# 2.8 Method of Weighted Residuals

## 2.8.4 Galerkin Method with New Basis

Measurable Outcome 2.12, Measurable Outcome 2.13, Measurable Outcome 2.14

Suppose we introduce another basis function \(\phi _3(x) = x^2(1-x)(1+x)\) into the example problem discussed in Section 2.8.3, which introduces an additional unknown \(a_3\). Use the Galerkin approach to solve for the new values \(a_1\), \(a_2\), and \(a_3\). (Hint: to check your solution, it might be a good idea to plot the estimate of \(\tilde{T}\) to check if it agrees well with the plot above.) Please include at least two decimal places in your answer.

\(a_1\):

Exercise 1
Numerical Response

\(a_2\):

Exercise 2
Numerical Response

\(a_3\):

Exercise 3
Numerical Response

Answer:

The three residual equations that we solve are:

\[R_1(\tilde{T}) = -8/3 a_1 - 8/15 a_3 + 200e^{-1} = 0\] | (2.195)

\[R_2(\tilde{T}) = -8/5 a_2 +100e^1 - 700e^{-1} = 0\] | (2.196)

\[R_3(\tilde{T}) = -8/15 a_1 - 88/105 a_3 -400e^{1}+ 3000e^{-1} = 0\] | (2.197)