Taylor Methods

Numerical Analysis · Axiom Academy

LESSON Taylor Series Methods for ODEs Using Taylor expansions to derive numerical methods for solving ordinary differential equations with controllable accuracy. Step 1: Taylor Series Foundation Consider the initial value problem: The Taylor series expansion of y(t + h) around t is: We can achieve any desired order of accuracy by including more terms The local truncation error is O(h n+1 ) for an n-th order method Provides a theoretical foundation for understanding other methods Step 2: First-Order Taylor Method (Euler's Method) Starting from the Taylor expansion: Truncating after the first derivative gives: The terms we dropped start with: So Euler's method has local error O(h 2 ) and global error O(h) . Step 3: Second-Order Taylor Method The second-order Taylor method includes the y'' term: But we need y''(t). Using the chain rule on y' = f(t, y): This gives us the complete second-order Taylor formula: Step 4: Computing Higher Derivatives We've seen that y'' requires f t and f y . For y''', we differentiate again: In general, differentiating y (n) to get y (n+1) requires: Take the total derivative of the expression for y (n) Apply the chain rule: replace dy/dt with f(t,y) Compute all required partial derivatives of f Example: y' = ty with y(0) = 1 Step 5: Higher-Order Taylor Methods Arbitrary accuracy - can achieve any order Theoretical foundation - basis for error analysis Clear error bounds - predictable behavior Single function evaluation per step (plus derivatives)

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