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

\begin{equation} \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) \end{equation}

(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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