Preamble

This work, following the developments in [1], investigates the properties of the function $h(x)$, defined as $h(x) = g(x+1) - g(x)$. We consider a smooth function $f(t)$ such that $f(t)f(1-t)dt$ is normalized over the interval $(0, 1)$. Specifically, we define the transition functions $g_n(x)$ and $h_n(x)$ which belong to the class $C^n$ and satisfy specific boundary conditions at the nodes $x_1$ and $x_2$.

The kernel $K(x)$ is constructed using the auxiliary function $g_n(x)$ as follows:
$$K(x) = g_n\left(1 - { |x| - 1 + |x| }\right)$$
For $n = 1, 2, 3$, the functions $h_n(x)$ are elements of $C^n(-1, +1)$ and satisfy $h_n(0) = 1$, while their derivatives vanish at the boundaries: $h_n^{(i)}(0) = h_n^{(i)}(\pm 1) = 0$. These functions are essential for constructing local approximations that maintain high degrees of smoothness across sub-intervals.

Local Approximation and Error Estimates

Let $f(x)$ be a function in $C^{n+1}$ on the interval $[x_1, x_2]$. We define a local interpolant using the basis functions $h_n(x)$ and the values of the function and its derivatives at the endpoints. The approximation formula is given by:
$$f(x) \approx \sum_{i=0}^n \frac{f^{(i)}(x_1)}{i!} (x - x_1)^i h_n\left(\frac{x - x_1}{h}\right) + \sum_{i=0}^n \frac{f^{(i)}(x_2)}{i!} (x - x_2)^i h_n\left(\frac{x - x_2}{h}\right)$$
where $h = x_2 - x_1$. The remainder term $R_{n+1}$ for this approximation can be estimated as:
$$|R_{n+1}| \leq \frac{M}{(n+1)!} h^{n+1}$$
where $M = \max |f^{(n+1)}(\xi)|$ for $\xi \in [x_1, x_2]$. This estimate ensures that the approximation converges as the mesh size $h$ decreases, provided the function $f(x)$ possesses sufficient smoothness.

Application to Boundary Value Problems

The proposed method is particularly effective for solving linear differential equations of the form $Lu = f$ with boundary conditions at $x=a$ and $x=b$. We represent the solution $u(x)$ as a linear combination of the basis functions $\phi_{i,m}(x, a)$ and $\phi_{i,m}(x, b)$. These basis functions are constructed to satisfy the homogeneous boundary conditions, allowing the global solution to be assembled from local components.

For a boundary value problem on $[a, b]$, we partition the interval into $k$ sub-intervals with nodes $x_i = a + ih$. In each sub-interval $(x_{i-1}, x_i)$, the solution is approximated by:
$$u(x) \approx \sum \alpha_i \Phi_i(x)$$
where the coefficients $\alpha_i$ are determined by matching the values and derivatives of the solution at the internal nodes. This approach leads to a system of linear algebraic equations.

Numerical Examples

To demonstrate the effectiveness of the method, we consider a fourth-order differential equation:
$$(x + 21)y^{IV} + gy - kx = 0$$
with boundary conditions $y(0) = \alpha_1$, $y'(0) = 0$, and $y(1) = 0$, $y'(1) = 0$. Using the basis functions $h_4(x)$ and a partition of the interval $[0, 1]$ with $h = 0.2$, we obtain a numerical solution. The coefficients $c_1$ and $c_2$ are determined by solving the resulting system of equations.

In a second example, we approximate the function $y = \sin x$ on a given interval. The numerical results at the nodes $x_1, x_2, x_3$ show high agreement with the exact values:
- At $x_1$: $y_{calc} = 0.4933$, $y_{exact} = 0.5000$
- At $x_2$: $y_{calc} = 0.7084$, $y_{exact} = 0.7033$
- At $x_3$: $y_{calc} = 0.8606$, $y_{exact} = 0.8659$

These results confirm that the method provides a robust framework for both function approximation and the numerical solution of differential equations, maintaining high accuracy with a relatively small number of nodes.