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numerical analysis

Questions and Answers of Numerical Analysis

The following chemical reactions take place in a closed system 2A + B ( C, A + D ( C. At equilibrium. They can be characterized by Where the nomenclature ci represents the concentration of
The Redlich-Kwong equation of state is given by Where R = the universal gas constant [= 0.518 kj/(kg K)], T = absolute temperature (K), p = pressure (kPa), and ?? = the volume of a kg of gas
The volume V of liquid in a hollow horizontal cylinder of radius r and length L is related to the liquid h by Determine h given r = 2m, L = 5m, and V = 8.5 m3. Note that if you are using a
The volume V of liquid in a spherical tank of radius r is related to the depth h of the liquid by V = πh2(3r – h)/3Determine h given r = 1 m and V = 0.75 m3.
For the spherical tank in Prob. 8.9, it is possible to develop the following two fixed-point formulas:If r = l m and V = 0.75 m3, determine whether either of these is stable, and the range of initial
The Ergun equation, shown below, is used to describe the flow of a fluid through a packed bed. ? P is the pressure drop, p is the density of the fluid, Go is the mass velocity (mass flow rate divided
The pressure drop in a section of pipe can be calculated as ?? = fL?V2/2D Where ? p = the pressure drop (Pa), ? = the friction factor, L = the length of pipe [m], p = density (kg/m3), V = velocity
The pH of water has great significance to environmental and chemical engineers. It can be related to processes ranging from pipe corrosion to acid rain. The pH is related to the hydrogen ion
The operation of a constant density plug flow reactor for the production of a substance via an enzymatic reaction is described by the equation below, where V is the volume of reactor, F is the flow
The displacement of a structure is defined by the following equation for a damped oscillation:y = 9 e –kt cost wtWhere k = 0.7 and ω = 4.(a) Use the graphical method to make an initial estimate of
In structural engineering, the secant formula defines the force per unit area, P/A, that causes a maximum stress σm in a column of given slenderness ratio L/k:P/A = σm/1 + (ec/k2) sec
A catenary cable is one that is hung between two points not in the same vertical line. As depicted in Figure a, it is subject to no loads other than its own weight. Thus, its weight (N/m) acts as a
Figure a, shows a uniform beam subject to a linearly increasing distributed load. The equation for the resulting elastic curve is (see Figure b) Use bisection to determine the point of maximum
In environmental engineering (a specialty area in civil engineering), the following equation can be used to compute the oxygen level c (mg/L) in a river downstream from a sewage discharge:c = 10 –
The concentration of pollutant bacteria c in a lake decreases according to c = 75e–1.5t + 20 e– 0.075tDetermine the time required for the bacteria concentration to be reduced to 15 using(a) The
In ocean engineering, the equation for a reflected standing wave in a harbor is given by ? = 16, t = 12, ? = 48: Solve for that lowest positive value of x if h = 0.5h0.
You buy a $25,000 piece of equipment for nothing down and $5,500 per year for 6 years. What interest rate are you paying? The formula relating present worth P, annual payments A, number of years n,
Many fields of engineering require accurate population estimated. For example, transportation engineers might find it necessary to determine separately the population growth trends of a city and
A simply supported beam is loaded as shown in Figure. Using singularity functions, the shear along the beam can be expressed by the equation: V (x) = 20[(x ? 0)1 ? (x ? 5)1] ? 15 (x ? 8)0 ? 57 By
Using the simply supported beam from Prob. 8.24, the moment along the beam, M(x), is given by:M(x) = – 10 [(x – 0)2 – (x – 5)2] + 15 (x – 8)1 + 150 (x – 7)0 + 57xUse a numerical method to
Using the simply supported beam from Prob. 8.24, the slope along the beam is given by: Use a numerical method to find the point(s) where the slope equalszero.
Using the simply supported beam from Prob. 8.24, the displacement along the beam is given by:(a) Find the point(s) where the displacement equals zero.(b) How would you use a root location technique
Perform the same computation as in Sec. 8.3, but determine the value of C required for the circuit to dissipate to 1% of its original value in t = 0.05 s, given R = 280 Ω, and L = 7.5 H. Use(a) A
An oscillating current in a electric circuit is described by i = 9e–1 cos(2πt), where t is in seconds. Determine all values of t such that i = 3.
