Numerical Method 2067

Attempt all questions:

1. Discuss methods of Half Interval and Newton’s for solving the nonlinear equation f(x) = 0. Illustrate the methods by figure and compare them stating their advantages and disadvantages. (8)

2. Derive the equation for Lagrange’s interpolating polynomial and find the value of f(x) at x = 1 for the following: (4+4)

3. Write Newton-cotes integration formulas in basic form for x = 1, 2, 3 and give their composite rules. Evaluate using the Gaussian integration three point formula. (4+4)

4. Solve the following algebraic system of linear equations by Gauss-Jordan algorithm. (8)

5. Write an algorithm and program to solve system of linear equations using Gauss-Seidel iterative method. (4+8)

6. Explain the Picard’s proves of successive approximation. Obtain a solution upto the fifth approximation of the equation such that y = 1 when x = 0 using Picard’s process of successive approximations . (2+6)

7. Define a difference equation to represent a Laplace’s equation. Solve the following Laplace equation  

For the rectangular plate given as:(3 + 5)

OR

Derive a difference equation to represent a Poison’s equation. Solve the Poison’s equation ∇² f = 2x²y² Over the domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1. (3+5)