Define and explain Absolute Error and Relative Error.
[3 marks]Derive General Error Formula for a function1 u = f ( x , y , z ) assuming that the errors in x , y and z are x , y and z04 .
[ marks]Explain Gauss Elimination for solution of a system of linear algebraic equations.
[7 marks]Define the operators Δ, Eand µ.
[3 marks]Use iteration method to solve the equation x e x = 1 . It is given that the solution lies in the range [0 1].
[4 marks]Explain LU Decomposition method for solution of a system of linear equations.
[7 marks]Find the cubic polynomial for y(x)which takes the following values and thus find y ( 8 ) .07 x 1 3 5 y 24 120 336 720
[7 marks]Use Newton Raphson method t o solve the non-linear equation x 3 − 2 x − 5 = 0 taking x0 = 203 . Perform two iterations.
[ marks]Define Forward Difference, Backward Difference and Central Difference.
[4 marks]1 Explain Simpson’s rule and rule for numerical integration.38
[7 marks]dy Explain the method to calculate and dx d d2 x y203 using Newton’s Forward Difference formula.
[ marks]Explain Taylor’s Series method of numerical integration.
[4 marks]11 Evaluate I = dx, correct to three decimal places using (a) Trapezoidal x+10 Rule (b) Simpson’s Rule Take h = 0 .5 and h = 0 .2 507
[ marks]Construct the Forward Difference Table for the following data. x 1 2 3 4 y=f(x) 4 6 9 12 17
[5 marks]Use Gauss’s backward interpolation formula to find the sales for the year 1986 from the following data. Year 1951 1961 1971 1981 1991 2001 Sales in 13 17 22 28 41 53 Thousands
[4 marks]Apply the Euler’s method to solve the ordinary differential equation2 d d y x = x + y using increments of h=0.2. It is given that y = 107 when x=0. Carry out at least five steps.
[ marks]Explain Modified Euler’s method for solution of Ordinary Differential Equation.
[3 marks]Explain Runge-Kutta method of second order.
[4 marks]It is given that d d y x = y − x where y ( 0 ) = 2 . Use fourth order Runge-Kutta method to find y(0.1) and y ( 0 .2 )07
[ marks]Explain the concept of predictor-corrector methods for solution of ordinary differential equations.
[3 marks]Derive Adams-Bashforth formula for computation of predictor while solving an ordinary differential equation.
[4 marks]Explain finite difference method for solution of an ordinary differential method.
[7 marks]Write general form of a second order Partial Differential Equation and explain classification of the equation.
[3 marks]Discuss solution of a wave equation.
[4 marks]Discuss solution of Laplace’s Equation and Leibmann’s iterative method to improve accuracy.
[7 marks]