Phi_4^{(j)}(x,\xi,t)\psi_2(\xi)\,d\xi, \]

whence it follows that

\[ |u_j(x,t)|\leq |\varphi_j(x)|+ \]

\[ +A_j\int_a^b |\varphi_1''(\xi)|\,d\xi+ B_j\int_a^b |\psi_1(\xi)|\,d\xi+ \]

\[ +C_j\int_b^c |\varphi_2''(\xi)|\,d\xi+ D_j\int_b^c |\psi_2(\xi)|\,d\xi, \]

where \(A_j,\ B_j,\ C_j,\ D_j\) \((j=1,2)\) are certain constants.

From the last inequality follows the continuous dependence of the solution on the initial conditions.

5. Computation of the total integral residue in one particular case. From §§ 2 and 3 it follows that the solution of problem (1.1.1)—(1.1.5) is represented in the form of a total integral residue

\[ u_j(x,t)=\frac{1}{2\pi i}\sum_\nu \int_{C_\nu} y_j(x,\lambda)\lambda e^{\lambda^2 t}\,d\lambda \qquad (j=1,2). \tag{1.5.1} \]

From § 2 it also follows that the singular points of the integrand in (1.5.1) are the essentially singular points
\(\lambda_{1,2}=\pm i\sqrt{\dfrac{a_1}{b_1}}\), \(\lambda_{3,4}=\pm i\sqrt{\dfrac{a_2}{b_2}}\), and an infinite set of poles, which are the roots of the equation

\[ a_0\beta_1\sqrt{a_1+b_1\lambda^2}\, \operatorname{sh}\frac{\lambda^2(b-a)}{\sqrt{a_1+b_1\lambda^2}}\, \operatorname{sh}\frac{\lambda^2(c-b)}{\sqrt{a_2+b_2\lambda^2}} + \]

\[ +a_1\beta_0\sqrt{a_2+b_2\lambda^2}\, \operatorname{ch}\frac{\lambda^2(b-a)}{\sqrt{a_1+b_1\lambda^2}}\, \operatorname{ch}\frac{\lambda^2(c-b)}{\sqrt{a_2+b_2\lambda^2}} =0. \]

It does not appear possible to solve the last equation in the general case with literal parameters. However, in certain particular cases its solutions can be found. Let, for example, \(\dfrac{a_2}{a_1}=\dfrac{b_2}{b_1}\), i.e. \(b_j=\alpha a_j\) \((j=1,2)\), and \(l_2=\beta l_1\), where \(l_1=b-a\), \(l_2=c-b\), \(\beta=\sqrt{\dfrac{a_2}{a_1}}\); then the poles take the form

\[ \lambda_k=\pm \frac{i}{p\sqrt{2}}\sqrt{\alpha^2(\gamma \pm k\pi)^2 \pm \sqrt{(\gamma \pm k\pi)^2\left[\alpha^2(\gamma \pm k\pi)^2-4p^2\right]}} \]

\[ (k=0,\;1,\;2,\;\ldots) \]

(four possible combinations of signs under the first radical), where \(p=\)

\[ =\frac{l_1}{\sqrt{a_1}},\qquad \gamma=\operatorname{arctg}\sqrt{\frac{\alpha_1\beta_0\sqrt{a_2}}{\alpha_0\beta_1\sqrt{a_1}}},\qquad 0<\gamma<\frac{\pi}{2}. \]

Let, further, \(b_j=\alpha a_j\) \((j=1,\;2)\) and

\[ \frac{\alpha_0\beta_1}{\alpha_1\beta_0}=\frac{\sqrt{a_2}}{\sqrt{a_1}}, \]

then the poles of the integrand in (1.5.1) have the form

\[ \lambda_k=\pm \frac{i}{\varkappa\sqrt{2}}\sqrt{\alpha\beta_k^2 \pm \beta_k\sqrt{\alpha^2\beta_k^2-4\varkappa^2}} \]

(four possible combinations of signs), where

\[ \varkappa=\frac{b-a}{\sqrt{a_1}}+\frac{c-b}{\sqrt{a_2}},\qquad \beta_k=\frac{(2k+1)\pi}{2}\quad (k=0,\;1,\;2\ldots). \]

Compute the complete integral residue in the second case. It is easy to see from equalities (1.2.9)—(1.2.14) that in this case

\[ G_{11}(x,\xi,\lambda)= \frac{\sqrt{a_1}}{z}\, \frac{ \operatorname{sh}\frac{\xi-a}{\sqrt{a_1}}z\, \operatorname{ch}\left(\frac{b-x}{\sqrt{a_1}}+\frac{c-b}{\sqrt{a_2}}\right)z }{ \operatorname{ch}\varkappa z }, \qquad G_{11}^{*}(x,\xi,\lambda)=G_{11}(\xi,x,\lambda), \tag{1.5.2} \]

\[ G_{12}(x,\xi,\lambda)= -\frac{\beta_0}{\alpha_0}\, \frac{\sqrt{a_2}}{z}\, \frac{ \operatorname{sh}\frac{x-a}{\sqrt{a_1}}z\, \operatorname{ch}\frac{c-\xi}{\sqrt{a_2}}z }{ \operatorname{ch}\varkappa z }, \tag{1.5.3} \]

\[ G_{21}(x,\xi,\lambda)= -\frac{\alpha_0}{\beta_0}\, \frac{\sqrt{a_1}}{z}\, \frac{ \operatorname{sh}\frac{\xi-a}{\sqrt{a_1}}z\, \operatorname{ch}\frac{c-x}{\sqrt{a_2}}z }{ \operatorname{ch}\varkappa z }, \tag{1.5.4} \]

\[ G_{22}(x,\xi,\lambda)= \frac{\sqrt{a_2}}{z}\, \frac{ \operatorname{ch}\frac{c-x}{\sqrt{a_2}}z\, \operatorname{sh}\left(\frac{\xi-b}{\sqrt{a_2}}+\frac{b-a}{\sqrt{a_1}}\right)z }{ \operatorname{ch}\varkappa z }, \qquad G_{22}^{*}(x,\xi,\lambda)=G_{22}(\xi,x,\lambda), \tag{1.5.5} \]

where

\[ z=\frac{\lambda^2}{\sqrt{1+\alpha\lambda^2}}. \]

