Comparison of Euler and backward Eulers methods with exact solution

Comparison of Euler and backward Euler s methods with exact solution 

One other way to solve stiff differential equations is to use orthogonal 

We expect that for large values of nn nn — oo), the solution in Equation 2.23 will approach the exact solution. The 9i x) denote the polynomial basis functions of x and the coefficients ai are obtained using the method of weighted residuals (MWR), which involves substituting the approximate solution given by Equation 2.23 in Equation 2.22 to obtain a residual function given below. 

The coefficients a are determined by minimizing the residual function R x, a) over the desired range of the independent variable x (for example, over 0 x 1) along with a choice of the weighting function, w(x). The residual minimization function may be written as follows  

Equation 2.25 describes the generalized form of all the methods of weighted residuals. The specific choice of the weighting function depends on the method used. The following list provides a few of these methods. 

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