08. Asymptotic Analysis for ODEs

Warning about Notes

Notes for this chapter aren’t as helpful as it’s lots of examples / learn as you go explore
It is advised to refer to lecture notes pages $84+$

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8.1 Regular Perturbations in ODEs

Finding Approximate Solutions to ODEs

Similar to Asymptotic approach for finding asymptotic expansions
As $y$ is a function of $x$ instead of constants use functions of $x$ so

Suppose solution $y(x)$ is expressed in integer powers of $ϵ$ so

$$y(x;ϵ)\sim y_0(x)+ϵy_1(x)+{ϵ}^2{y}_2(x)+\cdots$$

Continue with substitution into ODE and group powers of $ϵ$ and compare coefficients

Note that for examples refer to lecture notes page $85$

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8.2 Boundary Layers

8.2.1 A First Example

Boundary Layer

Boundary Layer refers to narrow region when approximating $y(x)$ where
Exact Solution $y(x)$ rapidly adjust to satisfy boundary conditions unlike approximation

Example

Consider Approximate Solution to IVP

$$ϵy^{\prime }(x)+y(x)=e^{-}x,x>0\text{ with }y(0)=0$$

Suppose solution to be regular asymptotic expansion of form

$$y\sim y_0+ϵy_1+\cdots$$

Then

$$\begin{array}{rl}{y}_0(x)& =e^{-x}\\ y_1(x)& =-y_0^{\prime }(x)=e^{-x}\\ & ⋮\end{array}$$

However cannot satisfy boundary condition as $ϵ$ multiplies higher derivative

With exact solution

$$y(x,ϵ)=\frac{e^{-x}}{1-ϵ}-\frac{e^{-\frac{x}{ϵ}}}{1-ϵ}$$

So $y(x)\sim e^{-x}$ provides a good approximation to exact for nearly all values of $x$
However close to $x=0$, the solution rapidly adjust to satisfy boundary condition

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8.2.2 Inner and Outer Expansions

Outer Expansion

Consider ODE

Outer Expansion of ODE generally consider

$$x=O(1)$$

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Inner Expansion

Consider ODE

Inner Expansion of ODE consider what order $x$ needs to be match boundary conditions
Then rescale $x$ accordingly

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8.2.3 Matching

Overlap Region

Overlap Region is where
Inner and Outer Expansions Approximations are valid

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Common Limit

Let $y_0$ and $Y_0$ denote the first term in the expansions of Outer and Inner respectively
Generally

$$\underset{x\to 0}{\lim }{y}_0(x)=\underset{X\to \infty }{\lim }{Y}_0(X)$$

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Composite Expansion

$$y_{\text{comp}}(x)=\underset{outer}{\underbrace{y_0(x)}}+\underset{inner}{\underbrace{Y_0(X)}}-\text{common limit}$$

where $X$ is in terms of $x$ and $ϵ$

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8.2.4 Getting the Expansion from the ODE

Getting Expansion from ODE

Consider ODE

Suppose $y$ has asymptotic expansion

$$y\sim y_0+ϵy_1+\cdots$$

Then $y_0$ is the Outer Expansion

If the expansion doesn’t satisfy boundary condition

$$y(a)=b$$

Then suppose there is a boundary layer at $x=a$
Hence rescale $x$ by

$$x=\delta X\text{ and }y(x)=Y(X)$$

where $\delta \ll 1$

Determine order of $\delta$ by seeking a dominant balance in ODE for $Y$
($\delta$ such that highest derivative term is no long negligible)

Then assume asymptotic expansion for $Y$

$$Y(X)\sim Y_0(X)+ϵY_1(X)+\cdots$$

to get Inner Expansion $Y_0$

If matching then find Common limit to get Composite expansion

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8.3 Boundary Layers in BVPs

Lecture Notes

Apologies, please refer to Lecture Notes pages $93$

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8.4 More General Perturbation Methods for ODEs

Lecture Notes

Apologies, please refer to Lecture Notes pages $96$