Questions and Answers of Applied Numerical Methods

Perform the same calculation as in Prob. 24.22, but for the case where the tube is also insulated (i.e., no convection) and the right-hand wall is held at a fixed boundary temperature of 200 K.Data
As in Fig. P24.22, an insulated metal rod has a fixed temperature (T0) boundary condition at its left end. On it right end, it is joined to a thin-walled tube filled with water through which heat is
Perform the same simulations as in Section 22.6 for the Lorenz equations but use a value of r = 99.96. Compare your results with those obtained in Section 22.6. 22.6 CASE STUDY An example of a simple
Repeat the the same simulations as in Section 22.6 for the Lorenz equations but generate the solutions with the midpoint method. 22.6 CASE STUDY An example of a simple nonlinear model based on
Develop a script to generate the same computations and plots as in Sec. 21.8, but for the MATLAB peaks function over ranges of both x and y from –3 to 3. 21.8 CASE STUDY VISUALIZING
Develop a script to generate the same computations and plots as in Sec. 21.8, but for the following functions (for x = –3 to 3 and y = –3 to 3): (a) f(x, y) = e−(x2 + y2) and (b) f(x, y) =
Employ the events option described in Section 23.1.2 to determine the period of a 1-m long, linear pendulum (see description in Prob. 23.10). Compute the period for the following initial
The following nonlinear differential equation was solved in Examples 24.4 and 24.7.Such equations are sometimes linearized to obtain an approximate solution. This is done by employing a first-order
Table P15.5 lists values for dissolved oxygen concentration in water as a function of temperature and chloride concentration.Table P15.5(a) Use quadratic and cubic interpolation to determine the
Use naive Gauss elimination to factor the following system according to the description in Section 10.2:7x1 + 2x2 − 3x3 = −122x1 + 5x2 − 3x3 = −20x1 − x2 − 6x3 = −26Then,
(a) Perform a Cholesky factorization of the following symmetric system by hand:(b) Verify your hand calculation with the built-in chol function. (c) Employ the results of the factorization [U]
(a) Use LU factorization to solve the system of equations in Prob. 10.3. Show all the steps in the computation. (b) Also solve the system for an alternative right-hand-side vector{b}T = ⌊12 18
Develop your own M-file to determine the Cholesky factorization of a symmetric matrix without pivoting. That is, develop a function that is passed the symmetric matrix and returns the matrix [U].
(a) Determine the LU factorization without pivoting by hand for the following matrix and check your results by validating that [L][U] = [A].(b) Employ the result of (a) to compute the determinant.(c)
Use Cholesky factorization to determine [U] so that [A] = [U]¹ [U] = [ 2 - 0 -1 2 -1 0 - 1 2
Compute the Cholesky factorization ofDo your results make sense in terms of Eqs. (10.15) and (10.16)?Equation (10.15) or (10.16) [A] = 0 25 9 0 0 0 0 0 16
Use the following LU factorization to (a) compute the determinant and (b) solve [A]{x} = {b} with {b}T = ⌊−10 50 −26⌋. [A] = [L][U] = 1 0.6667 -0.3333 [0 3 [₁ X 1 -0.3636
Determine the matrix inverse for the system described in Prob. 8.9. Use the matrix inverse to determine the concentration in reactor 5 if the inflow concentrations are changed to c01 = 10 and
Determine the matrix inverse for the system described in Prob. 8.10. Use the matrix inverse to determine the force in the three members (F1, F2 and F3) if the vertical load at node 1 is doubled to
Polynomial interpolation consists of determining the unique (n – 1)th-order polynomial that fits n data points. Such polynomials have the general form,where the p’s are constant coefficients. A
(a) Using the same approach as described in Sec. 11.3, develop steady-state mass balances for the room configuration depicted in Fig. P11.18.(b) Determine the matrix inverse and use it to calculate
(a) Determine the matrix inverse and condition number for the following matrix:(b) Repeat (a) but change a33 slightly to 9.1. 1 2 4 5 7 8 3 6 9.