Represent equality (1.5.1) in the form

\[ u_j(x,t)=u_j^{(1)}(x,t)+u_j^{(2)}(x,t), \tag{1.5.6} \]

where

\[ u_j^{(1)}(x,t)=\frac{1}{2\pi i}\sum_k \int_{C_k} y_j(x,\lambda)\lambda e^{\lambda^2 t}\,d\lambda . \tag{1.5.7} \]

(the sum on the right-hand side is extended over all poles of the integrand) and

\[ u_j^{(2)}(x,t)=\frac{1}{2\pi i}\sum_p \int_{C_p} y_j(x,\lambda)\lambda e^{\lambda^2 t}\,d\lambda \tag{1.5.8} \]

(the sum on the right-hand side is extended over the essentially singular points of the integrand).

Let us compute, for example, the total integral residue for \(j=1\). Substituting into the right-hand side of (1.5.7), in place of the function \(y_1(x,\lambda)\), its value (1.2.4), taking (1.2.6) into account, we obtain

\[ \begin{aligned} u_1^{(1)}(x,t)=\sum_{k=0}^{\infty}\Bigg\{& \int_a^b \left[\sum_{m=1}^{4}\operatorname*{Res}_{\lambda=\lambda_{km}} G_1(x,\xi,\lambda) \frac{\lambda^3 e^{\lambda^2 t}}{a_1+b_1\lambda^2}\right]\varphi_1(\xi)\,d\xi+\\ &+\int_a^b \left[\sum_{m=1}^{4}\operatorname*{Res}_{\lambda=\lambda_{km}} G_1(x,\xi,\lambda) \frac{\lambda e^{\lambda^2 t}}{a_1+b_1\lambda^2}\right]\,[\psi_1(\xi)-b_1\varphi_1''(\xi)]\,d\xi+\\ &+\int_b^c \left[\sum_{m=1}^{4}\operatorname*{Res}_{\lambda=\lambda_{km}} G_{12}(x,\xi,\lambda) \frac{\lambda^3 e^{\lambda^2 t}}{a_2+b_2\lambda^2}\right]\varphi_2(\xi)\,d\xi+\\ &+\int_b^c \left[\sum_{m=1}^{4}\operatorname*{Res}_{\lambda=\lambda_{km}} G_{12}(x,\xi,\lambda) \frac{\lambda e^{\lambda^2 t}}{a_2+b_2\lambda^2}\right]\,[\psi_2(\xi)-b_2\varphi_2''(\xi)]\,d\xi \Bigg\}, \end{aligned} \tag{1.5.9} \]

where

\[ \lambda_{k1}=i\sqrt{p_k+q_k},\qquad \lambda_{k2}=-i\sqrt{p_k+q_k},\qquad \lambda_{k3}=i\sqrt{p_k-q_k}, \]

\[ \lambda_{k4}=-i\sqrt{p_k-q_k},\qquad p_k=\frac{\alpha\beta_k^2}{2\chi^2},\qquad q_k=\frac{\beta_k\sqrt{\alpha^2\beta_k^2-4\chi^2}}{2\chi^2}. \]

Computing the residues on the right-hand side of equality (1.5.9), we have

\[ \operatorname*{Res}_{\lambda=\lambda_{km}} G_1(x,\xi,\lambda) \frac{\lambda^{2s+1}e^{\lambda^2t}}{a_1+b_1\lambda^2} = \]

\[ = \frac{(-1)^k\sqrt{\bar a_2}}{l}\, \frac{(1+\alpha\lambda_{km}^2)\lambda_{km}^{2s-2}}{2+\alpha\lambda_{km}^2} \times \]

\[ \times e^{\lambda_{km}^2 t} \begin{cases} \sin\gamma_k(\xi-a)\sqrt{\bar a_2}\,\cos\gamma_k(A-x\sqrt{\bar a_2}),\\ \qquad \text{for } \xi\le x,\\[4pt] \sin\gamma_k(x-a)\sqrt{\bar a_2}\,\cos\gamma_k(A-\xi\sqrt{\bar a_2}),\\ \qquad \text{for } \xi\ge x \end{cases} \tag{1.5.10} \]

\[ (s=0,1); \]

\[ \operatorname*{Res}_{\lambda=\lambda_{km}} G_{12}(x,\xi,\lambda) \frac{\lambda^{2s+1}e^{\lambda^2t}}{a_2+b_2\lambda^2} = \]

\[ = \frac{\beta_0}{\alpha_0}\, \frac{(-1)^k\sqrt{\bar a_1}}{l}\, \frac{(1+\alpha\lambda_{km}^2)\lambda_{km}^{2s-2}}{2+\alpha\lambda_{km}^2} \times \]

\[ \times e^{\lambda_{km}^2 t}\sin\gamma_k(x-a)\sqrt{\bar a_2}\cos\gamma_k(c-\xi)\sqrt{\bar a_1} \qquad (s=0,1), \tag{1.5.11} \]

where

\[ l=(b-a)\sqrt{a_2}+(c-b)\sqrt{a_1},\qquad \gamma_k=\frac{\beta_k}{l},\qquad A=b\sqrt{a_2}+(c-b)\sqrt{a_1}. \]

