Consider the following nonlinear second order ordinary

Question

Questions 1 - Consider the following nonlinear, second order ordinary differential equation:

ε(h³(x)uu'); = (h(x)u)', 0 < x < 1,

u(0; ε) = u(1; ε) = 1,

where h(x) > 0 for x ∈ [0, 1], h(1) = 1, and ε << 1. Determine the leading order inner, outer and composite expansion.

Question 2 - Determine a composite expansion to leading order plus one correction term for the following first order partial differential equation:

ε(∂u/∂t + ∂u/∂x) + tu = 1,    - ∞ < t < ∞, t > 0,

u(x, 0; ε) = sinx.

(Hint: Some the equation is non-homogeneous, consider a scaling of the dependent variable as well.)

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