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mathematics
numerical analysis

Numerical Methods For Engineers 5th Edition Steven C. Chapra, Raymond P. Canale - Solutions

Find the temperature distribution in a rod (Figure) with internal heat generation using the finite-element method. Derive the element nodal equations using Fourier heat conduction.qk = -kAdT∂xAnd heat conservation relationships∑[qk + ƒ(x)] = 0Where qk = heat flow (W), k = thermal conductivity
Perform the same computation as in Sec. 32.1, but use ∆x = 1.25.
Develop a finite-element solution for the steady-state system of Sec. 32.1.
Compute mass fluxes for the steady-state solution of Sec. 32.1 using Fick’s first law.
Compute the steady-state distribution of concentration for the tank shown in Figure. The PDE governing this system isD (∂2c/∂x2 + ∂2c/∂y2) – kc = 0And the boundary conditions are as shown. Employ a value of 0.5 for D and 0.l for k.
Two plates are 10 cm apart, as shown in Figure. Initially, both plates and the fluid are still. At t = 0, the top plate is moved at a constant velocity of 8 cm/s. The equations governing the motions of the fluids areAnd the following relationships hold true at the oil-water interface:What is the
Perform the same computation as in Sec. 32.2, but use ∆x = ∆y = 0.4 m.
The flow through porous media can be described by the Laplace equation∂2h/∂x2 + ∂2h/∂y2 = 0Where h is head. Use numerical methods to determine the distribution of head for the system shown in Figure.
The velocity of water flow through the porous media can be related to head by D’ Arcy’s lawqn = – K dh/dnWhere K is the hydraulic conductivity and qn is discharge velocity in the n direction. If K = 5 x 10–4 cm/s, compute the water velocities for Prob. 32.7.
Perform the same computation as in Sec. 32.3 but for the system depicted inFigure.
Perform the same computation as in Sec. 32.3 but for the system depicted inFigure.
Perform the same computation as in Sec. 32.4, but change the force to 1.5 and the spring constantsto
Perform the same computation as in Sec. 32.4, but use a force of 2 and five springswith
An insulated composite rod is formed of two parts arranged end to end, and both halves are of equal length. Part α has thermal conductivity kα, for 0 ≤ x ≤ 1/2, and part b has thermal conductivity kb, for 1/2 ≤ x ≤ 1. The nondimensional transient heat conduction equations that describe
Solve the nondimensional transient heal conduction equation in two dimensions, which represents the transient temperature distribution in an insulated plate. The governing equation is∂2u/∂x2 + ∂2u/∂y2 = ∂u/∂tWhere u = temperature, x and y are spatial coordinates, and t = time. The
Evaluate the following integral:(a) Analytically;(a) Single application of the trapezoidal rule;(c) multiple-application trapezoidal rule, with n = 2 and 4;(d) Single application of Simpson’s 1/3 rule;(e) Multiple-application Simpson’s 1/3 rule, with n 4;(f) Single application of Simpson’s
Evaluate the following integral:(a) Analytically;(a) Single application of the trapezoidal rule;(c) multiple-application trapezoidal rule, with n = 2 and 4;(d) Single application of Simpson’s 1/3 rule;(e) Multiple-application Simpson’s 1/3 rule, with n 4;(f) Single application of Simpson’s
Evaluate the following integral:(a) Analytically;(b) Single application of the trapezoidal rule;(c) Composite trapezoidal rule, with n = 2 and 4;(d) Single application of Simpson's 1/3 rule;(e) Simpson's 3/8 rule, and(f) Boole’s rule. For each of the numerical estimates (b) through (f) determine
Integrate the following function analytically and using the trapezoidal rule, with n = 1, 2, 3, and 4:Use the analytical solution to compute true percent relative errors to evaluate the accuracy of the trapezoidalapproximations.
Integrate the following function both analytically and using Simpson’s rules, with n = 4, and 5. Discuss the results.
Integrate the following function both analytically and numerically. Use both the trapezoidal and Simpson’s 1/3 rules to numerically integrate the function. For both cases, use the multiple-application version, with n = 4. Compute percent relative errors for the numerical results.