Since

\[ \frac{\lambda^2}{\sqrt{1+\alpha\lambda^2}}=\frac{\beta_k}{\chi}\, i, \]

then

\[ 1+\alpha\lambda^2=-\frac{\lambda^4\chi^2}{\beta_k^2} \]

and

\[ 2+\alpha\lambda^2=\frac{\beta_k^2-\chi^2\lambda^4}{\beta_k^2}, \]

whence

\[ \sum_{m=1}^{4}\frac{1+\alpha\lambda_{km}^2}{2+\alpha\lambda_{km}^2}\,e^{\lambda_{km}^2 t} = \sum_{m=1}^{4}\frac{\chi^2\lambda_{km}^4}{\beta_k^2-\chi^2\lambda_{km}^4}\,e^{\lambda_{km}^2 t} = 2\left(\operatorname{ch} q_k t-\frac{p_k}{q_k}\operatorname{sh} q_k t\right)e^{-p_k t}, \tag{1.5.12} \]

\[ \sum_{m=1}^{4}\frac{1+\alpha\lambda_{km}^2}{\lambda_{km}^2(2+\alpha\lambda_{km}^2)}\,e^{\lambda_{km}^2 t} = \sum_{m=1}^{4}\frac{\chi^2\lambda_{km}^2}{\beta_k^2-\chi^2\lambda_{km}^4}\,e^{\lambda_{km}^2 t} = \frac{2}{q_k}e^{-p_k t}\operatorname{sh} q_k t. \tag{1.5.13} \]

Substituting into the right-hand side of equality (1.5.9), in place of the expressions standing in square brackets, their values (1.5.10) and (1.5.11), taking into account (1.5.12) and (1.5.13), and noting that

\[ (-1)^k\cos\gamma_k(A-x\sqrt{a_2}) = (-1)^k\cos[\beta_k-\gamma_k(x-a)\sqrt{a_2}] = \sin\gamma_k(x-a)\sqrt{a_2} \quad (k=0,1,2,\ldots), \]

after a series of transformations we obtain

\[ \begin{aligned} u_1^{(1)}(x,t) &= \sum_{k=0}^{\infty} e^{-p_k t} \left[ A_k^{(1)}\operatorname{ch} q_k t + \frac{1}{q_k}\left(p_k A_k^{(1)}+B_k^{(1)}\right)\operatorname{sh} q_k t \right] \\ &\quad \times \sin\gamma_k(x-a)\sqrt{a_2} \\ &\quad + \frac{\beta_0}{\alpha_0}\sum_{k=0}^{\infty} e^{-p_k t} \left[ A_k^{(2)}\operatorname{ch} q_k t + \frac{1}{q_k}\left(p_k A_k^{(2)}+B_k^{(2)}\right)\operatorname{sh} q_k t \right] \\ &\quad \times \cos\gamma_k(A-x\sqrt{a_2}), \end{aligned} \tag{1.5.14} \]

where

\[ A_k^{(1)} = \frac{2\sqrt{a_2}}{l} \int_a^b \varphi_1(\xi)\sin\gamma_k(\xi-a)\sqrt{a_2}\,d\xi, \]

\[ B_k^{(1)} = \frac{2\sqrt{a_2}}{l} \int_a^b \psi_1(\xi)\sin\gamma_k(\xi-a)\sqrt{a_2}\,d\xi, \]

\[ A_k^{(2)}=\frac{2\sqrt{a_1}}{l}\int_b^c \varphi_2(\xi)\cos\gamma_k(c-\xi)\sqrt{a_1}\,d\xi, \]

\[ B_k^{(2)}=\frac{2\sqrt{a_1}}{l}\int_b^c \psi_2(\xi)\cos\gamma_k(c-\xi)\sqrt{a_1}\,d\xi. \]

Similarly one can show that

\[ \begin{aligned} u_2^{(1)}(x,t) &=\frac{\alpha_0}{\beta_0}\sum_{k=0}^{\infty} e^{-p_k t} \left[ A_k^{(1)}\operatorname{ch} q_k t+ \frac{1}{q_k}\left(p_k A_k^{(1)}+B_k^{(1)}\right)\operatorname{sh} q_k t \right]\sin\gamma_k(x\sqrt{a_1}-B) \\ &\quad+\sum_{k=0}^{\infty} e^{-p_k t} \left[ A_k^{(2)}\operatorname{ch} q_k t+ \frac{1}{q_k}\left(p_k A_k^{(2)}+B_k^{(2)}\right)\operatorname{sh}q_k t \right] \\ &\quad\times \cos\gamma_k(c-x)\sqrt{a_1}, \end{aligned} \tag{1.5.15} \]

where \(B=b\sqrt{a_1}-(b-a)\sqrt{a_2}\). The terms \(u_j^{(2)}(x,t)\) \((j=1,2)\) standing on the right-hand sides of equalities (1.5.6) are equal to zero, for only in this case, as is easy to verify, do the functions \(u_j(x,t)\) \((j=1,2)\) satisfy equations (1.1.1)—(1.1.2) and conditions (1.1.3)—(1.1.5). In particular, checking the fulfillment of the initial conditions, we obtain the expansions

