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Lesson 72 of 75
Easy practice

Applied mathematics synthesis

Objective. Connect PDEs, dynamics and numerical methods.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Applied mathematics synthesis connects partial differential equations, dynamical systems and numerical methods to analyse a single real system fully, recognising that a PDE's steady states correspond to a dynamical system's fixed points, and that a numerical scheme's stability depends on the equation's underlying mathematical structure. For example, modelling heat flow in a rod combines the heat PDE (governing equation), its long-time steady-state solution (a dynamical systems concept, the equilibrium), and a finite difference scheme (numerical method) whose stability requires checking the CFL condition tied to the PDE's diffusivity. Synthesis means using each area to check and inform the others, not applying them in isolation.

Worked example

In an applied mathematics synthesis problem, a damped oscillator dx/dt = −x has equilibrium x = 0, and an explicit Euler scheme xₙ₊₁ = xₙ(1−Δt) is used with Δt = 0.5 from x₀ = 8. Find x₂.

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Illustrated card showing 0 … 1
  1. 1Apply the scheme once: x₁ = x₀(1−Δt) = 8(1−0.5) = 8(0.5) = 4.
  2. 2Apply the scheme again: x₂ = x₁(1−Δt) = 4(0.5).
  3. 3Compute 4(0.5) = 2.
  4. 4So x₂ = 2, consistent with decay towards the dynamical system's equilibrium at 0.

Answer: 2

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Common mistake. A common mistake is discretising a PDE numerically without checking that the scheme preserves the qualitative long-time behaviour (such as decay to a known equilibrium) that the continuous dynamical system predicts.

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