Numerical Methods in Chemical Engineering — Summer 2023

Explain three sources of arising errors in numerical computation.

Re-arrange the given equations in diagonally dominant form and solve the new linear system by using Gauss Seidel Method with X = [0 0 0].0 Perform only three iterations. Calculate 𝜀 = max[𝜀 ,𝜀 ,𝜀 ] only in a a,x a,y a,z the last iteration. 2x +15y −3z = 16, 2x−3y+25z = 23, 12x +2y+z = 27.

Fit a second degree polynomial using least square method to the following data x 0 1 2 3 y 1 1.8 1.3 2.5 6.3

In calculating the area of a rectangle, an error of 3% is made in measuring each of its sides. Find the percentage error in calculating area of the rectangle.

Find a root of the function f(x) = cosx−xex using Bisection Method. Perform only four iterations.

Find the root of the equation x3 −2x −5 = 0 using Secant method correct up to three decimal places.

Find a positive root of x3 −4x +1 = 0 by the method of false position correct upto three decimal places.

Explain the Gauss Jordan method to solve the system of linear equations.

Fit a curve of the form y = a ebx to the following data: x 1 3 5 7 9 y 115 105 95 85 80

Using Newton- Raphson iterative method, find the real root of x log x = 1.2 correct to five decimal places.10

Determine the largest eigenvalue and the corresponding eigenvector of the matrix A = [ ].12

Fit a curve of the form y = axb to the following data: x 20 16 10 11 y 22 41 120 89 56

Examine the system of equations 3x+3y +2z = 1, x +2y = 4, 10y +3z = – 2, 2x – 3y – z = 5 for consistency and then solve it by Gauss Elimination method.

Construct the divided difference table with the arguments 2, 4, 9, 10 of the function f(x) = x3 −2x.

Using Newton’s forward interpolation formula, find the value of f(1.6). x 1 1.4 1.8 2.2 f(x) 3.49 4.82 5.96 6.5

Find the polynomial f (x) by using Lagrange’s interpolation formula and hence find f(3) for the below data: x 0 1 2 f(x) 2 3 12 147

Derive formula for Trapezoidal Rule of numerical integration.

The residents of a town are given below. Estimate the residents for the year 1830 using Newton’s backward interpolation. Year- x: 1791 1801 1811 1821 1831 residents –y: (in thousand)

Using Modified Euler’s method, find an approximate value of y when x = dy 0.6 with ℎ = 0.1 given that = x+3y, subject to y(0) = 1. dx

dy Find the approximate solution of = x +y, y(0) = 0 with ℎ = 2 using dx Euler’s Method in five steps.

3 1 Evaluate ∫ dx with n = 6 by using Simpson’s 3/8 rule. 0 1+x

Using Milne’s Predictor-Corrector Methods, find y(4.4) given that 5xy′ +y2 −2 = 0 with y(4) = 1,y(4.1) = 1.0049,y(4.2) = 1.0097, y(4.3) = 1.0143.2

Derive formula for Simpson’s 1/3 Rule of numerical integration.

Use second order Runge Kutta method to compute y(0.2) given that dy = x +√y, y(0) = 4 by taking h=0.1. dx

Use the Taylor series method to find y(0.2), given that dy = 2y+3ex, dx y(0)=1. Taking h=0.1.