The Fredholm equation is solved by the Nyström method using the two-point…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

The Fredholm equation u(x)=1+∫01x t u(t) dtu(x)=1+\displaystyle\int_0^1 x\,t\,u(t)\,\mathrm{d}t is solved by the Nyström method using the two-point Gauss-Legendre rule on [0,1][0,1], whose nodes are t1=12−123=0.2113t_1=\tfrac12-\tfrac{1}{2\sqrt3}=0.2113 and t2=12+123=0.7887t_2=\tfrac12+\tfrac{1}{2\sqrt3}=0.7887, each carrying weight 12\tfrac12. ((a)) Write the 2×22\times2 linear system for u1=u(t1)u_1=u(t_1) and u2=u(t2)u_2=u(t_2), and solve it. ((b)) Use the Nyström interpolation formula to evaluate u(1)u(1). ((c)) Check the result against the exact solution, found by substituting u(x)=1+Cxu(x)=1+Cx into the equation and solving for CC.

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