*3.2.1 Sources of nonuniformity*

There are two common reasons for nonuniformities in asymptotic expansions, they are


Consider the nonlinear Duffing equation

$$\begin{cases} \frac{d^2u}{dt^2} + u + \varepsilon u^3 = 0, \quad t \in [0, \infty) \\\\ u(0) = a, \quad \frac{du}{dt}(0) = 0. \end{cases} \tag{28}$$

Suppose the solution may be expanded using the standard asymptotic sequence

$$
\mu(t; \varepsilon) \sim u\_0(t) + \varepsilon u\_1(t) + \varepsilon^2 u\_2(t) + \dotsb \tag{29}
$$

On substituting this in (28) and in the initial conditions, we get

$$\begin{cases} \frac{d^2 u\_0}{dt^2} + \varepsilon \frac{d^2 u\_1}{dt^2} + \dots + u\_0 + \varepsilon u\_1 + \dots + \varepsilon u\_0^3 + \dots \sim \mathbf{0}, \\\ u\_0(\mathbf{0}) + \varepsilon u\_1(\mathbf{0}) + \dots = a + \mathbf{0} \cdot \varepsilon + \dots, \\\ \frac{d u\_0}{dt}(\mathbf{0}) + \varepsilon \frac{d u\_1}{dt}(\mathbf{0}) + \dots = \mathbf{0} + \mathbf{0} \cdot \varepsilon + \dots. \end{cases}$$

Equating like of powers of *ε* on both sides, we get

$$\begin{aligned} O(1): \quad \frac{d^2 u\_0}{dt^2} + u\_0 &= 0, \\ u\_0(0) &= a, \quad \frac{du\_0}{dt}(0) = 0, \end{aligned} \tag{30}$$

and

$$\begin{aligned} O(\epsilon): \quad \frac{d^2 u\_1}{dt^2} + u\_1 &= -u\_0^3, \\ u\_1(0) &= 0, \quad \frac{du\_1}{dt}(0) = 0. \end{aligned} \tag{31}$$

Solving Eqs. (30) and (31), we obtain

$$u \sim a \cos \left(t\right) + \varepsilon \left[\frac{a^3}{32} (\cos \left(3t\right) - \cos \left(t\right)) - \frac{3a^3}{8} t \sin \left(t\right)\right] + \cdots. \tag{32}$$

*Perturbation Expansion to the Solution of Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.94173*

The term *t*sin ð Þ*t* in the expansion (32) is called a *secular term*. It is an oscillating term of growing amplitude. All other terms are oscillating of fixed amplitude. The secular term leads to a nonuniformity for large *t*. The region of nonuniformity is obtained by equating the order of the first and second terms,

$$\cos\left(t\right) = O(\epsilon t \sin\left(t\right)), \quad \text{as } \epsilon \to \mathbf{0}.$$

The trigonometric functions are treated as *O*ð Þ1 terms. Thus, the region of nonuniformity is *t* ¼ *O*ð Þ 1*=ε* , as *ε* ! 0.

The second common source of nonuniformities is associated with the presence of singularities. Consider, the following initial-value problem:

$$\begin{cases} \varepsilon \frac{d\mathbf{y}}{d\mathbf{x}} + \mathbf{y} = e^{-\mathbf{x}}, & \mathbf{x} > \mathbf{0} \\\\ \mathbf{y}(\mathbf{0}) = \mathbf{2}, & \end{cases} \tag{33}$$

where *ε*> 0 is a small parameter. Suppose *y* has the expansion

$$
\partial\_0 \gamma \sim \mathcal{y}\_0(\varkappa) + \varepsilon \mathcal{y}\_1(\varkappa) + \varepsilon^2 \mathcal{y}\_2(\varkappa) + \cdots. \tag{34}
$$

