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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 114 "numerical_sol_ie.mws   10/27/04
\nNumerically solving an integral equation, representing function by a
 cubic spline." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?spline" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1032 "restart:\nwith(CurveFitti
ng):\nN:=8;\na:=0; b:=1;\ngridpoint:=i->a+i*(b-a)/N;\nk:=(x,y)->piecew
ise(x<y,x*(1-y),y*(1-x));\ngen_xydata:=ydata->[seq([gridpoint(i),\n   \+
                      ydata[i+1]],i=0..N)];\nmake_int:=proc(U)\n   loc
al spline_interpolant;\n   spline_interpolant:=unapply(Spline(gen_xyda
ta(U),y),y);\n   [seq( evalf(Int(k(x,y)*spline_interpolant(y),y=a..b) \+
)           ,x=[seq(gridpoint(i),i=0..N)])];\nend proc;\nIE:=U->op(exp
and(make_int(U)-lambda*U-[seq(f(x),\n              x=[seq(gridpoint(i)
,i=0..N)])]));\nlambda:=0.25;\nf:=x->x;\n# FSOLVE is going to call cal
lback with a SEQUENCE of args, not a LIST.\n# So use this wrapper to c
onvert sequence to list:\nIE_wrapper:=proc()\n   IE([args]);\nend proc
;\nUs:=seq(U[i],i=0..N);\nfsolve(\{IE_wrapper(Us)\},\{Us\});\nassign(%
);\nUs;\nspline_interpolant:=V->unapply(Spline(V,y),y);\nxy:=gen_xydat
a([Us]);\nnumsol:=spline_interpolant(xy);\n#residual:=int(k(x,y)*numso
l(y),y=0..1)-lambda*numsol(y)-f(x);\n#plot(residual,x=0..1);\nnumplot:
=plot(numsol(x),x=0..1,color=blue):\nplots[display](numplot);\n\n" }}
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