Picard and Taylor Series Method Calculator
Picard's method of successive approximations and Taylor's series method for dy/dx = f(x, y), y(x₀) = y₀: each approximation or derivative worked out exactly, compared with a fine Runge–Kutta solution.
About the Picard and Taylor Series Method Calculator
Taylor's series method expands y about x₀: y = y₀ + (x − x₀)y₀′ + (x − x₀)²y₀″/2! + …, where y′ = f, y″ = f_x + f·f_y, and so on, all evaluated at (x₀, y₀). Picard's method starts with y₀ and repeats yₙ₊₁(x) = y₀ + ∫ₓ₀ˣ f(t, yₙ(t)) dt.
Picard's method needs the integrals to be doable, so here f must be a polynomial in x and y.
The formula
Taylor: y = y₀ + hy₀′ + h²y₀″/2! + …; Picard: yₙ₊₁ = y₀ + ∫ₓ₀ˣ f(t, yₙ) dt
Worked example
dy/dx = x + y, y(0) = 1
- y′ = 1, y″ = 1 + y′ = 2, y‴ = 2, y⁗ = 2 at x = 0
- y(0.1) ≈ 1 + 0.1 + 0.01 + 0.000333 + 0.0000083 = 1.1103417
Questions people ask
Which method is better?
Both are accurate near x₀ and get worse further away. Taylor needs derivatives of f; Picard needs integrals. For long ranges use Runge–Kutta.
Why is there a difference from Runge–Kutta?
Both series are cut off after a few terms; the difference is the truncation error.