PHYS 598 hw11 - Here are some optional problems on integral...

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Physics 498/MMA Handout 10 Oct 10th 2002 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 305 Loomis Laboratory University of Illinois Here are some optional problems on integral equations. They are taken verbatim from Paul Goldbart’s homework sets. 1) Integral equations : a) Solve the inhomogeneous type II Fredholm integral equation u ( x ) = e x + λ Z 1 0 xy u ( y ) dy . b) Solve the homogeneous type II Fredholm integral equation u ( x ) = λ Z π 0 sin( x - y ) u ( y ) dy . c) Solve the inhomogeneous type II Fredholm integral equation u ( x ) = x + λ Z 1 0 y ( x + y ) u ( y ) dy to second order in λ using i) the Liouville-Neumann-Born series; and ii) the Fredholm series. d) By differentiating, solve the integral equation: u ( x ) = x + R x 0 u ( y ) dy . e) Solve the integral equation: u ( x ) = x 2 + R 1 0 xy u ( y ) dy . f) Find the eigenfunction(s) and eigenvalue(s) of the integral equation u ( x ) = λ Z 1 0 e x - y u ( y ) dy . g) Solve the integral equation: u ( x ) = e x + λ R 1 0 e x - y u ( y ) dy . 1
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2) Neumann Series : Consider the integral equation u ( x ) = g ( x ) + λ Z 1 0 K ( x, y ) u ( y ) dy , in which only u is considered unknown. a) Write down the solution u ( x ) to second order in the Liouville-Neumann-Born series. b) Suppose g ( x ) =
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