The resistivity ? of doped silicon is based on the charge q on an electron, the electron density n, and the electron mobility ?. The electron density is given in terms of the doping density N and the
A total charge Q is uniformly distributed around a ring-shaped conductor with radius ?. A charge q is located at a distance x from the center of the ring (Figure). The force exerted on the charge by
Figure shows a circuit with a resistor, an inductor, and a capacitor in parallel. Kirchhoff?s rules can be used to express the impedance of the system as Where Z = impedance (?) and ? = the angular
For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor ƒ. The Fanning friction factor is dependent on a number of parameter related to the size of the
Real mechanical system may involve the deflection of nonlinear springs. In Figure P8.34, a mass m is released a distance h above a nonlinear spring. The resistance force F of the spring is given by F
Mechanical engineers, as well as most other engineers, use thermodynamics extensively in their work. The following polynomial can be used to relate the zero-pressure specific heat of dry air, cp
Aerospace engineers sometimes compute the trajectories of projectiles like rockets. A related problem deals with the trajectory of a thrown ball. The trajectory of a ball is defined by the (x, y)
The upward velocity of a rocket can be computed by the following formula:v = v in m0/m0 – qt – gtWhere υ = upward velocity, u = the velocity at which fuel is expelled relative to the
In Sec. 8.4, the phase angle Ф between the forced vibration caused by the rough road and the motion of the car is given bytan Φ = 2(c/cc) (w/p)/1 – (w/p)2As a mechanical engineer, you would like
Two fluids at different temperatures enter a mixer and come out at the same temperature. The heat capacity of fluid A is given by:?cp = 3.381 + 1.804 x 10?1 T ? 4.300 x 10?6?T2 And the heat capacity
A compressor is operating at compression ratio Rc of 3.0 (the pressure of the gas at the outlet is three times greater than the pressure of the gas at the inlet). The power requirements of the
In the thermos shown in Figure, the innermost compartment is separated from the middle container by a vacuum. There is a final shell around the thermos. This final shell is separated from the middle
The general from for a three-dimensional stress field is given by Where the diagonal terms represent tensile or compressive stresses and the off-diagonal terms represent shear stresses. A stress
Figure shows three reservoirs connected by circular pipes. The pipes, which are made of asphalt-dipped cast iron (? = 0.0012 m), have the following characteristics: If the water surface elevations
A find is pumped into the network of pipes shown in Figure. At steady state, the following flow balances must hold, Q1 = Q2 + Q3 Q3 = Q4 + Q5 Q5 = Q6 + Q7 Where Qi = flow in pipe i [m3/s]. In
Repeat Prob. 8.44, but incorporate the fact that the friction factor can be computed with the von karman equation, 1/√f = 4 log 10 (Re√f) – 0.4. Where Re = the Reynolds number Re = ρVD/µ.
The space shuttle, at lift-off from the launch pad, has four forces acting on it, which are shown on the free-body diagram (Figure). The combined weight of the two solid rocket boosters and external
(a) Write the following set of equation in matrix from:50 = 5x3 + 2x210 – x1 = x33x2 + 8x1 = 20(b) Write the transpose of the matrix of coefficients.
A number of matrices are defined as Answer the following questions regarding these matrices;(a) What are the dimensions of the matrices?(b) Identify the square, column, and row matrices.(c) What
Three matrices are defined as (a) Perform all possible multiplications that can be computed between pairs of these matrices.(b) Use the method in Box PT3.2 to justify why the remaining pairs cannot
Use the graphical method to solve4x1 – 8x2 = – 24x1 + 6x2 = 34Check your results by substituting them back into the equation.
Given the system of equation–1.1 x1 + 10x2 = 120–2x1 + 17.4x2 = 174(a) Solve graphically and check your results by substituting them back into the equation.(b) On the basis of the graphical
For the set of equation2x2 + 5x3 = 92x1 + x2 + x3 = 93x1 + x2 = 10(a) Compute the determinant.(b) Use Cramer’s rule to solve for the x’s.(c) Substitute your results back into the original
Given the equation0.5x1 – x2 = – 9.51.02x1 – 2x2 = – 18.8(a) Solve graphically.(b) Compute the determinant.(c) On the basis of (a) and (b), what would you expect regarding the system’s
Given the equations10x1 + 2x2 – x3 = 27–3x1 – 6x2 + 2x3 = – 61.5x1 + x2 + 5x3 = – 21.5(a) Solve by naïve Guess elimination Show all steps of the computation.(b) Substitute your results
Use Guess elimination to solve:8x1 + 2x2 – 2x3 = – 210x1 + 2x2 + 4x3 = 412x1 + 2x2 + 2x3 = 6Employ partial pivoting and check your answers by substitute them into the original equations.
