Numerical Methods in Chemical Engineering — Winter 2022

Describe different types of errors.

Fit the straight line that best fits to the following data: x 1 2 3 4 6 y 2.4 3 3.6 4 5

Fit a second-degree parabola to the following data: x 0 1 2 3 y 1 1.8 1.3 2.5 6.3

Prepare Forward difference table for the following data: x 0 5 10 15 20 25 y 7 11 14 18 24 32

Using Newton Raphson method, find the root of x4 −x −10 = 0 correct up to three decimal places.

Using Secant method, find the root of x3 +x2 −3x −3 = 0 correct up to five decimal places starting from x = 1 and x = 201

Find the square root of 10 correct to three decimal places, by using Newton-Raphson iteration formula.

Find the percentage error in computing the parallel resistance Rof two resistances Rand Rif R , Rare each in error by 2%.

1 2 Find A−1 if A= [2 5 3]108

Using Bisection method find the root of the equation x3 −5x +3 = 0 , correct up to two decimal places.

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

Solve the following system of equations by Gauss Elimination method: x+3y +2z = 5 , 2x+4y−6z = −4 , x+5y+3z =1

Solve the following system of equations by Gauss-Jacobi method: 6x +2y−z = 4 , x+5y+z = 3 , 2x +y+4z = 27

Using Euler’s method find y (1.2), given that dy = x √y, y(1) = 1 , dx Taking ℎ = 0.1

Apply 4th order Runge Kutta Method to compute y for x = 0.5 , given dy that = √x +y, y(0.4) = 0.41, ℎ = 0.1 dx

Use the Taylor series method to find y(0.1), given that dy = x2 +y2, dx y(0) = 1. Taking ℎ = 0.1

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

Find the isothermal work done on the gas if it is compressed from v =1 22L to v = 2L. Use Trapezoidal rule to find W = −∫ v2p dv 2 v1 V(L) 2 7 12 17 22 P(atm) 12.20 3.49 2.049 1.44 1.11

3 dx Evaluate ∫ by using Simpson’s 3/8 Rule and hence calculate log 0 1+x

Prove that (1) ∆∇= (∆−∇) (2) ∆ = E∇= ∆E

Using Newton’s divided difference formula, evaluate f(9) from the following data: x 5 7 11 13 17 f(x) 150 392 1452 2366 5202

Use Lagrange’s interpolation formula to find the value of y when x = 4, if the values of x and y are given below: x 2 3 5 y 0.1506 0.3001 0.4517 0.6259

Discuss in brief about boundary value problems.

Find Y(2.36) from the following table using Newton’s backward interpolation method. x 1.6 1.8 2 2.2 2.4 2.6 y 4.95 6.05 7.39 9.03 11.02 13.46

Use Milne’s predictor-corrector method to find y(4.4). Given that 5xy’+ y2 −2 = 0 , y(4) = 1, y(4.1) = 1.0049 , y(4.2) = 1.0097 , y(4.3) = 1.0143 , with ℎ = 0.1