Discuss the distinction between perturbation problems that are ‘regular’ and those that are ‘singular’: Mathematics Assignment, OU, UK

• Discuss the distinction between perturbation problems that are ‘regular’ and those that are ‘singular’. Some inspiration to get you started∗: Consider the polynomial δx3 + 4x = 5E. First, set E = 1 and find a two-term approximation of the solution in the asymptotic limit that δ → 0. State whether this is a singular perturbation problem or a regular one. Second, set δ = 1 and find a two-term approximation of the solution in the asymptotic limit that E → 0. State whether this is a singular perturbation problem or a regular.
• Numerical and perturbation methods are both techniques used to find approximate solutions to a mathematical problem.
  Some inspiration to get you started∗: Discuss the benefits and drawbacks of the two techniques. Consider the initial value problem x 0 + 2Ex(1 + x) = t on t ∈ (0, 1) with x(0) = 1 and E = 0.01. Solve this using both perturbation methods and a numerical scheme of your choice, e.g., forward Euler and compare the approximate solutions obtained.

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