Numerical Methods in Chemical Engineering — Summer 2022

Explain following terms: 1) Significant figures, 2) Truncation Error.

Define: 1) Absolute error, 2) Relative error, 3) Percentage error, 4) Inherent Error.

Evaluate sum S = √4+√6+√8 to four significant digits and find absolute & relative errors.

Describe intermediate value properties.

Find the root of equation xlog x = 1.2 correct upto four decimal places10 using bisection method.

Enlist limitations of Newton-Raphson Method also find root of the function x4 −x = 10 upto three decimal places using Newton-Raphson method.

07 Solve following equation using Newton Raphson technique starting with x = 0.5 and y = 1.5, carry out two iterations.00 sinx −y = −0.9793 cosy−x =−0.6703

Explain Gauss elimination method with its pitfalls.

Solve the system of equation using Gauss Jordan method. 2x +y+z = 10;3x +2y+3z = 18;x +4y+9z = 16

Solve following set of equation using jacobi’s iteration method correct up to three decimal places. x = y = z = 0000 20x +y−2z = 17 3x +20y−z = −18 2x −3y+20z = 25

Give the normal equation to fit the straight line y = a+bx to n observations.

04 Find the eigen-values and eigenvectors of the matrix54 [ ]12

07 The pressure and volume of a gas are related by the equation pV𝛾 = k, γ and k being constants. Fit this equation to the following set of observations: p (kg/cm2) 0.5 1.0 1.5 2.0 2.5 3.0 V (lts) 1.62 1.00 0.75 0.62 0.52 0.461

Establish Newton’s backward interpolation formula.

04 If Pis pull required to lift a load Wby means of a pulley block, find a linear law of form P = mW+Cconnecting P & W, using following data. P 12 15 21 25 W 50 70 100 120

07 Obtain the density of a 26% solution of H PO in water at 20 ℃ during using34 Lagrange’s interpolation formula can we perform the same calculation using Newton’s forward difference interpolation formula? Yes or No? y (Density) 1.0764 1.1134 1.2160 1.3350 x % H PO 14 20 35 5034

Write an algorithm for trapezoidal rule.

Using Newton’s backward difference formula, construct an interpolating polynomial of degree 3 for the data: f(−0.75)= −0.0718125,f(−0.5)= −0.02475,f(−0.25)= 0.3349375,f(0)= 1.10100.

Evaluate ∫ 0.6 e−x2 using the trapezoidal rule and Simpson’s 1/3rd rule, taking0 h = 0.1

Discuss in brief about the boundary value problem.

Compute the value of ∫ 1.4 (sinx−logx+ex)dx using Simpson’s 3/8 rule. 0.2

Using Euler’s method, find an approximate value of y corresponding to x 1, given that dy/dx x y and y 1 when x 0.

Describe Milne’s predictor-corrector method.

Apply the Runge - Kutta fourth order method to find an approximate value of y when x 0.2 given that dy/dx x y and y 1 when x 0.

dy Solve by Taylor’s series method the equation = log(xy) for y(1.1) and dx y(1.2), given y(1) = 2.