Preamble

In 1967, V. T. Khorovtsev [1] investigated boundary value problems for second-order partial differential equations. Consider the following elliptic equation:
$$a_1(x) \frac{\partial^2 u}{\partial x^2} + a_2(x) \frac{\partial u}{\partial x} - a_3(x) u + a_0(x) \frac{\partial^2 u}{\partial y^2} = f(x, y)$$
subject to the boundary conditions:
$$\alpha_1 u(x, 0) - \beta_1 \frac{\partial u(x, 0)}{\partial y} = \phi_1(x), \quad \alpha_2 u(x, b) + \beta_2 \frac{\partial u(x, b)}{\partial y} = \phi_2(x)$$
where $\beta_i > 0$ ($i = 1, 2$). For discrete values $y = y_k$ ($k = 1, 2, \dots, n$), the problem can be approximated by a system of ordinary differential equations:
$$a_1(x) u''k + a_2(x) u'_k - a_3(x) u_k + a_0(x) \frac{1}{h^2} (B_k u) = f_k(x)$$} - A_k u_k + C_k u_{k-1
where $h$ is the step size in $y$. The boundary conditions at $x=0$ and $x=a$ are given by:
$$\alpha_3 u(0, y) - \beta_3 u'_x(0, y) = \phi_3(y), \quad \alpha_4 u(a, y) + \beta_4 u'_x(a, y) = \phi_4(y)$$
with $\beta_i > 0$ and $\alpha_i + \beta_i > 0$ for $i = 3, 4$.

Numerical Analysis and Stability

To solve the system of equations (7)–(8), we employ a separation of variables approach. Let $u_k(x) = y(k) v(x)$. Substituting this into the homogeneous part of the system leads to the eigenvalue problem:
$$B_k y(k+1) + (\lambda - A_k) y(k) + C_k y(k-1) = 0$$
with boundary conditions $Y(0) = \alpha Y(1)$ and $Y(n+1) = \beta_{n+1} Y(n)$. This results in a tridiagonal matrix whose characteristic equation $D_n(\lambda) = 0$ determines the eigenvalues $\lambda_j$. Under the conditions $B_k > 0$ and $C_k > 0$, the eigenvalues $\lambda_j$ are real and distinct [4].

The general solution for each mode $v_j(x)$ satisfies:
$$a_1(x) v''_j + a_2(x) v'_j - (a_3(x) + \lambda_j a_0(x)) v_j = 0$$
The complete solution $u_k(x)$ is then constructed as a linear combination of these eigenfunctions, satisfying the boundary conditions (4)–(5).

Convergence and Error Estimation

We define the operator $L_h u_k$ and establish a maximum principle for the discrete system. If $L_h u_k \geq 0$ in the domain $(0, a)$ and $u_k$ is not constant, then the maximum value of $u_k$ must occur on the boundary. This property ensures the stability of the numerical scheme.

Let $\epsilon_k(x)$ be the error of the approximation. The error satisfies the system $L_h \epsilon_k = R_k(x)$, where $R_k(x)$ is the truncation error. Using the properties of the coefficients $a_i(x)$ and the bounds on the derivatives of the exact solution $M_m = \max |\partial^m u / \partial x^m|$, we can estimate the error magnitude. Specifically, if $m = \min [a_3(x) + \lambda]$, the error is bounded by:
$$|\epsilon_k(x)| \leq \frac{1}{12m} \max |a_0(x) (b_{1k} M_4 + 2 |b_{2k}| M_3)| + \dots$$
This demonstrates that the numerical solution converges to the exact solution as the mesh size $h$ approaches zero, provided the coefficients satisfy the required smoothness and ellipticity conditions [5, 6].

Conclusion

The method of lines described above provides a robust framework for solving elliptic boundary value problems. By reducing the partial differential equation to a system of coupled ordinary differential equations, we leverage well-established techniques for tridiagonal systems and eigenvalue problems. The stability and convergence of the method are guaranteed under the specified constraints on the boundary coefficients and the operator $L_h$.

References

1. Khorovtsev, V. T. (1967). On certain boundary value problems for elliptic equations.
2. Faddeev, D. K., & Faddeeva, V. N. (1963). Computational Methods of Linear Algebra.
3. Givens, W. (1954). Numerical computation of the characteristic values of a real symmetric matrix. U.S. Atomic Energy Commission, ORNL-1574.
4. Krein, M. G. (1952). On the theory of symmetric operators. Progress in Mathematical Sciences, 30(3), 695–702.
5. Samarskii, A. A. (1965). Theory of Difference Schemes.
6. Richtmyer, R. D. (1956). Difference Methods for Initial-Value Problems.