Integrate the following function both analytically and numerically. For the numerical evaluations use (a) A single application of the trapezoidal rule,(b) Simpson’s 1/3 rule, (c) Simpson’s 3/8 rule, (d) Boole’s rule,(e) The midpoint method,(f) The 3-segment-2-point open
Integrate the following function both analytically and numerically. For the numerical evaluations use (a) Single application of, the trapezoidal rule; (b) Simpson’s 1/3 rule; (c) Simpson’s 3/8 rule; (d) Multiple application of Simpson’s rules with n = 5; (e) Bole’s
Suppose that the upward force of air resistance on a falling object is proportional to the square of the velocity. For this case, the velocity can be computed asWhere cd = a second-order drag coefficient.(a) If g = 9.8 m/s2, m = 68.1 kg and cd = 0.2.5 kg/m, use analytical integration to determine
Evaluate the integral of the following tabular data with(a) The trapezoidal rule and(b) Simpson’s rules:
Evaluate the integral of the following tabular data with(a) The trapezoidal rule and(b) Simpson’s rules:
Determine the mean value of the functionƒ(x) = - 46 + 45x – 14x2 + 2x3 – 0.075x4Between x 2 and 10 by(a) Graphing the function and visually estimating the mean value,(b) Using Eq. (PT6.4) and the analytical evaluation of the integral, and(c) Using Eq. (PT6.4) and a five-segment version of
The function f(x) = 2e-1.5x can be used to generate the following table of unequally spaced data:Evaluate the integral from Î± = 0 to b = 0.6 using(a) Analytical means,(b) The trapezoidal rule, and(c) A combination of the trapezoidal and Simpson's rules; employ Simpson's rules wherever possible
Evaluate the following double integral:(a) Analytically.(b) Using a multiple-application trapezoidal rule, with n = 2; and(c) Using single applications of Simpson€™s€™ 1/3 rule. For (b) and (c), compute the percent relative error(Îµl).
Evaluate the following triple Integral(a) Analytically and(b) Using single applications of Simpson€™s 1/3 rule. For(b) Compute the percent relative error(Îµl).
Develop a user-friendly computer program for the multiple-application trapezoidal rule based on Fig. Test your program by duplicating the computation from Example 21.2.
Develop a user-friendly computer program for the multiple application-version of Simpson’s rule based on Figure. Test it by duplicating the computations from Example 21.5.
Develop a user-friendly computer program for integrating unequally spaced data based on Figure. Test it by duplicating the computation from Example 21.8.
An 11-m beam is subjected to a load, and the shear force follows the equationV(x) = 5 + 0.25x2Where V is the shear force and x is length in distance along the beam. We know that V = dM/dx, and M is the bending moment. Integration yields the relationshipIf Mo, is zero and x = 11, calculate M
The work produced by a constant temperature, pressure-volume thermodynamic process can be computed asW = ∫ pdVWhere W is work, p is pressure, and V is volume. Using a combination of the trapezoidal rule, Simpson’s 1/3 rule, and Simpsons’ 3/8 rule, use the following data to compute the work in
Determine the distance traveled for the following data:(a) Use the trapezoidal rule.(b) The best combination of the trapezoidal and Simpson€™s rules, and(C) Analytically integrating second- and third-order polynomials determined byregression.
The total mass of a variable density rod is given byWhere m = mass, p (x) = density, Ac(x) = cross-sectional area, x = distance along the rod and L = the total length of the rod. The following data has been measured for a 10-m length rod. Determine the mass in kilograms to the best possibleaccuracy.
A transportation engineering study requires that you determine the number of cars that pass through an intersection traveling during morning rush hour. You stand at the side of the road and count the number of cars that pass every 4 minutes at several times as tabulated below. Use the best
Determine the average value for the data in Figure. Perform the integral needed for the average in the order shown by the followingequation:
Use Romberg integration to evaluateto an accuracy of Îµs = 0.5%. Your results should be presented in the form of Figure. Use the analytical solution of the integral to determine the percent relative error of the result obtained with Romberg integration. Check that Îµt is less than the stopping
Use order of h8 Romberg integration to evaluateCompare εα and εt.
Use Romberg integration to evaluateTo an accuracy of Îµs = 0.5%. Your results should be presented in the form ofFigure.
Obtain an estimate of the integral from Prob. 22.1, but using two-, three-, and four’ point Gauss-Legendre formulas. Compute εt for each case on the basis of the analytical solution.
Obtain an estimate of the integral from Prob. 22.2, but using two-, three-, and tour-point Gauss-Legendre formulas. Compute εt for each case on the basis of the analytical solution.
Obtain an estimate of the integral from Prob. 22.3 using the five-point Gauss-Legendre formula.
Perform the computation in Examples 21.3 and 22.5 for the falling parachutist, but use Romberg integration (εs = 0.05%)
Employ two-through six-point Gauss-Legendre formulas to solveInterpret your results in light of Eq.(22.26).
Use numerical integration to evaluate the following:
Develop a user-friendly computer program for the multiple-segment(a) Trapezoidal and(b) Simpson’s 1/3 rule based on Figure. Test it by IntegratingUse the true value of 0.602298 to compute εt for n = 4.
