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Perturbation Methods for Differential Equations

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Anharmonic oscillator: $\frac{d^2x}{dt^2} + x + \epsilon x^3 = 0$

Assume a solution of the form $x(t) = x_0(t) + \epsilon x_1(t) + \epsilon^2 x_2(t) + \cdots$ and solve for each order of $\epsilon$ .

Duffing equation: $\frac{d^2x}{dt^2} + x - \epsilon x^3 = 0$

Write $x(t) = x_0(t) + \epsilon x_1(t) + \cdots$ and find equations for $x_0(t)$ and $x_1(t)$ by equating powers of $\epsilon$ .

Perturbed harmonic oscillator: $\frac{d^2y}{dt^2} + \omega^2 y = \epsilon t y$

Express $y(t)$ as a series $y(t) = y_0(t) + \epsilon y_1(t) + \epsilon^2 y_2(t) + \cdots$ and solve for $y_0$ , $y_1$ , etc.

Van der Pol's oscillator: $\frac{d^2x}{dt^2} - \mu(1 - x^2)\frac{dx}{dt} + x = 0$

Use a small parameter $\epsilon$ to write $\mu = \epsilon$ and expand $x(t)$ as $x = x_0(t) + \epsilon x_1(t) + \cdots$ .

Perturbed linear growth: $\frac{dy}{dt} = r y + \epsilon y^2$

Expand the solution as $y(t) = y_0(t) + \epsilon y_1(t) + \epsilon^2 y_2(t) + \cdots$ and solve subsequently.

Boundary layer problem: $\epsilon \frac{d^2y}{dx^2} + (1 - x)\frac{dy}{dx} + y = 0$

Assume a solution of $y(x) = y_0(x) + \sqrt{\epsilon} y_1(x) + \epsilon y_2(x) + \cdots$ and solve for each term.

Weakly nonlinear pendulum: $\frac{d^2\theta}{dt^2} + \sin(\theta) = 0$

Introduce a small angle assumption to linearize the sine term, with $\sin(\theta) \approx \theta - \frac{\epsilon}{6}\theta^3$ , and solve using the perturbative series.

Perturbed wave equation: $\frac{d^2u}{dt^2} - c^2 \frac{d^2u}{dx^2} = -\epsilon u^2$

Develop a solution expansion as $u(x,t) = u_0(x,t) + \epsilon u_1(x,t) + \epsilon^2 u_2(x,t) + \cdots$ and solve order by order.

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