Questions and Answers of Applied Numerical Methods

Given (a) Estimate the step size required to maintain stability using the explicit Euler method.(b) If y(0) = 0, use the implicit Euler to obtain a solution from t = 0 to 2 using a step size of
Develop an M-file to solve a single ODE with the midpoint method. Design the M-file so that it creates a plot of the results. Test your program by using it to solve for population as described in
Evaluate ∂f/∂x, ∂f / ∂y, and ∂f(∂ x ∂y) for the following function at x = y = 1 (a) Analytically and (b) Numerically Δx = Δy = 0.0001:f(x, y) = 3xy + 3x − x3 − 3y3
Develop an M-file to solve a single ODE with the fourth-order RK method. Design the M-file so that it creates a plot of the results. Test your program by using it to solve Prob. 22.2. Employ a step
GivenIf y(0) = 0, use the implicit Euler to obtain a solution from t = 0 to 4 using a step size of 0.4. dy dt = 30(sin ty) + 3 cost
GivenIf x1(0) = x2(0) = 1, obtain a solution from t = 0 to 0.2 using a step size of 0.05 with the (a) Explicit and (b) Implicit Euler methods. dx1 dt dx2 dt = 999x1 +
Develop an M-file to solve a system of ODEs with Euler’s method. Design the M-file so that it creates a plot of the results. Test your program by using it to solve Prob. 22.7 with a step size of
Solve the following differential equation from t = 0 to 2with the initial condition y(0) = 1. Use the following techniques to obtain your solutions: (a) Analytically, (b) The explicit Euler
Repeat Prob. 23.11, but for the nonlinear pendulum described in Prob. 23.10.Data From Problem 23.11Employ the events option described in Section 23.1.2 to determine the period of a 1-m long, linear
Use ode45 to integrate the differential equations for the system described in Prob. 23.19. Generate vertically stacked subplots of displacements (top) and velocities (bottom). Employ the fft function
The following matrix is entered in MATLAB:>> A=[3 2 1;0:0.5:1;linspace(6, 8, 3)](a) Write out the resulting matrix.(b) Use colon notation to write a single-line MATLAB command to multiply the
Use calculus to solve Eq. (1.21) for the case where the initial velocity is Equation (1.21)(a) Positive and (b) Negative. (c) Based on your results for (a) and (b), perform the same
The following information is available for a bank account:The money earns interest which is computed as Interest = i Bi where i = the interest rate expressed as a fraction per month, and
Repeat Example 1.2. Compute the velocity to t = 12 s, with a step size of (a) 1(b) 0.5 s. Can you make any statement regarding the errors of the calculation based on the results?Example 1.2
Use calculus to verify that Eq. (1.9) is a solution of Eq. (1.8) for the initial condition ν(0) = 0.Equation (1.9)Equation (1.8) v(t) = gm Cd tanh 1 P38 gca m
Rather than the nonlinear relationship of Eq. (1.7), you might choose to model the upward force on the bungee jumper as a linear relationship:Equation (1.7)FU = −c′νwhere c′ = a first-order
For the free-falling bungee jumper with linear drag (Prob. 1.5), assume a first jumper is 70 kg and has a drag coefficient of 12 kg/s. If a second jumper has a drag coefficient of 15 kg/s and a mass
Figure P1.13 depicts the various ways in which an average man gains and loses water in one day. One liter is ingested as food, and the body metabolically produces 0.3 liters. In breathing air, the
For the second-order drag model (Eq. 1.8), compute the velocity of a free-falling parachutist using Euler’s method for the case where m = 80 kg and cd = 0.25 kg/m. Perform the calculation from t =
The amount of a uniformly distributed radioactive contaminant contained in a closed reactor is measured by its concentration c (becquerel/liter or Bq/L). The contaminant decreases at a decay rate
For the same storage tank described in Prob. 1.9, suppose that the outflow is not constant but rather depends on the depth. For this case, the differential equation for depth can be written
A fluid is pumped into the network shown in Fig. P1.16. If Q2 = 0.7, Q3 = 0.5, Q7 = 0.1, and Q8 = 0.3 m3/s, determine the other flows. Q₁ Q10 Q₂ Q3 Qg Q4 Q5 Q8 Q6 Q7
In our example of the free-falling bungee jumper, we assumed that the acceleration due to gravity was a constant value of 9.81 m/s2. Although this is a decent approximation when we are examining
Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area.where V = volume (mm3), t = time (min), k = the evaporation rate (mm/min), and A = surface
Use the linspace function to create vectors identical to the following created with colon notation:(a) t = 4:6:35(b) x = -4:2