A chemical constituent flows between three reactors as depicted in Fig. P11.15. Steady-state mass balances can be written for a substance that reacts with first-order kinetics. For example, the mass
Determine the matrix inverse for the following system:10x1 + 2x2 − x3 = 27−3x1 − 6x2 + 2x3 = −61.5x1 + x2 + 5x3 = −21.5Check your results by verifying that [A][A]−1 = [I ]. Do
Determine the matrix inverse for the following system:−8x1 + x2 − 2x3 = −202x1 − 6x2 − x3 = −38−3x1 − x2 + 7x3 = −34
As described in Examples 8.2 and 11.2, use the matrix inverse to answer the following: (a) Determine the change in position of the first jumper, if the mass of the third jumper is increased to 100
Of the following three sets of linear equations, identify the set(s) that you could not solve using an iterative method such as Gauss-Seidel. Show using any number of iterations that is necessary
(a) Use the Gauss-Seidel method to solve the following system until the percent relative error falls below εs = 5%:(b) Repeat (a) but use overrelaxation with λ = 1.2. 0.8 -0.4 -0.4 0.8
Solve the following system using three iterations with Gauss-Seidel using overrelaxation (λ = 1.25). If necessary, rearrange the equations and show all thea steps in your solution including your
The standard normal probability density function is a bell-shaped curve that can be represented asUse MATLAB to generate a plot of this function from z = −5 to 5. Label the ordinate as frequency
Determine the total flops as a function of the number of equations n for the (a) Factorization, (b) Forward substitution, (c) Back substitution phases of the LU factorization version
Determine the solution of the simultaneous nonlinear equationsy = −x2 + x + 0.5y + 5xy = x2Use the Newton-Raphson method and employ initial guesses of x = y = 1.2.
Develop your own M-file function for the Gauss- Seidel method without relaxation based on Fig. 12.2, but change the first line so that it returns the approximate error and the number of
As displayed in Fig. P13.6, an LC circuit can be modeled by the following system of differential equations:where L = inductance (H), t = time (s), i = current (A), and C = capacitance (F). Assuming
Develop your own M-file function for the Newton- Raphson method for nonlinear systems of equations based on Fig. 12.4. Test it by solving Example 12.4 and then use it to solve Prob. 12.8.Example
On average, the surface area A of human beings is related to weight W and height H. Measurements on a number of individuals of height 180 cm and different weights (kg) give values of A (m2) in the
The curvature of a slender column subject to an axial load P (Fig. P13.10) can be modeled bywherewhere E = the modulus of elasticity, and I = the moment of inertia of the cross section about its
Develop your own M-file function for Gauss-Seidel with relaxation. Here is the function’s first line:function [x,ea,iter] = ...GaussSeidelR(A,b,lambda,es,maxit) In the event that the user does not
A system of two homogeneous linear ordinary differential equations with constant coefficients can be written asIf you have taken a course in differential equations, you know that the solutions for
Water flows between the North American Great Lakes as depicted in Fig. P13.12. Based on mass balances, the following differential equations can be written for the concentrations in each of the lakes
Develop an M-file function to determine the largest eigenvalue and its associated eigenvector with the power method. Test the program by duplicating Example 13.3 and then use it to solve Prob.
The following data were gathered to determine the relationship between pressure and temperature of a fixed volume of 1 kg of nitrogen. The volume is 10 m3.Employ the ideal gas law pV = nRT to
Modify the linregr function in Fig. 14.15 so that it(a) Computes and returns the standard error of the estimate, and (b) Uses the subplot function to also display a plot of the residuals (the
Fit an exponential model toPlot the data and the equation on both standard and semilogarithmic graphs with the MATLAB subplot function. x 0.4 y 800 0.8 985 1.2 1490 1.6 1950 2 2850 2.3 3600
Develop an M-file function to compute descriptive statistics for a vector of values. Have the function determine and display number of values, mean, median, mode, range, standard deviation, variance,
A Monte Carlo analysis can be used for optimization. For example, the trajectory of a ball can be computed withwhere y = the height (m), θ0 = the initial angle (radians), v0 = the initial velocity
Develop an M-file function to fit a power model. Have the function return the best-fit coefficient α2 and power β2 along with the r2 for the untransformed model. In addition, use the subplot
Perform the same computation as in Example 14.3, but in addition to the drag coefficient, also vary the mass normally around its mean value with a coefficient of variation of 5.7887%.Example 14.3
Fit a parabola to the data from Table 14.1. Determine the r2 for the fit and comment on the efficacy of the result. TABLE 14.1 Experimental data for force (N) and velocity (m/s) from a wind
Fit a cubic polynomial to the following data:Along with the coefficients, determine r2 and sy/x. X 3 y 1.6 4 3.6 5 4.4 7 8 9 11 3.4 2.2 2.8 3.8 12 4.6
Develop an M-file to implement polynomial regression. Pass the M-file two vectors holding the x and y values along with the desired order m. Test it by solving Prob. 15.3.Data From Problem 15.3Fit a
