Interpolation Analysis and Numerical Solution of ODEs Using Euler's Method - MATLAB Implementation Examples

This program demonstrates MATLAB-based implementation examples for interpolation analysis and numerical solution of ordinary differential equations using Euler's method. For interpolation analysis, we employ polynomial interpolation and spline interpolation methods. Polynomial interpolation, implemented using MATLAB's polyfit and polyval functions, is suitable for datasets with fewer data points where we construct a single polynomial that passes through all given points. Spline interpolation, utilizing MATLAB's spline function, is more appropriate for larger datasets as it creates piecewise polynomials that maintain continuity and smoothness between segments. For solving ordinary differential equations, we introduce the fundamental Euler method. This numerical approach, implemented through iterative computations in MATLAB, calculates function values using a series of approximations based on initial conditions and the differential equation itself. The basic algorithm follows the formula: y_{n+1} = y_n + h*f(x_n, y_n), where h represents the step size. We also present enhanced versions including the improved Euler method (also known as Heun's method) which employs a predictor-corrector approach to achieve higher accuracy. These advanced methods typically implement a two-step process: first predicting a value using Euler's method, then correcting it using the average of slopes at the current and predicted points. The program's objective is to help beginners understand fundamental concepts and methods in interpolation analysis and ordinary differential equations while providing practical MATLAB implementation examples that demonstrate key functions and algorithmic approaches.