Enhancing fuzzy transform using PCHIP interpolation: A novel approach to function approximation and solving differential equations

In this study, a hybrid numerical method is presented, combining Fuzzy Transform (FT) and PCHIP (Piecewise Cubic Hermite Interpolating Polynomial) interpolation techniques in developing the accuracy and flexibility of function approximation and solutions to differential equations. The method operates in two stages: first, a low-dimensional fuzzy approximation is constructed using basis functions on a coarse grid, capturing global trends efficiently. Second, residuals between the fuzzy approximation and the true solution (or observed data) are interpolated using PCHIP, which preserves monotonicity and local shape characteristics while avoiding spurious oscillations. Numerical validation demonstrates a reduction of over 98% in mean square error compared to the standalone fuzzy transform, confirming the enhanced accuracy of the improved method across the tested cases. Theoretical error bounds are derived via the superposition principle, demonstrating that the total error is governed by the sum of FT approximation and PCHIP interpolation errors. Using this method, discrete measurements or sample observations can be mathematically modeled, and the method creates an interpolant in continuous space for any empirical data by using PCHIP, which makes it possible for any real-world data sets to be treated analytically (e.g., differentiated or integrated) over the observed values. So, this spoken itself satisfies the gap between measurements taken as discrete observations and using continuous representations in modeling. This would be highly useful for experimental science and engineering applications, such as when retrieving sensor data or performing irregularly sampled measurements that need intensive numerical treatment.