How can we calculate derivatives of continuous functions? Write down algorithm and program for differentiating continuous function using two point forward difference formula. [5]
2.
Find following integral using composite trapezoidal rule using 2 segments (k=2) and 4 segments (k=4). $\int_{2}^{8} (x+3)^2 dx$ [5]
Solution of Nonlinear Equations
1.
How Secant methods differs from Newton Raphson method? Derive the formula for Secant Method. Solve the equation $cosx+2sinx-x^2=0$ using Secant method. Assume error precision is 0.01. [10]
2.
Define the terms true error and relative error? Write down algorithm for Horner' method to evaluate polynomial and use the method to evaluate the polynomial $2x^3-3x^2+5x-2$ at x=3. [5]
Solution of Ordinary Differential Equations
1.
Approximate the solution of y'=2x+y, y(0)=1 using Euler's method with step size of 0.1. Approximate the value of y(0.4). [5]
2.
How boundary value problems differs from initial value problems? Discuss shooting method for solving boundary value problem. [5]
Solution of Partial Differential Equations
1.
Solve the Poisson's equation $\nabla^2 f = xy$ with $f=2$ on boundary by assuming square domain $0 \leq x \leq 3$, $0 \leq y \leq 3$ and h=1. [5]
Solving System of Linear Equations
1.
Explain the working of Jacobi Iteration method? Solve the following system of equations using the method. Assume error precision is 0.01. Compare Jacobi Iteration method with Gauss-Seidel method. $5x-2y+3z=-1$, $-3x+9y+z=2$, $2x-y-7z=-3$ [10]
2.
Derive formula for the Doolittle LU decomposition matrix factorization method. [5]