
Concept explainers
Use the Collocation Method with
a.
b.
Plot the approximate solutions together with the exact solutions (a)
a.

To find: Approximate solution for linear boundary value problem by collocation method.
Explanation of Solution
% Program for Finite element solution of linear BVP % Inputs are interval inter, boundary values bv, number of steps n % Output: solution values c function c=bvpfem(inter,bv,n) a=inter(1);b=inter(2);ya=bv(1);yb=bv(2); h=(b-a)/(n+1); alpha=(8/3)*h+2/h; beta = (2/3)*h-1/h; e=ones(n,1); M=spdiags([beta*e alpha*e beta*e],-1:1,n,n); d=zeros(n,1); d(1)= -ya*beta; d(n)= -yb*beta; c=M\d; c= bvpfem([0 1],[0 0.33e],8);
Collocation is applied
b.

To find: Approximate solution for linear boundary value problem by collocation method.
Explanation of Solution
% Program for Finite element solution of linear BVP % Inputs are interval inter, boundary values bv, number of steps n % Output: solution values c function c=bvpfem(inter,bv,n) a=inter(1);b=inter(2);ya=bv(1);yb=bv(2); h=(b-a)/(n+1); alpha=(8/3)*h+2/h; beta = (2)*h-4/h; e=ones(n,1); M=spdiags([beta*e alpha*e beta*e],-1:1,n,n); d=zeros(n,1); d(1)= -ya*beta; d(n)= -yb*beta; c=M\d; c= bvpfem([0 1],[1 e],16);
Similar to (a). The first and last equations are c1=1and c1+…+cn=e. The remaining n-2equations are
Substituting
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