Develop a user-friendly computer program for Romberg integration based on Figure. Test it by duplicating the results of Examples 22.3 and 22.4 and the function in Prob. 22.10.
Develop a user-friendly computer program for Gauss quadrature. Test it by duplicating the results of Examples 22.3 and 22.4 and the function in Prob. 21.10.
There is no closed form solution for the error function,Use the two-points Gauss quadrature approach to estimate erf(1.5). Note that the exact value is0.966105.
The amount of mass transported via a pipe over a period of time can be computed asWhere M = mass (mg), t1 = the initial time (min), t2 = the final time (min), Q(t) = flow rate (m3/min), and c(t) = concentration (mg/m3), The following functional representations define the temporal variations in flow
The depths of a river H are measured at equally spaced distances across a channel as tabulated below. The river’s cross- sectional area can be determined by Integration as inUse Romberg integration to perform the integration to a stopping criterion of 1%.
Compute forward and backward difference approximations of O(h) and O(h2), and central difference approximations of O(h2) and O(h4) for the first derivative of y = cos x at x = π/4 using a value of h = π/12. Estimate the true percent relative error εt for each approximation.
Repeat Prob. 23.1, but for y = log x evaluated at x = 25 with h = 2.
Use centered difference approximations to estimate the first and second derivatives of y = ex at x = 2 for h = 0.1. Employ both O(h2) and O(h4) formulas for your estimates.
Use Richardson extrapolation to estimate the first derivative of y = cos x at x = π/4 using step sizes of h1 = π/3 and = h2 = π/6. Employ centered differences of O(h2) for the initial estimates.
Repeat Prob. 23.4, but for the first derivative of In x at x = 5 using h1 = 2 and h2 = 1.
Employ Eq. (23.9) to determine the first derivative of y = 2x4 – 6x3 – 12x – 8 at x = 0 based on values at x0 = –0.5, x1= 1, and x2 = 2. Compare this result with the true value and with an estimate obtained using a centered difference approximation based on h = 1.
Prove that for equispaced data points, Eq. (23.9) reduces to Eq. (4.22) at x = xi.
Compute the first-order central difference approximations of O(h4) for each of the following functions at the specified locationand for the specified step size:(a) y = x3 + 4x – 15 at x = 0,
The following data was collected for the distance traveled versus time for a rocket:Use numerical differentiation to estimate the rocket’s velocity and acceleration at each time.
Develop a user-friendly program to apply a Romberg algorithm to estimate the derivative of a given function.
Develop a user-friendly program to obtain first-derivate estimates for unequally spaced data. Test it with the following data:Where ƒ(x) = 5e-2xx. Compare your results with the true derivatives.
Recall that for the falling parachutist problem, the velocity is given byAnd the distance traveled can be obtained byGiven g = 9.81, m = 70, and c = 12,(a) Use MATLAB to integrate Eq. (P23.12α) from t = 0 to 10.(b) Analytically integrate Eq. (P23.12b) with the initial condition that d = 0 at γ =
The normal distribution is defined as(a) Use MATLAB to integrate this function from x = - 1 to 1 and from -2 to 2.(b) Use MATLAB to determine the inflection points of thisfunction.
The following data was generated from the normal distribution:(a) Use MATLAB to integrate this data from x = -1 to 1 and -2 to 2 with the trap function.(b) Use MATLAB to estimate the inflection points of this data.
Use IMSL to integrate the normal distribution (see Prob. 23.13) from x = -1 to 1, from -2 to 2, and from -3 to 3.
Write a MATLAB program tointegrate
Write a MATLAB program to integrateusing both the quad and quadL functions. To learn more about quadL, type help guadL at the MATLAB prompt.
Use the diff(y) command in MATLAB and compute the finite-difference approximation to the first and second derivative at each x-value in the table below, excluding the two end points. Use finite-difference approximations that are second-order correct,O(ˆ†x2).
The objective of this problem is to compare second-order accurate forward, backward, and centered finite-difference approximations of the first derivative of a function to the actual value of the derivative. This will be done for(a) Use calculus to determine the correct value of the derivative at x
Use a Taylor series expansion to derive a centered finite- difference approximation to the third derivative that is second- order accurate. To do this, you will have to use four different expansions for the points xi-2, xi-1, xi+1, and xi+2. In each case, the expansion will be around the point xi.
Use the following data to find the velocity and acceleration at t = 10 seconds:Use second-order correct(a) Centered finite-difference,(b) Forward finite-difference, and(c) Backward finite-differencemethods.
