Hello Everybody:
I am learning about NDSolve and how to apply it to "stiff" boundary value problems. The following example appears to show that NDSolve of Mathematica 10 "struggles". Could you please let me know how can I make NDSolve work? Thanks for any help!!!
\[Lambda] = .62;
yn1 = NDSolve[{1*\[Lambda]*y''[x] - y[x] == 0., y[0] == 1, y[1] == 0},
y[x], {x, 0, 1}];
yn2 = NDSolve[{.1*\[Lambda]*y''[x] - y[x] == 0., y[0] == 1,
y[1] == 0}, y[x], {x, 0, 1}];
yn3 = NDSolve[{.01*\[Lambda]*y''[x] - y[x] == 0., y[0] == 1,
y[1] == 0}, y[x], {x, 0, 1}];
yn4 = NDSolve[{.001*\[Lambda]*y''[x] - y[x] == 0., y[0] == 1,
y[1] == 0}, y[x], {x, 0, 1}];
Plot[{Evaluate[y[x] /. yn1], Evaluate[y[x] /. yn2],
Evaluate[y[x] /. yn3], Evaluate[y[x] /. yn4]}, {x, 0, 1},
PlotRange -> {{0, 1}, {-.2, 1}},
PlotStyle -> {Red, Blue, Green, Black}, AxesOrigin -> {0, 1}
]
NDSolve::bvluc: The equations derived from the boundary conditions are numerically ill-conditioned. The boundary conditions may not be sufficient to uniquely define a solution. If a solution is computed, it may match the boundary conditions poorly. >>
NDSolve::berr: The scaled boundary value residual error of 1.0003677141389873` indicates that the boundary values are not satisfied to specified tolerances. Returning the best solution found. >>