# Techniques of applied maths: Eigenfunction expansions and integral equations

Question from Week 2 revisions (added May 3, 2013): What is the correct expansion for the solution of an inhomogeneous denegrate Fredholm integral equation? See the small note within the solution to Q2d.

## 1. Application of the Fredholm Alternative Theorem

Use the Fredholm Alternative Theorem to discuss existence of solutions to the following BVPs, depending on the values of (A, B).

### (a) BVP 1

\begin{gather*} y'' + y = A\sin x + B\cos x + \sin(x + \pi/3) + \sin^3 x \\ y(0) = y(2\pi) \\ y'(0) = y'(2\pi) \end{gather*}

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### (b) BVP 2

\begin{gather*} y'' + 2y' + y = 1 \\ y'(0) + y(0) = A \\ y'(1) + y(1) = 3. \end{gather*}

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## 2. Degenerate Fredholm Integral Equations

• Determine all eigenfunctions and eigenvalues of finite multiplicity. State the eigenvalue with infinite multiplicity $(\lambda_\infty)$?
• Solve the inhomogeneous equation. Are solutions unique? If not, then find the simplest homogeneous solution that may be added to the particular solution.

### (a) FIE of first kind

$\mathcal{L}y \equiv \frac{1}{2}\int_{-1}^1 (x + t)y(t)dt = 6x + 7 = f(x).$

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### (b) FIE of first kind

$\mathcal{L}y \equiv \int_{-1}^1 (\tfrac{1}{5} + xt)y(t)dt = \sin(\pi x) = f(x).$

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### (c) FIE of second kind

$\mathcal{L}y = y + \int_0^1 (\tfrac{1}{5} + xt) y(t) \, dt = 31\sin(\pi x) \equiv f(x)$

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### (d) FIE of second kind

$$\label{fie4a} \mathcal{L}y = \frac{6}{5\pi} y + \int_0^1 \sin(2\pi x - 3\pi t) y(t) \, \de{t} = f(x)$$

(i) $f(x) = 2\sin(2\pi x - \pi/6)$

(ii) $f(x) = \sin(2\pi x)$

(iii) General $f$

Because part d) is quite long, we split it into two parts. This first part finds the eigenvalues and eigenfunctions.

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This second part addresses the uniqueness of solutions to the inhomogeneous problem.

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