The following model is used to represent the effect of solar radiation on the photosynthesis rate of aquatic plants:where P = the photosynthesis rate (mg m−3d−1),Pm = the maximum photosynthesis
Enzymatic reactions are used extensively to characterize biologically mediated reactions. The following is an example of a model that is used to fit such reactions:where v0 = the initial rate of the
The following data represent the bacterial growth in a liquid culture over of number of days:Find a best-fit equation to the data trend. Try several possibilities—linear, quadratic, and
Dynamic viscosity of water μ(10–3 N · s/m2) is related to temperature T(◦C) in the following manner:(a) Plot this data.(b) Use linear interpolation to predict μ at T = 7.5 °C.(c) Use
The pH in a reactor varies sinusoidally over the course of a day. Use least-squares regression to fit Eq. (16.11) to the following data. Use your fit to determine the mean, amplitude, and time of
Duplicate Example 16.3, but for 64 points sampled at a rate of Δt = 0.01 s from the functionf (t) = cos[2 π (12.5)t] + cos[2 π (25)t]Use fft to generate a DFT of these values and plot the
An investigator has reported the data tabulated below. It is known that such data can be modeled by the following equation x = e(y−b)/a where a and b are parameters. Use nonlinear regression
It is known that the data tabulated below can be modeled by the following equationUse nonlinear regression to determine the parameters a and b.Based on your analysis predict y at x = 1.6. = (a +
Use MATLAB to generate 32 points for the sinusoid depicted in Fig. 16.2 from t = 0 to 6 s. Compute the DFT and create subplots of (a) The original signal, (b) The real part, and (c)
Use the Maclaurin series expansions for ex, cos x and sin x to prove Euler’s formula (Eq. 16.21). etix = cos x + isin.x
Develop a plot of a cubic spline fit of the following data with(a) Natural end conditions and (b) Not-a-knot end conditions. In addition, develop a plot using (c) piecewise cubic Hermite
Employ inverse interpolation using a cubic interpolating polynomial and bisection to determine the value of x that corresponds to f (x) = 1.7 for the following tabulated data: X 1 f(x)
The following data for the density of nitrogen gas versus temperature come from a table that was measured with high precision. Use first- through fifth-order polynomials to estimate the density at a
Develop an M-file to compute a cubic spline fit with natural end conditions. Test your code by using it to duplicate Example 18.3.Data From Example 18.3 Natural Cubic Splines Problem Statement. Fit
Repeat Example 17.6 but using first-, second-, third-, and fourth order interpolating polynomials to predict the population in 2000 based on the most recent data. That is, for the linear prediction
The following data were generated with the fifth order polynomial: f(x) = 0.0185x5 − 0.444x4 + 3.9125x3 − 15.456x2 + 27.069x − 14.1:(a) Fit these data with a cubic spline with not-a-knot end
The drag coefficient for spheres such as sporting balls is known to vary as a function of the Reynolds number Re, a dimensionless number that gives a measure of the ratio of inertial forces to
The following data are sampled from the step function depicted in Fig. 18.1:Fit these data with a (a) Cubic spline with not-a-knot end conditions,(b) Cubic spline with zero-slope clamped end
Develop an M-file function that uses the fft function to generate a power spectrum plot. Use it to solve Prob. 16.9. Data From Problem16.9 Duplicate Example 16.3, but for 64 points sampled at a
The cross-sectional area of a channel can be computed aswhere B = the total channel width (m), H = the depth (m), and y = distance from the bank (m). In a similar fashion, the average flow Q (m3/s)
The average concentration of a substance c̅ (g/m3) in a lake where the area As(m2) varies with depth z(m) can be computed by integration:where Z = the total depth (m). Determine the average
The function f (x) = e−x can be used to generate the following table of unequally spaced data:Evaluate the integral from a = 0 to b = 0.6 using (a) Analytical means, (b) The trapezoidal
As specified in the following table, the earth’s density varies as a function of the distance from its center (r = 0):Use numerical integration to estimate the earth’s mass (in metric tonnes) and
Determine the distance traveled from the following velocity data:(a) Use the trapezoidal rule. In addition, determine the average velocity.(b) Fit the data with a cubic equation using polynomial
Evaluate the following integral (a) Analytically,(b) Romberg integration (εs = 0.5%),(c) The three-point Gauss quadrature formula, and (d) MATLAB quad function: I= 0 -0.055x4 +0.86x³
Use MATLAB to generate eight points from the function f(t) = sin2 t from t = 0 to 2π. Fit these data using(a) Cubic spline with not-a-knot end conditions,(b) Cubic spline with derivative end
As specified in the following table, a manufactured spherical particle has a density that varies as a function of the distance from its center (r = 0):Use numerical integration to estimate the
The following function describes the temperature distribution on a rectangular plate for the range −2 ≤ x ≤ 0 and 0 ≤ y ≤ 3T = 2 + x − y + 2x2 + 2xy + y2Develop a script to: (a)
Evaluate the following integral with (a) Romberg integration (εs = 0.5%),(b) the two-point Gauss quadrature formula, and (c) MATLAB quad and quadl functions: I = 3 h xe dx
As specified in the following table, the earth’s density varies as a function of the distance from its center (r = 0):Develop a script to fit these data with interp1 using the pchip option.