Use colon notation to create vectors identical to the following created with the linspace function:(a) v = linspace (-2,1.5,8)(b) r = linspace (8,4.5,8)
The command linspace(a, b, n) generates a row vector of n equally spaced points between a and b. Use colon notation to write an alternative one-line command to generate the same vector. Test your
The following equation can be used to compute values of y as a function of x:y = be−ax sin (bx) (0.012x4 − 0.15x3 + 0.075x2 + 2.5x)where a and b are parameters. Write the equation for
A simple electric circuit consisting of a resistor, a capacitor, and an inductor is depicted in Fig. P2.6. The charge on the capacitor q(t) as a function of time can be computed aswhere t = time,
If a force F (N) is applied to compress a spring, its displacement x (m) can often be modeled by Hooke’s law:F = kxwhere k = the spring constant (N/m). The potential energy stored in the spring U
It is general practice in engineering and science that equations be plotted as lines and discrete data as symbols. Here are some data for concentration (c) versus time (t) for the photodegradation of
The semilogy function operates in an identical fashion to the plot function except that a logarithmic (base-10) scale is used for the y axis. Use this function to plot the data and function as
Here are some wind tunnel data for force (F) versus velocity (ν):These data can be described by the following function:F = 0.2741ν1.9842Use MATLAB to create a plot displaying both the data (using
The density of freshwater can be computed as a function of temperature with the following cubic equation:ρ = 5.5289 × 10−8T3C − 8.5016 × 10−6T2C + 6.5622 × 10−5 TC +
The loglog function operates in an identical fashion to the plot function except that logarithmic scales are used for both the x and y axes. Use this function to plot the data and function as
The Maclaurin series expansion for the cosine isUse MATLAB to create a plot of the sine (solid line) along with a plot of the series expansion (black dashed line) up to and including the term x8/8!.
The trajectory of an object can be modeled aswhere y = height (m), θ0 = initial angle (radians), x = horizontal distance (m), g = gravitational acceleration (= 9.81 m/s2), v0 = initial velocity
You contact the jumpers used to generate the data in Table 2.1 and measure their frontal areas. The resulting values, which are ordered in the same sequence as the corresponding values in Table 2.1,
The following parametric equations generate a conical helix.x = t cos(6t)y = t sin(6t)z = tCompute values of x, y, and z for t = 0 to 6π with Δt = π/64. Use subplot to generate a two-dimensional
Exactly what will be displayed after the followingMATLAB commands are typed?(a) >> x = 5;>> x ^ 3;>> y = 8 – x(b) >> q = 4:2:12;>> r = [7 8 4; 3 6 –5];>>
The temperature dependence of chemical reactions can be computed with the Arrhenius equation:k = Ae−E/(RTa)where k = reaction rate (s−1), A = the preexponential (or frequency) factor, E =
Figure P2.21a shows a uniform beam subject to a linearly increasing distributed load. As depicted in Fig. P2.21b, deflection y (m) can be computed withwhere E = the modulus of elasticity and I = the
The butterfly curve is given by the following parametric equations:Generate values of x and y for values of t from 0 to 100 with Δt = 1/16. Construct plots of (a) x and y versus t and (b) y versus
The sine function can be evaluated by the following infinite series:Create an M-file to implement this formula so that it computes and displays the values of sin x as each term in the series is
Develop an M-file function that is passed a numeric grade from 0 to 100 and returns a letter grade according to the scheme:The first line of the function should befunction grade =
The butterfly curve from Prob. 2.22 can also be represented in polar coordinates as Generate values of r for values of θ from 0 to 8π withΔθ = π/32. Use the MATLAB function polar to generate
A simply supported beam is loaded as shown in Fig. P3.10. Using singularity functions, the displacement along the beam can be expressed by the equation:By definition, the singularity function can be
Develop an M-file to determine polar coordinates as described in Prob. 3.6. However, rather than designing the function to evaluate a single case, pass vectors of x and y. Have the function display
Modify the function function odesimp developed at the end of Sec. 3.6 so that it can be passed the arguments of the passed function. Test it for the following case: >> dvdt=@ (v, m, cd) 9.81- (cd/m)