\[ \varphi_1(x)=\sum_{k=0}^{\infty} \left[ A_k^{(1)}\sin\gamma_k(x-a)\sqrt{a_2} +\frac{\beta_0}{\alpha_0}A_k^{(2)}\cos\gamma_k(A-x\sqrt{a_2}) \right], \]

\[ \psi_1(x)=\sum_{k=0}^{\infty} \left[ B_k^{(1)}\sin\gamma_k(x-a)\sqrt{a_2} +\frac{\beta_0}{\alpha_0}B_k^{(2)}\cos\gamma_k(A-x\sqrt{a_2}) \right], \]

\[ \varphi_2(x)=\sum_{k=0}^{\infty} \left[ \frac{\alpha_0}{\beta_0}A_k^{(1)}\sin\gamma_k(x\sqrt{a_1}-B) +A_k^{(2)}\cos\gamma_k(c-x)\sqrt{a_1} \right], \]

\[ \psi_2(x)=\sum_{k=0}^{\infty} \left[ \frac{\alpha_0}{\beta_0}B_k^{(1)}\sin\gamma_k(x\sqrt{a_1}-B) +B_k^{(2)}\cos\gamma_k(c-x)\sqrt{a_1} \right], \]

which are also obtained if, in the problem under consideration, one sets \(\alpha=0\), \(\lambda^2=z\), i.e., considers the problem of vibrations of a composite rod consisting of elastic sections (in this case the Green’s function of the spectral problem has no essentially singular points).

1. Differential equations

Thus, \(u_j(x,t)=u_j^{(1)}(x,t)\) \((j=1,2)\), where \(u_1^{(1)}(x,t)\) and \(u_2^{(1)}(x,t)\) are given by the equalities (1.5.14) and (1.5.15).

§ 2. THE PROBLEM OF LONGITUDINAL VIBRATIONS OF A COMPOSITE ROD CONSISTING OF ELASTIC AND ELASTIC-VISCOUS SECTIONS

1. Formulation of the problem. It is required to find solutions of the equations

\[ \frac{\partial^2 u_1}{\partial t^2} = a_1 \frac{\partial^2 u_1}{\partial x^2} \qquad (a<x<b;\quad 0<t<T) \tag{2.1.1} \]

and

\[ \frac{\partial^2 u_2}{\partial t^2} = a_2 \frac{\partial^2 u_2}{\partial x^2} + r \frac{\partial^3 u_2}{\partial x^2 \partial t} \qquad (b<x<c,\ 0<t<T) \tag{2.1.2} \]

under the initial conditions

\[ \begin{aligned} u_1\big|_{t=0} &= \varphi_1(x), & \left.\frac{\partial u_1}{\partial t}\right|_{t=0} &= \psi_1(x) \qquad (a<x<b), \\ u_2\big|_{t=0} &= \varphi_2(x), & \left.\frac{\partial u_2}{\partial t}\right|_{t=0} &= \psi_2(x) \qquad (b<x<c), \end{aligned} \tag{2.1.3} \]

boundary conditions

\[ u_1\big|_{x=a}=0,\qquad \left. a_2\frac{\partial u_2}{\partial x} + r\frac{\partial^2 u_2}{\partial x\partial t} \right|_{x=c} =0 \qquad (0<t<T) \tag{2.1.4} \]

and conjugation conditions

\[ \begin{aligned} a_0 u_1\big|_{x=b} &= \beta_0 u_2\big|_{x=b} \qquad (0<t<T), \\ a_1 \left.\frac{\partial u_1}{\partial x}\right|_{x=b} &= \beta_1 \left.\frac{\partial u_2}{\partial x}\right|_{x=b} \qquad (0<t<T) \end{aligned} \tag{2.1.5} \]

on the assumption that the following conditions are satisfied:

a) the function \(\varphi_1(x)\) on the interval \([a,b]\) has its first three derivatives continuous and its fourth derivative piecewise continuous;

b) the function \(\psi_1(x)\) on the same interval has its first two derivatives continuous and its third derivative piecewise continuous;

c) the function \(\varphi_2(x)\) on the interval \([b,c]\) has its first five derivatives continuous and its sixth derivative piecewise continuous;

d) the function \(\psi_2(x)\) on the same interval has its first three derivatives continuous and its fourth derivative piecewise continuous;

e) \(\varphi_1(a)=0,\ \varphi_1''(a)=0,\ \psi_1(a)=0,\ \varphi_2'(c)=0,\ \varphi_2''(c)=0,\ \psi_2'(c)=0\)

and

\[ \alpha_0\varphi_1(b)=\beta_0\varphi_2(b),\qquad \alpha_1\varphi_1'(b)=\beta_1\varphi_2'(b),\qquad \alpha_0\psi_1(b)=\beta_0\psi_2(b),\qquad \alpha_1\psi_1'(b)=\beta_1\psi_2'(b), \]
\[ \alpha_0 a_1 \varphi_1''(b) = \beta_0\left[a_2\varphi_2''(b)+r\psi_2''(b)\right]. \]

The constants \(a_k\) \((k=1,2)\), \(r\), \(\alpha_k\), \(\beta_k\) \((k=0,1)\) are assumed to be nonzero and positive.

