Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: $y^{\prime \prime }\left(x\right)+Q\left(x\right)y^{\prime }\left(x\right)+P\left(x\right)y\left(x\right)=R\left(x\right)$ with the Dirichlet’s boundary conditions, where $P\left(x\right)$ and $Q\left(x\right)$ are smooth, but $R\left(x\right)$ may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of $y^{\prime \prime }$ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergenct based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology.

分形插值函数（FIF）补充和包含所有经典插值。使用分形函数的主要优点是它们可以捕获与函数相关的不规则性或平滑度。这项工作提出了通过矩来使用三次样条FIF来解决两点边值问题（BVP）的问题，该问题涉及非齐次二阶微分方程中的复杂非光滑函数。特别地，我们采用了二阶线性BVP：${ÿ}^{\prime \prime }（X）+问（X）{ÿ}^{\prime }（X）+P（X）ÿ（X）=\mathrm{[R}（X）$ 与狄利克雷的边界条件， $P（X）$ 和 $问（X）$ 很顺利，但是 $\mathrm{[R}（X）$可能是连续无处可微的功能。使用微分方程的离散形式，通过三对角线系统计算矩，该对角线系统由内部网格的连续性条件和端点条件通过导数函数获得。然后将这些矩用于构造BVP的三次分形样条曲线解，其中BVP的非光滑性质${ÿ}^{\prime \prime }$可以通过分形方法来捕获。当与分形样条相关的比例因子设为零时，分形解减小为BVP的经典三次样条解。我们基于网格点的截断误差分析证明了该方法是收敛的。数值例子证明了分形方法的优势。