Introduction

This work, following the foundational research of M. L. Rasulov [1] and G. I. Pirmamedov [2], investigates the solution of boundary value problems for linear differential equations. We consider the problem defined by the following equation:

$$c_2(x) \frac{\partial^2 y}{\partial t^2} + c_1(x) \frac{\partial y}{\partial t} + c_0(x) y = \sum_{k=0}^{2} a_k \frac{\partial^k y}{\partial x^k} \tag{1}$$

subject to the boundary conditions:

$$a_0 y + a_1 \frac{\partial y}{\partial x} + a_2 \frac{\partial^2 y}{\partial x^2} = \psi(x) \tag{2}$$

where $c_k(x)$ ($k=0, 1, 2$) are given coefficients and $\Phi(x)$ represents the initial distribution. Following the methodology established by Rasulov [1], the problem (1)–(2) can be analyzed by examining the spectral properties of the associated differential operator.

Spectral Analysis and Asymptotic Representations

As noted by M. Z. Itskovich [3] and G. I. Pirmamedov [2], the behavior of the solution depends significantly on the roots of the characteristic equation:

$$\det(\lambda^2 E - C_2(x)) = 0 \tag{6}$$

We assume that the roots $\lambda_k(x)$ of equation (6) are distinct and satisfy specific regularity conditions for all $x \in [a, b]$. Specifically, we require that the real parts of the eigenvalues satisfy a strict ordering:

$$\text{Re } \lambda \phi_1(x) < \text{Re } \lambda \phi_2(x) < \dots < \text{Re } \lambda \phi_r(x) < 0 < \text{Re } \lambda \phi_{r+1}(x) < \dots < \text{Re } \lambda \phi_{2r}(x) \tag{7}$$

Under these conditions, the fundamental system of solutions $y_k(x, \lambda)$ for the homogeneous equation associated with (1) admits the following asymptotic representation for large values of the spectral parameter $\lambda$:

$$\frac{d^s y_k(x, \lambda)}{dx^s} = \lambda^s [\eta_{k0s}(x) + E(x, \lambda)] \exp\left(\lambda \int_a^x \phi_k(t) dt\right) \tag{9}$$

where $s=0, 1, 2$ and $k=1, 2, \dots, 2r$. Here, $E(x, \lambda)$ denotes a term that vanishes as $|\lambda| \to \infty$.

Green's Function and Solution Representation

The solution to the boundary value problem (1)–(2) can be expressed using the Green's function $U(x, \xi, \lambda)$. The construction of this function involves the determinant of the boundary condition matrix, $\Delta(\lambda) = \det D(\lambda)$. For large $|\lambda|$, the determinant $\Delta(\lambda)$ can be expanded as:

$$\Delta(\lambda) = \lambda^k \exp\left(\lambda \sum_{s=1}^{\nu_s} \omega_{ks}\right) [M_0(\lambda) + E(\lambda)] \tag{19}$$

where $M_0(\lambda)$ is a leading-order term and $E(\lambda)$ represents higher-order corrections. The asymptotic behavior of the solution in various regions of the complex $\lambda$-plane is determined by the signs of $\text{Re } \lambda \omega_k$.

By applying the residue theorem and integrating over the appropriate contours in the $\lambda$-plane, we obtain the final representation of the solution $y(x, t)$. The integral representation accounts for the initial data $\Phi(x)$ and the boundary constraints. The resulting formulas allow for the effective calculation of the system's dynamics, extending the classical results found in [1] and [5] to a broader class of second-order differential operators with variable coefficients.

References

1. Rasulov, M. L., The Contour Integral Method, Nauka, Moscow, 1964.
2. Pirmamedov, G. I., Scientific Notes of Azerbaijan State University, No. 3, 43–51, 1963.
3. Itskovich, M. Z., Mathematical Modeling and Differential Equations, 1967.
4. Tamarkin, J. D., General Problems of the Theory of Ordinary Linear Differential Equations, Petrograd, 1917.
5. Tikhonov, A. N., and Samarskii, A. A., Equations of Mathematical Physics, Moscow, 1964.