# Chebyshev Collocation Method for Linear and Nonlinear Boundary Value Problems

Chebyshev Collocation Method for Linear and Nonlinear Boundary Value Problems

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

The linear problem is defined by , with the boundary conditions and . This problem admits an analytical solution given by , displayed in blue when you click the linear BVP tab.

u''(y)+πu(y)=π

u(-1)=0

u(1)=1

u(y)=[2-cos(

1

2

π

y)sec(π

)+csc(π

)sin(π

y)]The nonlinear problem statement is , where again, and . This problem is solved numerically and the solution is shown in blue when you click the nonlinear BVP tab.

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

u(-1)=0

u(1)=1

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

N>16