Use modified Euler’s method to find the approximate solution of IVP with h = 0.1
If the exact solution is , find the error.
We will use the Modified Euler’s Method (Heun's method) to solve the initial value problem (IVP): \[ y' = 2xy, \quad y(1) = 1 \] at \( x = 1.5 \) with step size \( h = 0.1 \). The __________ ___ _________ __________ _______.
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Derive a suitable numerical differentiation formula of 0(h2) to find (2.4) with h= 0.1 given the table
| x | 0.1 | 1.2 | 2.4 | 3.9 |
| 3.41 | 2.68 | 1.37 | -1.48 |
Find the inverse of the matrix
using Gauss Jordan method.
Determine the constants in the differentiation formula
so that the method is of the highest possible order. Find the order and the error term of the method.
Show that where
and
are the average and central differences operators, respectively.
Using as an initial approximation find an approximation to one of the zeros of
by using Birge-Vieta method. Perform two iterations.
The position f (x) of a particle moving in a line at various times xk is given in the following table. Estimate the velocity and acceleration of the particle at x =1.2.
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 |
| f (x) | 2.72 | 3.32 | 4.06 | 4.96 | 6.05 | 7.39 | 9.02 |
Compute the values of
by using the trapezoidal rule with h = 0.5, 0.25, 0.125 . Improve this value by using the Romberg’s method. Compare your result with the true value.