Picard Iteration for Delay Equations

ISEF Category: Mathematics

Subcategory: Analysis  ·  Difficulty: Advanced  ·  Setup: University Lab  ·  Time: Full Year

The Hook

Some equations do not react to the present alone. They also remember the past, and that memory can change the answer. If you can measure how fast a solution method settles down, you can tell when a model is reliable and when it needs more work. That is the heart of this project.

What Is It?

Picard iteration is a repeat-the-process method for solving equations. You start with a guess, plug it into the equation, get a better guess, and repeat. Think of it like refining a blurry photo, each pass should make the answer clearer. For delay differential equations, the rule depends on past values, not just the current one, so the feedback loop gets more complicated.

A state-dependent delay means the amount of delay changes with the system itself. That is like a thermostat that waits longer to respond when the room is already warm. This makes the math harder, because the delay is part of the unknown solution, not just a fixed input. Your project asks how fast Picard iteration converges when the delay changes with the state, and how well theory matches numerical experiments.

Why This Is a Good Topic

This is a strong math project because you can test a clear claim, the rate of convergence, with real computations and proofs. You can compare theory to numerical output, which gives your project depth. The topic connects to models in economics, biology, and engineering, where delayed feedback shows up all the time. You can learn how to turn an abstract theorem into a measurable prediction.

Research Questions

• How does the size of the delay sensitivity change the convergence rate of Picard iteration?
• What is the effect of the initial guess on the number of iterations needed for convergence?
• Does a larger Lipschitz constant in the delay term slow convergence in the expected way?
• To what extent do numerical results in Julia match the theoretical convergence bounds?
• Which household-economy parameter choices make the iteration fail to converge quickly?
• How does the state-dependent delay compare with a fixed-delay version under the same model structure?

Basic Materials

• Laptop with Julia installed.
• DifferentialEquations.jl package.
• Jupyter notebook or Pluto notebook for notes and plots.
• Spreadsheet software for tracking iteration counts and error values.
• Graphing calculator or online graphing tool for quick checks.
• Access to journal articles or lecture notes on delay differential equations.
• Scratch paper or a tablet for writing out proofs and recursions.

Advanced Materials

• University library access for analysis papers on delay differential equations.
• Julia with DifferentialEquations.jl, Plots.jl, and DataFrames.jl.
• Symbolic algebra tool such as Mathematica, Maple, or SymPy for checking derivations.
• Access to a high-performance laptop or workstation for repeated simulations.
• LaTeX editor for writing proofs and organizing theorem statements.
• Version control software such as Git for tracking code and proof drafts.

Software & Tools

• Julia: Runs numerical experiments for delay differential equations and iteration tests.
• DifferentialEquations.jl: Solves delay differential equation models and supports state-dependent delay workflows.
• Pluto: Helps you keep code, notes, and plots in one reproducible notebook.
• DataFrames.jl: Organizes iteration counts, errors, and parameter sweeps.
• Python: Gives you a backup option for plotting, fitting, and statistical checks.

Experiment Steps

1. Define the exact delay differential equation and state-dependent delay rule you want to study.
2. Identify the theorem or bound you want to test, then write down the quantities that should control convergence.
3. Choose one model parameter to vary first, so you can see how the rate changes in a clean way.
4. Plan a numerical workflow that computes successive Picard iterates and measures error between iterations.
5. Build a comparison table that places the theoretical bound next to the observed convergence rate.
6. Add a second model version, such as a fixed-delay case, so you can compare how state dependence changes the behavior.

Common Pitfalls

• Using a delay rule that is not smooth enough for the theorem you want to test, which breaks the link between theory and numerics.
• Comparing iteration runs with different step sizes or solver settings, which makes the convergence rate look better or worse for the wrong reason.
• Measuring only final solution values instead of iteration-to-iteration error, which hides the actual Picard convergence pattern.
• Changing several model parameters at once, which makes it impossible to tell which one affected the rate.
• Assuming the numerical solver output is the Picard iteration itself, when the solver may use a different internal method.

What Makes This Competitive

A strong version of this project does more than confirm that the iteration works. You compare a real convergence bound with observed rates across several parameter regimes, then explain when the bound is tight and when it is loose. You can also test a fixed-delay model against a state-dependent one, which gives a sharper mathematical story. Careful error metrics, clean parameter sweeps, and clear proof structure will matter more than flashy code.

Project Variations

• Test the same convergence question on a simple biological delay model instead of a household-economy system.
• Replace the state-dependent delay with a fixed delay, then compare the observed rate of convergence.
• Study how different norms or error measures change the apparent convergence rate of Picard iteration.

Learn More

• MIT OpenCourseWare: Search for real analysis, differential equations, and dynamical systems lecture notes that support the proof side of the project.
• NIH PubMed: Search for review articles on delay differential equations in biological and medical models.
• arXiv: Search for preprints on state-dependent delay and Picard iteration to see current research directions.
• DifferentialEquations.jl Documentation: Read the official examples for delay differential equations and state-dependent delay setup.
• SIAM Review: Look for survey articles on numerical methods for delay differential equations through your library or journal access.