Chapter 7

Use singular perturbation concepts to find accurate approximations for three solutions of \(e x^{3}-x+1=0\) as \(\varepsilon \downarrow 0\).

Use a matched asymptotic procedure, as \(\varepsilon \downarrow 0\), to find uniformly valid solutions of the following and compare your results with the exact solution: (i) \(\varepsilon u^{\prime}+u=1, u(0)=0\). (ii) \(\varepsilon u^{\prime \prime}-u=-1, u(-1)=0=u(1)\). (The stretching involves \(\varepsilon^{1 / 2}\) in this problem.) (iii) \(\varepsilon u^{\prime \prime}+u^{\prime}=a+b x, a, b\) constants, \(u(0)=0, \quad u(1)=1\).

Find the first term in a uniformly valid asymptotic solution, as positive \(\varepsilon \rightarrow 0\), of $$ \frac{\mathrm{d}^{2} u}{\mathrm{~d} t^{2}}+u=c\left(u^{2}-1\right) \frac{\mathrm{d} u}{\mathrm{~d} t}, \quad u(0 ; \varepsilon)=a, \quad \frac{\mathrm{d} u}{\mathrm{~d} t}(0 ; \varepsilon)=0 $$

Find the first term in the uniformly valid asymptotic solution of $$ \frac{\mathrm{d}^{2} u}{\mathrm{~d} t^{2}}+u=-c u^{3}, \quad u(0 ; \varepsilon)=a, \quad \frac{\mathrm{d} u}{\mathrm{~d} t}(0 ; \varepsilon)=0, \quad 0<\varepsilon \ll 1 $$