Taylor Series Method for ODE’s

Source inspiration: (Mathew 2000-2019).

Animations

Each animation below shows the Taylor series solution trace building step by step for the IVP \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\) over \([0,5]\). Each frame adds one new solution point. The Taylor method uses the ODE to compute higher derivatives analytically, then advances via a truncated Taylor series \(y_{n+1} = y_n + hy'_n + \frac{h^2}{2}y''_n + \cdots\) — giving local accuracy of \(O(h^{p+1})\) for order \(p\).

Julia source scripts that generated these animations are linked under each case.

Case 1 — Taylor Order 3, \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\)

Behavior: With order \(p=3\), three derivative terms are used per step. The local truncation error is \(O(h^4)\), matching RK4 in order. The Taylor method requires explicit symbolic derivatives of the ODE right-hand side.

Julia source

Taylor Order 3 animation: numerical solution trace builds from left to right for y’ = 1 - t·y^(1/3) with y(0)=1; each frame adds one Taylor order-3 step

Case 2 — Taylor Order 4, \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\)

Behavior: With order \(p=4\), four derivative terms are used per step. The local truncation error is \(O(h^5)\), giving slightly better accuracy than order 3 with the same step size \(h=0.25\).

Julia source