Substituting (34) in (33), we have

$$\begin{cases} \varepsilon \left( \frac{d\boldsymbol{y}\_0}{d\boldsymbol{\omega}} + \varepsilon \frac{d\boldsymbol{y}\_1}{d\boldsymbol{\omega}} + \cdots \right) + (\boldsymbol{y}\_0 + \varepsilon \boldsymbol{y}\_1 + \cdots) = \boldsymbol{\varepsilon}^{-\boldsymbol{x}}, \\\\ \boldsymbol{y}\_0(\mathbf{0}) + \varepsilon \boldsymbol{y}\_1(\mathbf{0}) + \cdots = \boldsymbol{2}. \end{cases} \tag{35}$$

Equating coefficients of like powers of *ε* on both sides, we get

$$\begin{array}{ll} O(1): & \mathcal{Y}\_0 = e^{-\mathbf{x}}, \quad \mathcal{Y}\_0(\mathbf{0}) = \mathbf{2}, \\\\ O(\varepsilon): & \mathcal{Y}\_1 = -\frac{d\mathcal{Y}\_0}{d\mathbf{x}} = e^{-\mathbf{x}}, \quad \mathcal{Y}\_1(\mathbf{0}) = \mathbf{0}, \\\\ O\left(\varepsilon^2\right): & \mathcal{Y}\_2 = -\frac{d\mathcal{Y}\_1}{d\mathbf{x}} = e^{-\mathbf{x}}, \quad \mathcal{Y}\_2(\mathbf{0}) = \mathbf{0}. \end{array}$$

Clearly, *y*<sup>0</sup> cannot satisfy the boundary condition *y*0ð Þ¼ 0 2 as no constant of integration is available because the equation determining *y*<sup>0</sup> is an algebraic equation not a differential equation, and no additional conditions are required. Thus, we have obtain the expression

$$
\mathcal{Y} \sim e^{-\mathbf{x}} + \epsilon e^{-\mathbf{x}} + \epsilon^2 e^{-\mathbf{x}} + \cdots,\tag{36}
$$

but the initial condition *y*ð Þ¼ 0 2 has not been satisfied.

The unperturbed problem, obtained by setting *ε* ¼ 0 is not a DE, but an algebraic equation *<sup>y</sup>* <sup>¼</sup> *<sup>e</sup>*�*<sup>x</sup>*. This cannot satisfy an arbitrarily imposed condition at *<sup>x</sup>* <sup>¼</sup> 0. For any nonzero value of *ε*, (33) becomes a first-order DE which can satisfy an initial condition. This is an example of a singular perturbation problem (SPP), where the behavior of the perturbed problem is very different from that of the unperturbed problem.

Thus, the perturbation expansion (36) is a good approximation of the exact solution away from the region *x* ¼ 0. To see this, let us compare (36) with the following exact solution:

$$y\_{\varepsilon\mathbf{x}} = \frac{\mathbf{1} - \mathbf{2}\varepsilon}{\mathbf{1} - \varepsilon} e^{-\mathbf{x}/\varepsilon} + \frac{e^{-\mathbf{x}}}{\mathbf{1} - \varepsilon} = \left[ (\mathbf{1} - \varepsilon - \varepsilon^2 - \cdots) e^{-\mathbf{x}/\varepsilon} \right] + \left[ (\mathbf{1} + \varepsilon + \varepsilon^2 + \cdots) e^{-\mathbf{x}} \right]. \tag{37}$$

The perturbation expansion (36) generates the second member of (37), but not the first member. The coefficient *<sup>e</sup>*�*x=<sup>ε</sup>*Þis a rapidly varying function which takes the value of unity at *x* ¼ 0, and rapidly decays to zero for *x*> 0. Clearly, *y*<sup>0</sup> provides a good approximation away from the region *x* ¼ 0. The region near *x* ¼ 0 is called the *boundary layer*. These regions usually occur when the highest order derivative of a DE is multiplied by a small parameter. The unperturbed problem, obtained by setting *ε* ¼ 0 is of lower order and consequently cannot satisfy all the boundary conditions. This leads to boundary layer regions where the solution varies rapidly in order to satisfy the boundary condition.

Boundary layers are regions of nonuniformity in perturbation expansions of the form (36).