The root-mean-square current can be computed asFor T = 1, suppose that i(t) is defined asEvaluate the IRMS using (a) Romberg integration to a tolerance of 0.1%, (b) The two- and three-point
Develop an M-file function to implement adaptive quadrature based on Fig. 20.6. Test the function by using it to determine the integral of the polynomial from Example 20.1. Then use it to solve Prob.
Develop an M-file function to implement Romberg integration based on Fig. 20.2. Test the function by using it to determine the integral of the polynomial from Example 20.1. Then use it to solve Prob.
Develop an M-file function that computes first and second derivative estimates of order O(h2) based on the formulas in Figs. 21.3 through 21.5. The function’s first line should be set up as
The following data were generated from the normal distribution:Use MATLAB to estimate the inflection points of these data. x -2 f(x) 0.05399 -1.5 -0.5 O 0.12952 0.24197 0.35207
Develop an M-file to obtain first-derivative estimates for unequally spaced data. Test it with the following data:where f(x) = 5e−2x x. Compare your results with the true derivatives. X 0.6 f(x)
The heat required, ΔH(cal), to induce a temperature change, ΔT (°C), of a material can be computed asH = mCp (T) ΔTwhere m = mass (g), and Cp(T) = heat capacity [cal/(g .°C)].The heat capacity
The specific heat at constant pressure cp [J/(kg · K)] of an ideal gas is related to enthalpy bywhere h = enthalpy (kJ/kg), and T = absolute temperature (K). The following enthalpies are provided
Compute the power absorbed by an element in a circuit as described in Sec. 20.5, but for a simple sinusoidal current i = sin(2πt/T) where T = 1 s.(a) Assume that Ohm’s law holds and R = 5 Ω.(b)
The sediment oxygen demand [SOD in units of g/(m2 · d)] is an important parameter in determining the dissolved oxygen content of a natural water. It is measured by placing a sediment core in a
Fourier’s law is used routinely by architectural engineers to determine heat flow through walls. The following temperatures are measured from the surface (x = 0) into a stone wall:If the flux at x
The following data for the specific heat of benzene were generated with an nth-order polynomial. Use numerical differentiation to determine n. T, K 300 400 500 600 Cp, kJ/(kmol. K) 82.888 112.136
Solve the following problem over the interval from t = 0 to 2 using a step size of 0.5 where y(0) = 1. Display all your results on the same graph.Obtain your solutions with (a) Heun’s method
The growth of populations of organisms has many engineering and scientific applications. One of the simplest models assumes that the rate of change of the population p is proportional to the existing
An nth-order rate law is often used to model chemical reactions that solely depend on the concentration of a single reactant:where c = concentration (mole), t = time (min), n = reaction order
Solve the following pair of ODEs over the interval from t = 0 to 0.4 using a step size of 0.1. The initial conditions are y(0) = 2 and z(0) = 4. Obtain your solution with (a) Euler’s method
The van der Pol equation is a model of an electronic circuit that arose back in the days of vacuum tubes:Given the initial conditions, y(0) = y′(0) = 1, solve this equation from t = 0 to 10 using
Develop an M-file to solve a single ODE with Heun’s method with iteration. Design the M-file so that it creates a plot of the results. Test your program by using it to solve for population as
Solve the following initial-value problem over the interval from t = 2 to 3:Use the non-self-starting Heun method with a step size of 0.5 and initial conditions of y(1.5) = 5.222138 and y(2.0) =

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