Given the system of equations –3x2 + 7x3 = 2 x1 + 2x2 – x3 = 3 5x1 – 2x2 = 2(a) Compute the determinant.(b) Use Cramer’s rule to solve for the x’s.(c) Use Guess elimination with
Given the equations2x1 – 6x2 – x3 = – 38– 3x1 – x2 + 7x3 = – 34– 8x1 + x2 – 2x3 = – 20(a) Solve by Gauss elimination with partial pivoting. Show all steps of the computation.(b)
Use Gauss-Jordan elimination to solve:2x1 + x2 – x3 = 15x1 + 2x2 + 2x3 = – 43x1 + xc2 + x3 = 5Do not employ pivoting. Check your answer by substitute them into the original equation.
Solve:x1 + x2 – x3 = – 36x1 + 2x2 + 2xc3 = 2– 3x1 + 4x2 + x3 = 1With (a) naïve Gauss elimination,(b) Gauss elimination with partial pivoting, and(c) Gauss-Jordan without partial pivoting.
Perform the same computation as in Example 9.11, but use five parachutists with the following characteristics: The parachutists have a velocity of 9m/s.
Solve
Develop, debug, and test a program in either a high-level language or macro language of your choice to multiply two matrices that is, [X] = [Y] [Z], where [Y] is m by n and [Z] is n by p. Test the
Develop, debug, and test a program in either a high-level language or macro language of your choice to generate the transpose of a matrix. Test it on the matrices from Prob. 9.3.
Develop, debug, and test a program in either a high-level language or macro language of your choice to solve a system of equation with Gauss elimination with partial pivoting. Base the program on the
Use the rules of matrix multiplication to prove that the Eqs. (10.7) and (10.8) follow from Eq. (10.6).
(a)Use naïve Gauss elimination to decompose the following system according to the description in Sec. 10.2.10x1 + 2x2 – x3 = 27–3x1 – 6x2 + 2x3 = –21.5x1 + x2 + 5x3 = – 21.5Then, multiply
(a) Solve the following system of equation by LU decomposition without pivoting8x1 + 4x2 – x3 = 11–2x1 + 5x2 + x3 = 42x1 – x2 + 6x3 = 7(b) Determine the matrix inverse. Check your results by
Solve the following system of equation using LU decomposition with partial pivoting:2x1 – 6x2 – x3 = – 38–3x1 – x2 + 7x3 = – 34–8x1 + x2 – 2xc3 = – 20
Determine the total flops as a function of the number of equation n for the(a) Decomposition,(b) Forward-substitution, and(c) Back- substitution phases of the LU decomposition version of Gauss
Use LU decomposition to determine the matrix inverse for the following system. Do not use a pivoting strategy, and check your results by verifying that [A] [A]-1 = [I].10x1 + 2x2 – x3 = 27–3x1
Perform Crout decomposition on2x1 – 6x2 + x3 = 12–x1 + 7x2 – x3 = – 8x1 – 3x2 + 2x3 = 16Then, multiply the resulting [L] and [U] matrices to determine that [A] is produced.
The following system of equations is designed to determine concentration (the c’s in g/m3) in a series of coupled reactors as a function of the amount of mass input to each reactor (the right-hand
Determine ||A||e, ||A||1, and ||A||? for Scale the matrix by making the maximum elements in each row equal toone.
Determine the Euclidean and the row-sum norms for the systems in Probs. 10.3 and 10.4. Scale the matrices by making the maximum element in each row equal to one.
A matrix [A] is defined as Using the row-sum norm, compute the condition number and how many suspect digits would be generated by thematrix.
(a) Determine the condition number for the following system using the row-sum norm. Do not normalize the system. How many digits of precision will be lost due to ill-conditioning?(b) Repeat (a),
Determine the condition number based on the row-sum norm for the normalized 5 x 5 Hilbert matrix. How many significant digits of precision will be lost due to ill-conditioning?
Besides the Hilbert matrix, there are other matrices that are inherently ill-conditioned. One such case is the vandermonde matrix, which has the following form: (a) Determine the condition number
Develop a user-friendly program for LU decomposition based on the pseudocode from Figure.