A plane is being tracked by radar, and data is taken every second in polar coordinates Î¸ and r.At 206 seconds, use the centered finite difference (second-order correct) to find the vector expressions for velocity υ, and acceleration α. The velocity and acceleration given in polar coordinates
Develop an Excel VBA macro program to read in adjacent columns of x and y values from a worksheet. Evaluate the derivatives at each point using Eq. 23.9, and display the results in a third column adjacent to the x and y values back on the spreadsheet. Test your program by applying it to evaluate
Use regression to estimate the acceleration at each time for the following data with second-, third-, and fourth-order polynomials. Plot theresults.
You have to measure the flow rate of water through a small pipe. In order to do it, you place a bucket at the pipe’s outlet and measure the volume in the bucket as a function of time as tabulated below. Estimate the flow rate at t = 7 s.
The velocity υ (m/s) of air flowing past a flat surface is measured at several distances y (m) away from the surface. Determine the shear stress τ (N/m2) at the surface (y = 0),Assume a value of dynamic viscosity μ = 1.8 x l0-5 N ∙ s/m2.
Chemical reactions often follow the model:dc/dt = – kcnwhere c = concentration, t = time. k = reaction rate, and n = reaction order. Given values of c and dc/dt, k and n can be evaluated by a linear regression of the logarithm of this equation:Use this approach along with the following data to
Perform the same computation as Sec. 24.1, but compute the amount of heat requited to raise the temperature of 1200 g of the material from -150 to 100°C. Use Simpson’s rule for your computation, with values of T at 50°C increments.
Repeat Prob. 24.1, but use Romberg integration to εs = 0.0 1%.
Repeat Prob. 24.1, but use a two- and a three-point Gauss Legendre formula. Interpret your results.
Integration provides a means to compute how much mass enters or leaves a reactor over a specified time period, as inwhere t1 and t2 = the initial and final times, respectively. This formula makes intuitive sense if you recall the analogy between integration and summation. Thus, the integral
The outflow chemical concentration from a completely mixed reactor is measured asFor an outflow of Q = 0.3 m3/s, estimate the mass of chemical in grams that exits the reactor from t = 0 to 20min.
Fick’s first diffusion law states thatWhere mass flux = the quantity of mass that passes across a unit area per unit time (g/cm2/s), D = a diffusion coefficient (cm2/s), c = concentration, and x = distance (cm). As environmental engineer measures the following concentration of a pollutant in the
The following data was collected when a large oil tanker was loading:Calculate the flow rate Q (that is, dV/dt) for each time to the order ofh2.
You are interested in measuring the fluid velocity in a narrow rectangular open channel carrying petroleum waste between locations in an oil refinery. You know that, because of bottom friction, the velocity varies with depth in the channel. If your technician has time to perform only two velocity
Soft tissue follows an exponential deformation behavior in uniaxial tension while it is in the physiologic a normal range of elongation. This can be expressed asWhere σ = stress, ε = strain, and Eo and α are material constants that are determined experimentally. To evaluate the two material
The standard technique for determining cardiac output is the indicator dilution method developed by Hamilton. One end of a small catheter is inserted into the radial artery and the other end is connected to a densitometer, which can automatically record the concentration of the dye in the blood. A
Glaucoma is the second leading cause of vision loss world-wide. High intraocular pressure (pressure inside the eye) almost always accompanies vision loss. It is postulated that the high pressure damages a subset of cells in the eye that are responsible for vision. One investigator theorizes that
One of your colleagues has designed a new transdermal patch to deliver insulin through the skin to diabetic Patients in a controlled way, eliminating the need for painful injections. She hasCollected the following data on the mass flux of insulin being delivered through the patch (and skin) as a
Videoangiography is used to measure blood flow and determine the status of circulatory function. In order to quantify the videoangiograms, blood vessel diameter and blood velocity are needed such that total blood flow is determined. Below is the densitometric profile taken from a videoangiogram of
Perform the same computation as in Sec. 24.2, but use O(h8) Romberg integration to evaluate the integral.
Perform the same computation as in Sec. 24.2, but use Gauss quadrature to evaluate the integral.
As in Sec. 24.2, compute F using the trapezoidal rule and Simpson€™s 1/3 and Simpson€™s 3/8 rules but use the following force. Divide the mast into 5-ftintervals.
Stream cross-sectional areas (A) are required for a number of tasks in water resources engineering, including flood forecasting and reservoir designing. Unless electronic sounding devices are available to obtain continuous profiles of the channel bottom, the engineer must rely on discrete depth
As described in Prob. 24.17, 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) can be computed asWhere U = water velocity (m/s). Use these relationships
During a survey, you are required to compute the area of the field shown in Figure. Use Simpson’s rules to determine the area.

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