Chebyshev Collocation Method for Linear and Nonlinear Boundary Value Problems

linear BVP

nonlinear BVP

interior points

16

Consider two boundary-value problems (BVP), one linear and the other nonlinear.

The linear problem is defined by

u''(y)+πu(y)=π

, with the boundary conditions

u(-1)=0

and

u(1)=1

. This problem admits an analytical solution given by

u(y)=

1

2

[2-cos(

π

y)sec(

π

)+csc(

π

)sin(

π

y)]

, displayed in blue when you click the linear BVP tab.

The nonlinear problem statement is

u''(y)-u'(y)u(y)=1

, where again,

u(-1)=0

and

u(1)=1

. This problem is solved numerically and the solution is shown in blue when you click the nonlinear BVP tab.

This Demonstration compares these two solutions to those obtained using the Chebyshev collocation method. You can set the number of interior points,

N

, and the results of the Chebyshev collocation technique are plotted as red solid squares. The agreement between the two solutions is very good if you choose a relatively large number of interior points (e.g.,

N>16

).