# 2.8 Method of Weighted Residuals

## 2.8.4 Galerkin Method with New Basis

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

$$a_2$$:

Exercise 2

$$a_3$$:

Exercise 3

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