Write an algorithm and program to compute the interpolation using Lagrange Interpolation. [10]
2.
The temperature of a metal strip was measured at various time intervals during heating and the values are given in the table below. If the relation between the time 't' and temperature 'T' is of the form: $T = be^{t/4} + a$ . Estimate the temperature at t = 6 minute.
The table below gives the values of distance travelled by a car at various time intervals during the initial running. Estimate the velocity and acceleration at time t = 7sec.
Solve the following integral using trapezoidal rule form = 8, $l = \int_{2}^{4} (x^4 + 1)dx$. [5]
Solution of Nonlinear Equations
1.
Derive the formula for integration using simpsons 3/8 rule. Use Secant Method to estimate the root of equation with initial estimate x₁ = 4 and x₂ = 2, $x^2 - 4x - 10 = 0$. [10]
2.
Show that the rate of convergence of Newtons Raphson method is quadratic. [5]
Solution of Ordinary Differential Equations
1.
What do you mean by boundary value problem? Use shootuing method, solve the equation: y'' = 6x², with y(0) = 1 and y(1) = 2 in the interval (0, 1) for y(0.5) taking h = 0.5 [10]
2.
Given the equation $y' = 3x^2 + 1$ with y(1) = 2, estimate y(2) by Euler's Method using h = 0.2. [5]
Solution of Partial Differential Equations
1.
Solve the Poisson's Equation $\nabla^2 f = 2x^2 y^2$ over the square domain $0 \leq x \leq 3$ and $0 \leq y \leq 3$ with $f = 0$ on the boundary and $h = 1$. [5]
Solving System of Linear Equations
1.
Solve the following system of linear equation by Gauss Elimination with Pivoting $2x + 2y + z = 6$, $4x + 2y + 3z = 4$,$x - y + 1 = 0$. [5]
2.
Determine the Eigen Values and corresponding Eigen Vectors for the matrix.