Taylor Series Methods Explained Visually | Numerical Methods – Lecture 38

Euler's method steps along the tangent line, the Taylor series cut after the h term. Taylor series methods keep more terms: the order-n method is yₙ₊₁ = yₙ + h yₙ′ + (h²/2) yₙ″ + … + (hⁿ/n!) yₙ⁽ⁿ⁾, a polynomial that bends with the solution. The higher derivatives come from differentiating y′ = f(t, y) along the solution with the chain rule, y″ = f_t + f_y f. A method of order p has local error O(h^(p+1)) and global error O(h^p). We work the examples from the notes: y′ = −2ty, the test problem y′ = t + y (h = 0.1: error 0.0084 at t = 1 against 0.249 for Euler), and the nonlinear y′ = −2ty² with h = 0.2 (y(0.6) ≈ 0.72485 against 0.73529). We compare orders 1 to 4 on a log–log plot, count the steps needed for an error of 10⁻⁶, look at stability on y′ = λy and at the nearly closed orbit on y″ = −y, and see the price: the derivatives grow messy fast. 📌 Chapters 0:00 Introduction 0:30 Recap: Euler's method 1:12 The idea: polynomials that bend 1:52 The Taylor method of order n 2:46 Order of a method 3:19 Higher derivatives: the chain rule 4:01 The total derivative as a picture 4:39 Example: y′ = −2ty 5:35 Worked example: y′ = t + y 6:32 Euler vs Taylor-2: the table 7:08 Why Taylor-2 stays on the curve 7:42 Local truncation error O(h³) 8:35 Global error O(h²) 9:14 Orders 1 to 4 9:55 Steps for an error of 10⁻⁶ 10:25 Nonlinear example: y′ = −2ty² 11:25 Parabola steps vs Euler 12:06 The price: messy derivatives 12:57 Stability: y′ = λy 13:52 Oscillations: a nearly closed orbit 14:28 Algorithm 15:05 Comparison 15:44 Summary 📝 Key results • Taylor method of order n: yₙ₊₁ = yₙ + h yₙ′ + (h²/2) yₙ″ + … + (hⁿ/n!) yₙ⁽ⁿ⁾; Euler is order 1 • Higher derivatives from the equation: y″ = f_t + f_y f (total derivative along the solution) • Local error hⁿ⁺¹/(n+1)! y⁽ⁿ⁺¹⁾(η) = O(hⁿ⁺¹); global error O(hⁿ) • y′ = t + y, y(0) = 1, h = 0.1: Taylor-2 gives y(1) ≈ 3.42816 (error 0.0084) against Euler's 3.18748 (error 0.249) • Error 10⁻⁶ at t = 1: Euler about 2.7 million steps, Taylor order 2: 952, order 3: 61, order 4: 15 • y′ = −2ty², h = 0.2: y(0.6) ≈ 0.72485 against 1/1.36 = 0.73529 • y′ = λy: factor 1 + hλ + (hλ)²/2, stable for −2 ≤ hλ ≤ 0; price: derivatives of f grow messy 💻 Practise this lecture Self-paced practice with live plots, hints and instant feedback: 👉 https://biswarupbiswas.github.io/nume... 📚 Course website (all lectures, notes and practice): https://biswarupbiswas.github.io/nume... ▶ Previous lecture: Euler's Method    • Euler's Method Explained Visually | Numeri...   Next lecture: second-order Runge–Kutta methods, the same order as Taylor-2 using only values of f. #NumericalMethods #TaylorSeries #DifferentialEquations #ODE #Mathematics #Engineering