Although problem (2.1.1)—(2.1.5) is a particular case of problem (1.1.1)—(1.1.5) (it is obtained if in problem (1.1.1)—(1.1.5) one sets \(b_1=0,\ b_2=r\)), nevertheless it can be shown that its solution in the form of the contour integral (1.3.1) does not exist. The point is that the poles of the solution of the spectral-

problem obtained from problem (1.2.1)—(1.2.3) for \(b_1=0\) and \(b_2=r\), for sufficiently large \(|\lambda|\), have the form

\[ \lambda_k=\pm \frac{(2k+1)\pi i}{2m_2}+O\left(\frac{1}{2k+1}\right)\qquad (k=0,1,2,\ldots), \]

\[ \lambda_m=\pm \frac{1}{2}\sqrt{\frac{(2m+1)\pi}{m_1}}(1\pm i)+ \]

\[ +\,O\left(\frac{1}{\sqrt{2m+1}}\right)\qquad (m=0,1,2,\ldots) \]

(four possible combinations of signs); and in this case it is impossible to choose a strip \(D_h\) of bounded width \(h\) in which all the poles of the solution of the spectral problem would lie. Consequently, the arguments used in proving the theorem on the existence of a solution [4] do not apply here (in proving the satisfaction of the initial conditions one cannot pass from the integral over a sequence of closed contours \(\Gamma_n\) \((n=1,2,3,\ldots)\) to the integral over a sequence of circles \(O_n\) \((n=1,2,3,\ldots)\)). To find the solution of problem (2.1.1)—(2.1.5), we must construct here, in another way, the spectral problem corresponding to problem (2.1.1)—(2.1.5), and impose on the functions \(\varphi_j(x)\) and \(\psi_j(x)\) \((j=1,2)\) stronger conditions than those imposed on the initial functions in the first problem.

2. A residue representation of the solution of problem (2.1.1)—(2.1.5). Let us associate with problem (2.1.1)—(2.1.5) the following spectral problem:

\[ a_1\frac{d^2y_1}{dx^2}-\lambda^2y_1=-f_1(x,\lambda)\qquad (a<x<b), \tag{2.2.1} \]

\[ (a_2+r\lambda)\frac{d^2y_2}{dx^2}-\lambda^2y_2=-f_2(x,\lambda)\qquad (b<x<c), \tag{2.2.2} \]

\[ y_1|_{x=a}=0,\qquad \left.\frac{dy_2}{dx}\right|_{x=c}=0, \tag{2.2.3} \]

\[ \alpha_k\left.\frac{d^ky_1}{dx^k}\right|_{x=b} = \beta_k\left.\frac{d^ky_2}{dx^k}\right|_{x=b} \qquad (k=0,1), \tag{2.2.4} \]

where

\[ f_1(x,\lambda)=\lambda\varphi_1(x)+\psi_1(x),\qquad f_2(x,\lambda)=\lambda\varphi_2(x)+\psi_2(x)-r\varphi_2''(x) \]

and where, in contrast to problem (1.1.1)—(1.1.5), the operation of differentiation with respect to \(t\) is put in correspondence with the complex parameter \(\lambda\), and not \(\lambda^2\).

The solution of problem (2.2.1)—(2.2.4) can be represented in the same form as the solution of problem (1.2.1)—(1.2.4), if in (1.2.4) and (1.2.5) we put

\[ F_1(x,\lambda)=\frac{f_1(x,\lambda)}{a_1},\qquad F_2(x,\lambda)=\frac{f_2(x,\lambda)}{a_2+r\lambda} \tag{2.2.5} \]

and take into account that

\[ z_1=\frac{\lambda}{\sqrt{a_1}},\qquad z_2=\frac{\lambda}{\sqrt{a_2+r\lambda}}. \tag{2.2.6} \]

Let the following conditions be satisfied:

a) the function \(\varphi_1(x)\) on the interval \([a,b]\) has two first continuous derivatives and a third piecewise-continuous derivative;

b) the function \(\psi_1(x)\) on the same interval has a first continuous derivative and a second piecewise-continuous derivative;

c) the function \(\varphi_2(x)\) has on the interval \([b,c]\) three first continuous derivatives and a fourth piecewise-continuous derivative;

d) the function \(\psi_2(x)\) on the same interval has a first continuous derivative and a second piecewise-continuous derivative;

e) \(\varphi_1(a)=0,\quad \psi_1(a)=0,\quad \varphi_2(c)=0\) and \(\alpha_0\varphi_1(b)=\beta_0\varphi_2(b),\quad \alpha_1\varphi'_1(b)=\beta_1\varphi'_2(b),\quad \alpha_0\psi_1(b)=\beta_0\psi_2(b)\).

Then the following theorem is valid.

Theorem 5. Under conditions a)—e) of the present subsection, the expansion formulas hold
\[ \frac{1}{2\pi i}\sum_\nu \int_{C_\nu} y_j(x,\lambda)\lambda^s\,d\lambda = \begin{cases} \varphi_j(x), & \text{for } s=0,\\ \psi_j(x), & \text{for } s=1, \end{cases} \tag{2.2.7} \]
where the convergence of the series is uniform with respect to \(x\in[a,b]\) for \(j=1\) and \(x\in[b,c]\) for \(j=2\), where \(C_\nu\) is a simple closed contour containing inside it only one singular point of the integrand, and where the singular points are numbered in the order of increase of their moduli, and \(y_j(x,\lambda)\) \((j=1,2)\) are determined by the equalities (1.2.4), (1.2.5) and (1.2.7)—(1.2.13), taking into account (2.2.5) and (2.2.6).