Develop a vectorized version of the following code: tend=20; ni=8; tstart=0; t (1)=tstart; y (1) = 12 + 6*cos (2*pi*t (1) / (tend-tstart)); for i=2:ni+1 t(i)=t (1-1) + (tend-tstart) /ni; y (1) = 10 +
For the hypothetical base-10 computer in Example 4.2, prove that the machine epsilon is 0.05. Problem Statement. Suppose that we had a hypothetical base-10 computer with a 5-digit word size. Assume
The following infinite series can be used to approximate ex:(a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion (Eq. 4.13) with xi = 0 and h = x.(b)
The “divide and average” method, an old-time method for approximating the square root of any positive number a, can be formulated asWrite a well-structured function to implement this algorithm
The derivative of f (x) = 1/(1 − 3x2) is given byDo you expect to have difficulties evaluating this function at x = 0.577? Try it using 3- and 4-digit arithmetic with chopping. 6х (1 - 3x²)²
Develop an M-file function called rounder to round a number x to a specified number of decimal digits, n. The first line of the function should be set up asfunction xr = rounder(x, n)Test the program
Develop an M-file function called rounder to round a number x to a specified number of decimal digits, n. The first line of the function should be set up as function nd = days(mo, da, leap) where mo
Develop an M-file function to determine the elapsed days in a year. The first line of the function should be set up as function nd = days(mo, da, year) where mo = the month (1–12), da = the day
A Cartesian vector can be thought of as representing magnitudes along the x-, y-, and z-axes multiplied by a unit vector (i, j, k). For such cases, the dot product of two of these vectors {a} and {b}
Develop a function function M-file that returns the difference between the passed function’s maximum and minimum value given a range of the independent variable. In addition, have the function
Develop a script to produce a movie for the butterfly plot from Prob. 2.22. Use a particle located at the x-y coordinates to visualize how the plot evolves in time.Data From Problem 2.22Develop a
Based on Example 3.6, develop a script to produce an animation of a bouncing ball where ν0 = 5 m/s and θ0 = 50°. To do this, you must be able to predict exactly when the ball hits the ground. At
Convert the following base-8 numbers to base 10:61,565 and 2.71.
In a fashion similar to Prob. 4.4, develop your own M-file to determine the smallest positive real number used in MATLAB. Base your algorithm on the notion that your computer will be unable to
Convert the following base-2 numbers to base 10:(a) 1011001, (b) 0.01011, and (c) 110.01001.
Prove that Eq. (4.11) is exact for all values of x if f (x) =ax2 + bx + c.
Derive Eq. (4.30). hopt 38 M
Repeat Example 4.5, but for the forward divided difference (Eq. 4.21).Example 4.5 f'(x₁) = f(x₁+1) = f(xi) h + 0(h)
Repeat Example 4.5, but for f (x) = cos(x) at x = /6.Example 4.5 Problem Statement. In Example 4.4, we used a centered difference approximation of O(h²) to estimate the first derivative of the
One common instance where subtractive cancellation occurs involves finding the roots of a parabola, ax2 + bx + c, with the quadratic formula:For cases where b2 >> 4ac, the difference
Develop your own M-file for bisection in a similar fashion o Fig. 5.7. However, rather than using the maximum iterations and Eq. (5.5), employ Eq. (5.6) as your stopping criterion. Make sure to round
Use a centered difference approximation of O(h2) to estimate the second derivative of the function examined in Prob. 4.13. Perform the evaluation at x = 2 using step sizes of h = 0.2 and 0.1. Compare
You buy a $35,000 vehicle for nothing down at $8,500 per year for 7 years. Use the bisect function from Fig. 5.7 to determine the interest rate that you are paying. Employ initial guesses for the
Perform the same computation as in Prob. 5.22, but for the frustrum of a cone as depicted in Fig. P5.23. Employ the following values for your computation: r1 = 0.5 m, r2 = 1 m, h = 1 m, ρf =
The Michaelis-Menten model describes the kinetics of enzyme mediated reactions:where S = substrate concentration (moles/L), νm = maximum uptake rate (moles/L/d), and ks = the half-saturation
Repeat Prob. 5.1, but use the false-position method to obtain your solution.Data From Problem 5.1Use bisection to determine the drag coefficient needed so that an 80-kg bungee jumper has a velocity
