Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2005 • Question 5

Question

Math 6420 • PDE Fall 2005 • Question 5

Prove that the nonlinear integral equation \[ u(x)-\int _0^1 \frac {u^2(y)}{4+x^2-y}\,dy=\frac {x^3}{3},\qquad 0\le x\le 1, \] has exactly one continuous solution \(u\) satisfying \(\max _{0\le x\le 1}|u(x)|\le 1\). Describe, but do not carry out, an iterative procedure for obtaining an approximation to this solution.

Raw LaTeX

Prove that the nonlinear integral equation
\[
  u(x)-\int_0^1 \frac{u^2(y)}{4+x^2-y}\,dy=\frac{x^3}{3},\qquad 0\le x\le 1,
\]
has exactly one continuous solution \(u\) satisfying
\(\max_{0\le x\le 1}|u(x)|\le 1\). Describe, but do not carry out, an iterative procedure
for obtaining an approximation to this solution.

Topic Tags

Cauchy Integral Formula and Integral Theorems Contraction Mapping and Nonlinear Equations Distributions and Weak Solutions Linear Programming and Optimization