Question

Obtain an approximate solution of the integral equation

y(x)=x²+∫₀¹(x+ξ)y(ξ)dξ
using the method of collocation at the points x=0,1/2, and 1 .
Assume a three term polynomial approximation of the form
y(x)=c₁+c₂x+c₃x².
(Note: Use fractions instead of decimals).

Answer 1

To solve the integral equation using collocation at points x=0, 1/2, and 1 with a polynomial approximation y(x)=c1+c2x+c3x^2, one must set up and solve a system of equations resulting from substituting the approximation into the integral and evaluating at each collocation point.

Step-by-step explanation:

The student is asking to solve an integral equation using the method of collocation at the points x=0, 1/2, and 1 with a three term polynomial approximation of the form y(x)=c1+c2x+c3x2. To apply the method of collocation, we substitute our approximation into the integral equation and set up a system of equations by evaluating at the collocation points.

At x=0:

• y(0) = c1 = 0 + ∫01 ξ(c1+c2ξ+c3ξ2)dξ

At x=1/2:

• y(1/2) = c1 + (1/2)c2 + (1/4)c3 = (1/4) + ∫01 (1/2+ξ)(c1+c2ξ+c3ξ2)dξ

At x=1:

• y(1) = c1 + c2 + c3 = 1 + ∫01 (1+ξ)(c1+c2ξ+c3ξ2)dξ

Solving the resulting system of linear equations will yield the values of c1, c2, and c3. These equations require evaluating the integrals, which involves basic integration techniques, and solving for the constants which is straightforward linear algebra.