Develop an M-file for the false-position method. Test it by solving Prob. 5.1.Data From Problem 5.1Use bisection to determine the drag coefficient needed so that an 80-kg bungee jumper has a velocity
Locate the first nontrivial root of sin(x) = x2 where x is in radians. Use a graphical technique and bisection with the initial interval from 0.5 to 1. Perform the computation until εa is less than
The “divide and average” method, an old-time method for approximating the square root of any positive number a, can be formulated asProve that this formula is based on the Newton-Raphson
Differentiate Eq. (E6.4.1) to get Eq. (E6.4.2).Equation (6.4.1) or (6.4.2) f(m) = gm Cd tanh gcdt-v(t) m
Real mechanical systems may involve the deflection of nonlinear springs. In Fig. P6.20, a block of mass m is released a distance h above a nonlinear spring. The resistance force F of the spring is
Perform the identical MATLAB operations as those in Example 6.8 to manipulate and find all the roots of the polynomialf5(x) = (x + 2) (x + 5) (x − 6) (x − 4) (x − 8)Example 6.8 Using MATLAB to
In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a system’s input to its output. A transfer function for a robotic positioning system is
Employ fixed-point iteration to locate the root of f(x) = sin (√x) − x.Use an initial guess of x0 = 0.5 and iterate until εa ≤ 0.01%. Verify that the process is linearly convergent as
Use (a) fixed-point iteration and (b) the Newton- Raphson method to determine a root of f(x) = −0.9x2 + 1.7x + 2.5 using x0 = 5. Perform the computation until εa is less than εs = 0.01%. Also
Determine the lowest positive root of f(x) = 7 sin(x)e−x − 1:(a) Graphically.(b) Using the Newton-Raphson method (three iterations, xi = 0.3).(c) Using the secant method (three iterations,
Use (a) the Newton-Raphson method and (b) the modified secant method (δ = 0.05) to determine a root of f(x) = x5 − 16.05x4 + 88.75x3 − 192.0375x2 + 116.35x + 31.6875 using an
Employ the Newton-Raphson method to determine a real root for f (x) = −2 + 6x − 4x2 + 0.5x3, using an initial guess of (a) 4.5,(b) 4.43. Discuss and use graphical and analytical methods to
Perform three iterations of the Newton-Raphson method to determine the root of Eq. (E7.1.1). Use the parameter values from Example 7.1 along with an initial guess of t = 3 s.Equation (7.1.1)Example
Given the formula f(x) = −x2 + 8x − 12(a) Determine the maximum and the corresponding value of x for this function analytically (i.e., using differentiation).(b) Verify that Eq. (7.10) yields the
(a) Develop an M-file function to implement Brent’s root-location method. Base your function on Fig. 6.10, but with the beginning of the function changed to Make the appropriate modifications so
An oscillating current in an electric circuit is described by I = 9e−t sin(2πt), where t is in seconds. Determine all values of t such that I = 3.5.
Develop a single script to (a) generate contour and mesh subplots of the following temperature field in a similar fashion to Example 7.4: T (x, y) = 2x2 + 3y2 − 4xy − y − 3x and (b) determine
See if you can develop a foolproof function to compute the friction factor based on the Colebrook equation as described in Sec. 6.7. Your function should return a precise result for Reynolds number
As electric current moves through a wire (Fig. P7.14), heat generated by resistance is conducted through a layer of insulation and then convected to the surrounding air. The steady-state temperature
Use the Newton-Raphson method to find the root off(x) = e−0.5x (4 − x) − 2
Given f(x) = −2x6 − 1.5x4 + 10x + 2 Use a root-location technique to determine the maximum of this function. Perform iterations until the approximate relative error falls below 5%. If you
The head of a groundwater aquifer is described in Cartesian coordinates byDevelop a single script to (a) Generate contour and mesh subplots of the function in a similar fashion to Example 7.4,
Consider the following function: f(x) = 3 + 6x + 5x2 + 3x3 + 4x4Locate the minimum by finding the root of the derivative of this function. Use bisection with initial guesses of xl =
Three matrices are defined as(a) Perform all possible multiplications that can be computed between pairs of these matrices.(b) Justify why the remaining pairs cannot be multiplied.(c) Use the results
Repeat Prob. 7.5, except use parabolic interpolation. Employ initial guesses of x1 = 0, x2 = 1, and x3 = 2, and perform three iterations.Data From Problem 7.5Solve for the value of x that maximizes

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