Proof. Taking into account equality (1.2.16) for \(j=2\), which in our case takes the form
\[ e^{\frac{a\lambda}{\sqrt{a_2+r\lambda}}} = e^{\frac{a\sqrt{\lambda}}{\sqrt{r}}} \left\{1+O\left(\frac{1}{\lambda}\right)\right\}, \]
one can show that, for sufficiently large \(|\lambda|\),
\[ \Delta(\lambda)=\frac{1}{4}M\lambda e^{\mu_1m_1\lambda+\mu_2m_2\sqrt{\lambda}}H(\lambda), \tag{2.2.8} \]
where
\[ H(\lambda)=\left[1+e^{-2\mu_1m_1\lambda}\right]\left[1+e^{-2\mu_2m_2\sqrt{\lambda}}\right] +O\left(\frac{1}{\sqrt{\lambda}}\right), \]
\[ M=\frac{\alpha_1\beta_0}{\sqrt{a_1}},\qquad m_1=\frac{b-a}{\sqrt{a_1}},\qquad m_2=\frac{c-b}{\sqrt{r}}, \]
the quantities \(\mu_j\) \((j=1,2)\) are equal to \(1\) or \(-1\), depending on which of the sectors
\[ \frac{m\pi}{2}\leq \arg\lambda \leq \frac{(m+1)\pi}{2}\qquad (m=0,1,2,3) \]
of the complex \(\lambda\)-plane contains \(\lambda\), and for the function \(z=\sqrt{\lambda}\) that branch is chosen for which \(\operatorname{Im}\sqrt{\lambda}\geq 0\).

Following [8], one can show that the roots of the equation \(H(\lambda)=0\), for sufficiently large \(|\lambda|\), have the form
\[ \lambda_k=\pm \frac{(2k+1)\pi i}{2m_1}+O\left(\frac{1}{2k+1}\right)\qquad (k=0,1,2,\ldots), \]
\[ \lambda_n=-\frac{(2n+1)^2\pi^2}{4m_2^2}+O(1)+O\left(\frac{1}{(2n+1)^2}\right)\qquad (n=0,1,2,\ldots). \]

From [8] and Sec. 2 § 1 it follows that for every \(\delta>0\) one can specify a number \(A_\delta>0\) such that the inequality

\[ |H(\lambda)| \geqslant A_\delta, \tag{2.2.9} \]

holds in the \(\lambda\)-plane outside the circles \(Q_\delta^{(k)}\) of radius \(\delta\) with centers at the zeros \(\lambda_k\), and outside certain curves \(C_\delta^{(n)}\) of the fourth order into which the circles \(|z-z_n|=\delta\) pass under the mapping \(z=\sqrt{\lambda}\) \((z_n=\sqrt{\lambda_n})\), with diameters increasing as \(n\) increases.

Now let us take in the \(z\)-plane a sequence of circles \(|z|=r_n\) \((n=1,2,3,\ldots)\) with radii \(r_n\to\infty\) as \(n\to\infty\), lying away from the roots \(z_n=\sqrt{\lambda_n}\) by a distance greater than \(\delta\). On passing to the \(\lambda\)-plane, the circles \(|z|=r_n\) pass into the circles \(|\lambda|=r_n^2=\rho_n\), which do not intersect the curves \(C_\delta^{(n)}\) and which we shall denote by \(O_n\). Clearly, the circles \(|z|=r_n\) can be chosen so that, after the transition to the \(\lambda\)-plane, they do not intersect the circles \(Q_\delta^{(k)}\) (Fig. 2).

Fig. 2.

Integrating by parts the terms standing on the right-hand sides of equalities (1.2.4) and (1.2.5), taking into account equalities (2.2.5) and (2.2.6) and conditions a)—d) of the present item, after a series of transformations we obtain:

\[ y_1(x,\lambda)=\frac{\varphi_1(x)}{\lambda} +\frac{\psi_1(x)}{\lambda^2} +\frac{1}{\lambda^2\sqrt{\lambda}}\,W_1^{(1)}(x,\lambda) + \]

\[ +\frac{1}{\lambda^3}W_2^{(1)}(x,\lambda) +\frac{1}{\lambda^3\sqrt{\lambda}}W_3^{(1)}(x,\lambda), \tag{2.2.10} \]

\[ y_2(x,\lambda)=\frac{\varphi_2(x)}{\lambda} +\frac{\psi_2(x)}{\lambda^2} +\frac{1}{\lambda^2\sqrt{\lambda}}W_1^{(2)}(x,\lambda) +\frac{1}{\lambda^3}W_2^{(2)}(x,\lambda)+ \]

\[ +\frac{1}{\lambda^3\sqrt{\lambda}}W_3^{(2)}(x,\lambda) +\frac{1}{\lambda^4}W_4^{(2)}(x,\lambda), \tag{2.2.11} \]

where the functions \(W_p^{(1)}(x,\lambda)\) \((p=1,2,3)\) and \(W_p^{(2)}(x,\lambda)\) \((p=1,2,3,4)\) are bounded in modulus for sufficiently large \(|\lambda|\) outside the circles \(Q_\delta^{(k)}\) and the curves \(C_\delta^{(n)}\).

Substituting (2.2.10) and (2.2.11) into the integrals

\[ \lim_{n\to\infty}\frac{1}{2\pi i}\int_{O_n} y_j(x,\lambda)\lambda^s\,d\lambda \quad (j=1,2;\ s=0,1) \]

and evaluating them, we obtain, as in Sec. 2 § 1, that

\[ \lim_{n\to\infty}\frac{1}{2\pi i}\int_{O_n} y_j(x,\lambda)\lambda^s\,d\lambda = \begin{cases} \varphi_j(x), & \text{for } s=0,\\ \psi_j(x), & \text{for } s=1. \end{cases} \]

Since, by the well-known residue theorem,

\[ \lim_{n\to\infty}\frac{2}{2\pi i}\int_{O_n} y_j(x,\lambda)\lambda^s\,d\lambda = \frac{1}{2\pi i}\sum_\nu \int_{C_\nu} y_j(x,\lambda)\lambda^s\,d\lambda, \tag{2.2.12} \]

where the sum over \(\nu\) on the right-hand side of (2.2.12) is extended over all singular points of the integrand, the theorem is proved.

