1st Kind Difficulty 1-D Reminder Denote the integral operator as K Ko= lc-a'lo(a')dS= Ko=y The integral operator is singular: K has a null space 斗a0(x)=0,x≠0,a0(0) Koo ax-c'o(ax)ds′=0 fKo=业theK(+ao)=业 SMA+HPC⊙2003M First and Second Kind 10
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1st Kind Difficulty 1-D Reminder Numerical results n=10 来 n=40 20 Eigenvalues accumulating at zero SMA+HPC⊙2003M First and Second Kind 11
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1st Kind Difficulty 1-D Reminder Eigenvalues As the discretization is refined, oo(a) becomes better approximated xo M x2 十H十十十十H十十HH x=-1x As the discretization is refined, Ks null space can be more accurately represented SMA+HPC⊙2003M First and Second Kind 12
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2nd Kind 1-D Reminder Second Kind equation (I+K)o=0(x)+/1x-a1(x)d.y→(I+K)o=业 (I+K)(00+a)≠(I+K)a o0(x)=0,x≠0,o(0)=1 (+K)| 十 ≠ SMA+HPC⊙2003M First and Second Kind 13
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