Develop a user-friendly program for LU decomposition, including the capability to evaluate the matrix inverse. Base the program on Figs. 10.2 and 10.5.
Use inerative refinement techniques to improve x1 = 2, x2 = –3, and x3 = 8, which are approximate solution of2x1 + 5x2 + x3 = – 56x1 + 2x2 + x3 = 12x1 + 2x2 + x3 = 3
Consider vectors:A = 2i – 3j + akB = bi + j – 4kC = 3i + cj + 2kVector A is perpendicular to B as well as to C. It is also known that B.C = 2. Use any method studies in this chapter to solve for
Consider the following vectors:A = ai + bj + ckB = – 2i + j – 4kC = I + 3j + 2kWhere A is an unknown vector. If(A x B) + (A x C) = (5a + b)i + (3b - 2)cj + 9(-4c + 1)kUse any method learned in
Let the function be defined on the interval [0, 2] as follows: Determine the constants a, b, c, and d so that function ? satisfies the following: (i) ? (0) = ? (2) = 1. (ii) ??is continuous on the
(a) Create a 3 x 3 Hilbert matrix. This will be your matrix [A]. Multiply the matrix by the column vector {x} = [1, 1, 1]T. The solution of [A] {x} will be another column vector {b}. Using any
Perform the same calculations as in Example 11.1, but for the tridiagonalsystem,
Determine the matrix inverse for Example 11.1 based on the LU decomposition and unit vectors.
The following tridiagonal system must be solved a part of a larger algorithm (Crank-Nicolson) for solving partial differential equations: Use the Thomas algorithm to obtain asolution.
Confirm the validity of the Cholesky decomposition of Example 11.2 by substituting the results into Eq. (11.2) to see if the product of [L] and T yields [A].
Perform a Cholesky decomposition of the following symmetric system byhand,
Perform the same calculations as in Example 11.2, but for the symmetric system, In addition to solving for the Cholesky decomposition, employ it to solve for the a?s
(a) Use the Gauss-Seidel method to solve the tridiagonal system from Prob. 11.1 (εS = 5%).(b) Repeat (a) but use over relaxation with λ = 1.2.
Recall from Prob. 10.8, that the following system of equation is designed to determine concentrations (the c’s in g/m3) in a series of coupled reactors as a function of amount of mass input to each
Repeat Prob. 11.8, but use Jacobi iteration.
Use the Gauss-Seidel method to solve the following system until the percent relative error falls below εS = 5%,10x1 + 2x2 – x3 = 27–3x1 – 6x2 + 2x3 = – 61.5x1 + x2 + 5x3 = – 21.5
Use the Gauss-Seidel method(a) Without relaxation and(b) With relaxation (λ = 0.95) to solve the following system to a tolerance of εS = 5%. If necessary, rearrange the equations to achieve
Use the Gauss-Seidel method(a) Without relaxation and(b) With relaxation (λ = 1.2) to solve the following system to a tolerance of εS = 5%. If necessary, rearrange the equations to achieve
Redraw figure for the case where the slopes the equations are 1 and -1. What is the result of applying Gauss-Seidel to such a system?
Of the following three sets of linear equations, identify the set(s) that you could not solve using an iterations that is necessary that your solution does not converge. Clearly state your
Use the software library or package of your choice to obtain a solution, calculate the inverse, and the condition number (without scaling) based on the row-sum norm for (a) (b) In both cases, the
Given the pair of nonlinear simultaneous equation:f(x, y) = 4 – y – 2x2g(x, y) = 8 – y2 – 4x(a) Use the Excel Solver to determine the two pairs of values of x and y that satisfy these
An electronics company produces transistors, resistors, and computer chips. Each transistor requires four units of copper, one unit of zinc, and two units of glass. Each resistor requires three,
Use MATLAB software to determine the spectral condition number for a 10-dimensional Hilbert matrix. How many digits of precision are expected to be lost due to ill-conditioning? Determine the
Repeat Prob. 11.18, but for the case of a six-dimensional Vandermonde matrix (see Prob. 10.14) where x1 = 4, x2 = 2, x3 = 7, x4 = 10, x5 = 3, and x6 = 5.
In Sec. 9.2.1, we determine the number of operations required for Gauss elimination without partial pivoting. Make a similar determination for the Thomas algorithm (Figure). Develop a plot of

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