Theorem 6. If problem (2.1.1)—(2.1.5) has a solution \(u_j(x,t)\) \((j=1,2)\), continuous for \(x\in[a,b]\) \((j=1)\), \(x\in[b,c]\) \((j=2)\), \(t\in[0,T]\), and having the continuous derivatives entering into the problem \(\left[\dfrac{\partial u_j}{\partial t}\right.\) for

\[ a<x<b\quad (j=1),\qquad b<x<c\quad (j=2),\qquad 0\le t<T; \]

\[ \left.\frac{\partial u_j}{\partial x}\ \text{for } a<x\le b\ (j=1),\qquad b\le x\le c\ (j=2),\qquad 0<t<T;\quad \frac{\partial^2 u_2}{\partial x\,\partial t}\ \text{for } b<x\le c,\ 0<t<T,\right. \]

the remaining derivatives for \(a<x<b\) \((j=1)\), \(b<x<c\) \((j=2)\), \(0<t<T\)], and if the expansion formulas (2.2.7) hold, then this solution is representable in the form of the complete integral residue

\[ u_j(x,t)=\frac{1}{2\pi i}\sum_\nu \int_{C_\nu} y_j(x,\lambda)e^{\lambda t}\,d\lambda \qquad (j=1,2). \tag{2.2.13} \]

The validity of the theorem is verified by substituting the last formula into equations (2.1.1), (2.1.2) and into the conditions (2.1.3)—(2.1.5).

3. Proof of the existence of a solution. In the complex \(\lambda\)-plane, construct a sequence of expanding closed contours \(\Gamma_n\) \((n=1,2,3,\ldots)\), consisting of arcs \(b_n c_n a_n\) of concentric circles \(O_n\), described in item 2, § 2, and of segments \(a_n b_n\) of the straight line \(\operatorname{Re}\lambda=\gamma\), situated to the right of all the circles \(Q_\delta^{(k)}\) and curves \(C_\delta^{(n)}\) (see Fig. 2). Then the following is valid.

Theorem 7. Under fulfillment of conditions a)—d) of item 1, § 2, problem (2.1.1)—(2.1.5) has a solution possessing the properties indicated in Theorem 6 and representable by the formula

\[ u_j(x,t)=\frac{1}{2\pi i}\lim_{n\to\infty}\int_{\Gamma_n} y_j(x,\lambda)e^{\lambda t}\,d\lambda \qquad (j=1,2), \tag{2.3.1} \]

where \(\Gamma_n\) \((n=1,2,3,\ldots)\) is the sequence of closed contours described above.

Proof. To prove the theorem it is sufficient to prove the uniform convergence of the integrals

\[ \lim_{n\to\infty}\int_{\Gamma_n}\lambda^s\frac{d^m y_1}{dx^m}e^{\lambda t}\,d\lambda \tag{2.3.2} \]

\[ (s=0,\ m=0,1,2;\ s=1,\ m=0;\ s=2,\ m=0), \]

\[ \lim_{n\to\infty}\int_{\Gamma_n}\lambda^s\frac{d^m y_2}{dx^m}e^{\lambda t}\,d\lambda \tag{2.3.3} \]

\[ (s=0,\ m=0,1,2;\ s=1,\ m=0,1,2;\ s=2,\ m=0) \]

with respect to \(x\) and \(t\), varying over the intervals indicated for each of the integrals (2.3.2) and (2.3.3) in Theorem 6.

The uniform convergence of the integrals (2.2.2) and (2.3.3) is proved in the same way as in [9], with the aid of the asymptotic representations

\[ \frac{d^m y_1}{dx^m} = \gamma_1^{(m)}(x,\lambda) + \sum_{k=1}^{4}\alpha_k^{(m)}(x)\lambda^{m-k} + \sum_{k=1}^{3}(\sqrt{\lambda})^{2m-k-6}W_k^{(1,m)}(x,\lambda) + (\sqrt{\lambda})^{2m-11}W_4^{(1,m)}(x,\lambda); \tag{2.3.4} \]

\[ \frac{d^m y_2}{dx^m} = \gamma_2^{(m)}(x,\lambda) + \sum_{k=1}^{5}\beta_k^{(m)}(x)(\sqrt{\lambda})^{m-2k} + \sum_{k=1}^{6}(\sqrt{\lambda})^{m-k-6}W_k^{(2,m)}(x,\lambda), \tag{2.3.5} \]

where the functions \(W_k^{(1,m)}(x,\lambda)\) \((k=1,2,3,4)\) and \(W_k^{(2,m)}(x,\lambda)\) \((k=1,2,3,4,5,6)\) are bounded in modulus for sufficiently large \(|\lambda|\) outside the circles \(Q_\delta^{(k)}\) and the curves \(C_\delta^{(n)}\), and moreover
\(\alpha_1^{(0)}(x)=\varphi_1(x)\), \(\alpha_2^{(0)}(x)=\psi_1(x)\),
\(\beta_1^{(0)}(x)=\varphi_2(x)\), \(\beta_2^{(0)}(x)=\psi_2(x)\),

\[ \gamma_1^{(m)}(x,\lambda)= \begin{cases} 0, & \text{for } m=0,1,\\ -F_1(x,\lambda), & \text{for } m=2, \end{cases} \]

\[ \gamma_2^{(m)}(x,\lambda)= \begin{cases} 0, & \text{for } m=0,1,\\ -F_2(x,\lambda), & \text{for } m=2. \end{cases} \]

