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Department of Mathematics

Question

Math 6420 • PDE

Fall 1995 • Question 4

Question

Math 6420 • PDE Fall 1995 • Question 4
Consider the integral equation \[ u(x)-\lambda\int_{-1}^{1}\sum_{j=0}^{2}P_j(x)P_j(t)u(t)\,dt=f(x), \qquad\text{for } -1\le x\le 1, \] where \(P_j(x)\) is the \(j\)th Legendre function. **(a)** For which values of \(\lambda\) does the integral equation have a unique solution? **(b)** Calculate this solution for the case that \(f(x)=x^2+1\).

Raw LaTeX

Consider the integral equation
\[
  u(x)-\lambda\int_{-1}^{1}\sum_{j=0}^{2}P_j(x)P_j(t)u(t)\,dt=f(x),
  \qquad\text{for } -1\le x\le 1,
\]
where \(P_j(x)\) is the \(j\)th Legendre function.
\begin{enumerate}
\item[(a)] For which values of \(\lambda\) does the integral equation have a unique solution?
\item[(b)] Calculate this solution for the case that \(f(x)=x^2+1\).
\end{enumerate}
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Distributions and Weak Solutions Linear Programming and Optimization Principal Value Integrals