Calculate exact or Picard iteration approximations of y(t) for first-order initial value problems from y′=f(t,y), t₀, y(t₀), and target t.
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Picard Iteration Formula
For an IVP y′ = f(t, y) with y(t₀) = y₀, Picard iteration constructs a sequence of functions converging to the unique solution on some interval around t₀, provided f is continuous near the initial point and locally Lipschitz continuous in y there.
| Symbol | Meaning |
|---|---|
| t₀ | Initial point |
| y₀ = y(t₀) | Initial value |
| f(t, y) | Right-hand side of y′ = f(t, y) |
| yk(t) | k-th iterate; y0(t) = y₀ (constant starting guess) |
| L | A sufficient Lipschitz bound is sup |∂f/∂y| when that derivative exists and is bounded |
| h | Guaranteed interval of existence: min(a, b/M) |
What is Picard’s Theorem?
The Picard-Lindelöf theorem (also called the Cauchy-Lipschitz theorem) states: if f(t, y) is continuous on a closed rectangle R centered at (t₀, y₀) and satisfies a Lipschitz condition in y with constant L, then the IVP y′ = f(t, y), y(t₀) = y₀ has exactly one solution on |t − t₀| ≤ h, where R = {|t −t₀| ≤a, |y −y₀| ≤b}, h = min(a, b/M) and M = sup|f| on R (take h =a when M =0).
The theorem is named after Emile Picard (1856–1941), who formalized the iterative proof in 1890, and Ernst Lindelöf (1870–1946), who extended it to local Lipschitz conditions in 1894. Earlier foundational work by Cauchy (1820s) and Lipschitz (1876) yields the alternate name Cauchy-Lipschitz theorem.
Existence and Uniqueness Theorem Comparison
| Theorem | Existence | Uniqueness | Requirement on f |
|---|---|---|---|
| Picard-Lindelöf | Yes | Yes | Continuous + Lipschitz in y |
| Peano | Yes | No | Continuous in (t, y) only |
| Carathéodory | Yes (a.e.) | No | Measurable in t, continuous in y almost everywhere, with a locally integrable bound |
Lipschitz Constants for the Built-In Equations
| Equation | ∂f/∂y | Lipschitz Constant L | Scope |
|---|---|---|---|
| y′ = y | 1 | L = 1 | Global |
| y′ = t | 0 | L = 0 | Global |
| y′ = y + t | 1 | L = 1 | Global |
| y′ = 1 + y² | 2y | L = 2 max |y| on the chosen bounded interval | Local only; solution blows up at finite t |
When Uniqueness Fails
The Lipschitz condition is not merely technical. When it fails at a point, multiple distinct solutions can pass through the same initial condition, or solutions can escape to infinity in finite time.
| ODE | IVP | Issue | Result |
|---|---|---|---|
| y′ = y1/3 | y(0) = 0 | ∂f/∂y unbounded at y = 0 | Infinitely many solutions: y = 0 and y = (2t/3)3/2 |
| y′ = y² | y(0) = 1 | Locally Lipschitz; global condition fails | Unique but blows up at t = 1: y = 1/(1 − t) |
| y′ = |y|1/2 | y(0) = 0 | Hölder continuous but not Lipschitz | Infinitely many forward solutions: remain zero until any c ≥0, then y = (t −c)²/4 for t ≥c |
| y′ = sign(y) | y(0) = 0 | f discontinuous at y = 0 | With sign(0) =0, zero is the unique classical differentiable solution; delayed departures are only almost-everywhere solutions |
Convergence of Picard Iterates
For y′ = y, y(0) = 1 (exact solution et), each Picard iterate equals a partial Taylor sum. The table below shows exact convergence values at t = 1.
| Iterate k | yk(t) closed form | yk(1) | Absolute error at t = 1 |
|---|---|---|---|
| 0 | 1 | 1.000000 | 1.718282 |
| 1 | 1 + t | 2.000000 | 0.718282 |
| 2 | 1 + t + t²/2 | 2.500000 | 0.218282 |
| 3 | 1 + t + t²/2 + t³/6 | 2.666667 | 0.051616 |
| 5 | Partial sum to t⁵/120 | 2.716667 | 0.001616 |
| 7 | Partial sum to t⁷/5040 | 2.718254 | 0.000028 |
| 10 | Partial sum to t10/3628800 | 2.718282 | 2.7313 ×10⁻⁸ |
For stiff equations or large intervals, Runge-Kutta methods reach comparable accuracy in far fewer function evaluations. Picard iteration is used primarily as a theoretical tool; method choice depends on the equation, stability and error requirements; stiff problems often need suitable implicit methods.
How to Calculate Using Picard’s Theorem
- Write the IVP in standard form: y′ = f(t, y) with y(t₀) = y₀.
- Verify the Lipschitz condition: compute |∂f/∂y| on the target domain and confirm it is bounded above by some constant L.
- Choose target point t and start with y₀(t) = y₀ as the constant initial guess.
- Apply the Picard integral formula repeatedly. For numerics, increase integration steps first (per-iterate accuracy) before increasing the iterate count (convergence depth).
- Stop when successive iterates differ by less than your error tolerance, then compare with the exact reference shown in the results. A small endpoint change is not a certified error bound.
Example Problem:
Equation: y′ = y + t, t₀ = 0, y(0) = 1, target t = 0.5. Lipschitz constant L = 1 (global). Exact solution: y(t) = 2et − t − 1, so y(0.5) ≈ 1.797443. Using Picard iteration mode with 5 iterations and 100 integration steps gives approximately 1.797421 with composite trapezoidal integration, an absolute error of about 0.0000216. Both values round to 1.7974.
Applications
| Domain | Example ODE | Role of Picard-Lindelöf |
|---|---|---|
| Population dynamics | Logistic: P′ = rP(1 − P/K) | Globally Lipschitz on compact sets; unique population trajectory |
| Circuit analysis | RC circuit: V′ = −V/(RC) + E(t)/(RC) | Global Lipschitz; predictable transient response guaranteed |
| Numerical solvers | Euler, RK4, adaptive methods | Existence and uniqueness support the IVP model; numerical convergence also requires method-specific consistency and stability conditions |
| Control theory | State-space: x′ = Ax + Bu | Bounded matrix A implies global Lipschitz; unique state trajectory |
| Epidemiology | SIR model: S′ = −βSI/N | Locally Lipschitz on feasible domain; unique epidemic curve on finite horizon |