The equalities (2.3.4), (2.3.5) can be obtained in the same way as the equalities (2.2.10), (2.2.11), by differentiating twice with respect to \(x\) the solutions (1.2.4) and (1.2.5) of the spectral problem (2.2.1)—(2.2.4), taking into account, when integrating by parts, conditions a)—d) of item 1, § 2, and using inequality (2.2.9). It turns out that the integrals (2.3.2) and (2.3.3) converge uniformly with respect to \(x\in[a,b]\), \(t\in[0,T]\), and respectively \(x\in[b,c]\), \(t\in[0,T]\).

4. Uniqueness of the solution and its continuous dependence on the initial conditions

Theorem 8. If conditions a)—d) of item 1, § 2 are fulfilled, the solution of problem (2.1.1)—(2.1.5) is unique in the class of functions representable in the form (2.2.13), and depends continuously on the initial conditions.

The proof of the formulated theorem is carried out analogously to the proof of Theorem 4.

We present the inequalities from which the continuous dependence of the solution on the initial conditions follows. Substituting into the right-hand sides of equalities (1.2.4) and (1.2.5), instead of the functions \(F_j(x,\lambda)\) \((j=1,2)\), their values (2.2.5), and computing twice by parts the integrals whose integrands contain the functions \(\varphi_j(x)\) \((j=1,2)\), and once the integrals whose integrands contain the functions \(\psi_j(x)\) \((j=1,2)\), after a series of transformations we obtain:

\[ y_1(x,\lambda) = \frac{\varphi_1(x)}{\lambda} + \frac{\psi_1(x)}{\lambda^2} + \frac{1}{\lambda^2}\int_a^b \omega_1^{(1)}(x,\xi,\lambda)\varphi_1''(\xi)\,d\xi + \]

\[ + \frac{1}{\lambda^2}\int_a^b \omega_2^{(1)}(x,\xi,\lambda)\psi_1'(\xi)d\xi + \]

\[ + \frac{1}{\lambda^2}\int_b^c \omega_3^{(1)}(x,\xi,\lambda)\varphi_2''(\xi)\,d\xi + \frac{1}{\lambda^2\sqrt{\lambda}}\int_b^c \omega_4^{(1)}(x,\xi,\lambda)\psi_2'(\xi)\,d\xi, \tag{2.4.1} \]

\[ y_2(x,\lambda)=\frac{\varphi_2(x)}{\lambda}+\frac{\psi_2(x)}{\lambda^2} +\frac{1}{\lambda^2}\int_a^b \omega_1^{(2)}(x,\xi,\lambda)\varphi_1''(\xi)\,d\xi+ \]

\[ +\frac{1}{\lambda^2}\int_a^b \omega_2^{(2)}(x,\xi,\lambda)\psi_1'(\xi)\,d\xi+ \]

\[ +\frac{1}{\lambda\sqrt{\lambda}}\int_b^c \omega_3^{(2)}(x,\xi,\lambda)\varphi_2''(\xi)\,d\xi +\frac{1}{\lambda^2}\int_b^c \omega_4^{(2)}(x,\xi,\lambda)\psi_2'(\xi)\,d\xi, \tag{2.4.2} \]

where the functions \(\omega_p^{(j)}(x,\xi,\lambda)\) \((j=1,2;\ p=1,2,3,4)\) are bounded in modulus for sufficiently large \(|\lambda|\).

With the aid of equalities (2.4.1), (2.4.2), just as in item 4 of § 1, one can obtain the inequalities

\[ |u_j(x,t)|\le |\varphi_j(x)|+T|\psi_j(x)|+A_j\int_a^b |\varphi_1''(\xi)|\,d\xi+ \]

\[ +B_j\int_a^b |\psi_1'(\xi)|\,d\xi +C_j\int_b^c |\varphi_2''(\xi)|\,d\xi +D_j\int_b^c |\psi_2'(\xi)|\,d\xi, \]

where \(A_j,\ B_j,\ C_j,\ D_j\ (j=1,2)\) are certain constants.

References

1. Ishlinskii A. Yu. Journal of Applied Mathematics and Mechanics, 4, no. 1, 79—92, 1940.
2. Lazaryan V. A. Proceedings of the Dnepropetrovsk Institute of Railway Transport Engineers, vol. XX, 3—32, 1950.
3. Tvertin A. I. Proceedings of the Dnepropetrovsk Institute of Railway Transport Engineers, vol. XXIII, 1953.
4. Gaiduk S. I. Differential Equations, 2, no. 8, 1061—1071, 1966.
5. Rasulov M. L. The Method of Contour Integral. Moscow, Nauka Publishing House, 1964.
6. Ivanov A. V. In the collection Thermal Physics in Foundry Production. Publishing House of the Academy of Sciences of the BSSR, 11—52, 1963.
7. Tamarkin Ya. D. On Certain General Problems in the Theory of Ordinary Linear Differential Equations and on the Expansion of Arbitrary Functions in Series. Petrograd, 1917.
8. Naimark M. A. Linear Differential Operators. Gostekhizdat, 1954.
9. Gaiduk S. I. Differential Equations, 1, no. 10, 1366—1382, 1965.

Received by the editors
December 28, 1965

Institute of Mathematics, Academy of Sciences of the BSSR