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What are the different types of errors? Write an algorithm and a C-Program to obtain the roots of non-linear equation using Newton Raphson method.
Define ordinary differential equation. Why numerical differential equation is required? Derive Newton forward difference formula with suitable diagram.
Solve the following ordinary differential equation using shooting method.
y" + xy' − xy = 2x ,with boundary conditions y(0) = 1; and y(2) = 9
Attempt any Eight questions
[8x5=40]How would you choose two initial values that are required for Bisection method? Use Bisection method to estimate the root of the equation log x −cos x =0.
Solve the following equations using Gauss Elimination Method with partial pivoting.
x + 2y + 3z = 5
2x + 8y + 22z = 6
3x + 22y +82z = -10
Fit a second order polynomial to the data in the table below:
| X | 1 | 2 | 3 | 4 | 5 |
| F(x) | 2 | 6 | 12 | 20 | 30 |
Estimate f(3) from the following data using cubic spline interpolation.
| X | 1 | 2.5 | 4 | 5.7 |
| F(x) | -2.0 | 4.2 | 14.4 | 31.2 |
Use Gauss Legendre three-point formula to evaluate the integral: I = ∫₂⁴(x⁴ + 4)dx.
Solve the following differential equation dy/dx = 3x + y/2 with y(0) = 1 for x = 0.2 (h = 0.1) using Euler's Method.
From the following table find the value of X, correct to 3-decimal places for Which Y is minimum and find this value of Y.
| X | 0.60 | 0.65 | 0.70 | 0.75 |
| Y | 0.6221 | 0.6155 | 0.6138 | 0.6170 |
Find the Eigen values and Eigen vectors of the Matrix:
A = ⎛48 8⎞
⎝6 26⎠
The steady- state two-dimensional heat flow in a metal plate is defined by:
∂²T/∂x² + ∂²T/∂y² = 0
A steel plate of size 30 × 30cm is given. Two adjacent sides are placed at 100° and other side held at 0°. Find the temperature at interior points, assuming the grid size of 10 × 10cm.