10.3.1 Solution Using Microsoft Excel Software

Because the roots of an equation represented by f(x) = 0 are located at the intersections of the function f(x) and the x-coordinate axis as illustrated in Figure 10.1, one may use the widely available tool of spreadsheets (such as Microsoft Office Excel) to locate the roots of the equation by evaluating the function f(x) with for various values of the variable x. The range in which the roots of the equation f(x) = 0 lie can be identified as the values where f(x) change sign from positive to negative or vice versa, as illustrated for the range (x_i−1–x_i+1) in Figure 10.1. More accurate values of the roots may be found by repeating the computation of the function values with smaller increments of the variable x within that range. This method, though sounds tedious, involves straightforward evaluation of the function f(x) with an initially estimated value of x_i by using the Excel software that has enormous computational speed to identify the range in which the roots are located, as will be illustrated by Example 10.1.

10.3.2 The Newton–Raphson Method

Perhaps the most widely used method for finding the roots of nonlinear equations is the Newton–Raphson method. This method offers rapid convergence to the roots of many nonlinear equations from the initially estimated roots. The fast convergence to true roots from the estimated roots is achieved by means of computation of both the function f(x_i) and the corresponding slope f′(x_i) of the function at x_i, as illustrated in Figure 10.3.

Figure 10.3 illustrates the principle of the Newton–Raphson method of solving nonlinear equations. As in many other numerical solution methods, the user has to estimate a root at x = x_i for the equation f(x) = 0, from which they may compute the function f(x_i) and at the same time the slope of the curve generated by the function f(x). This slope may be expressed f′(x_i). The graphical representation of this situation indicates that the slope f′(x_i) may be expressed by Equation 10.4:

10.4

which leads to the following expression for the next estimated root at x = x_i+1:

10.5

One readily sees from Figure 10.3 that the computed approximate next root x_i+1 is much closer to the real root (shown as a solid circle) than the previous estimated value at x_i.

10.4.1 The Trapezoidal Rule for Numerical Integration

We learned in Section 2.2.6 that the value of a definite integral of a function y(x) is equal to the area under the curve produced by this function between the upper and lower limits of the integration as illustrated in Figure 10.6. Mathematically, the integral of function y(x) can be expressed as

10.6

The value of the integral of a function may thus be determined by computing the area covered by the function between the two specified limits. For example, the value of the function y(x) in Figure 10.7 may be approximated by the sum of the three trapezoidal plane areas A₁, A₂, and A₃.

The area of a trapezoidal plane may be evaluated by the formula of half of the sum of the length of two parallel sides multiplied by the distance between these two sides. Mathematically, the plane areas A₁, A₂, and A₃ with equal distance h between two parallel sides in Figure 10.7 may be computed by the following formulae:

The sum of A₁, A₂,and A₃ is given by

10.7

in which h is the assigned increment of variable x, and y₀, y₁, y₂, and y₃ are the values of the function evaluated at x₀, x₁, x₂, and x₃, respectively.

This approximate value of the integral I obtained by the above formula is less than the analytical solution of 191.45 obtained from a mathematical handbook (Zwillinger, 2003). The difference between the value of the integral of 180.92 obtained using the trapezoidal rule and that of 191.45 from the analytical solution method is to be expected. This difference in results represents the errors inherent in any numerical method. In the trapezoidal rule method in this example, the discrepancy is introduced by the approximation of the curve representing the given function by straight line segments (shown dashed) that were the edges of the trapezoids in computing the approximate area under the curve in Figure 10.8. One may readily observe that this discrepancy between the curved edges and the straight edges shown as dashed lines can be reduced by the reduction of the size of the increment h, as can be observed from the graphic illustration in Figure 10.8. Consequently, closer approximation, or more accurate results from numerical integration, may be achieved by reducing the size of increment h in this particular numerical analysis.

Figure 10.9 shows the area under function y(x), which can be approximated as the sum of the areas of (n − 1) trapezoids. Many more trapezoids under the function curve such as shown in Figure 10.9 entails many more increments of h along the x-coordinate between the upper and lower integration limits, with concomitant more accurate results.

The approximate value of the integral of function y(x) in Figure 10.9 is equal to the sum of all the trapezoids in the figure according to the following equation derived from the same principle as in the earlier case with three trapezoids:

10.8

where h is the increment along the x-coordinate axis in the numerical integration.

This value of 188.88 for the same integral now with h = 0.5 is much closer to the analytical value of 191.45 than that of 180 obtained with h = 1.0 in Example 10.5. It has thus been demonstrated the fact that the smaller the increment one uses in numerical integration, the more accurate a result will be obtained.

10.4.2 Numerical Integration by Simpson's One-third Rule

In Section 10.4.1, we refreshed the principle of evaluating an integral as the plane area under the curve representing the function (the integrand) in the integral between two limits of the variable in the integration. This area is graphically expressed in Figure 10.11a.

The value of an integral may be obtained by numerical methods such as the trapezoidal rule described in Section 10.4.1. The trapezoidal method is a simple but a relatively “crude” method that will result in an approximate value of the integral with minimal computational effort. Graphical representation of the trapezoidal method is illustrated in Figure 10.11b for the simplest possible numerical approximation of the area under the curve of y(x) as the area of one trapezoid (shown as the cross-hatched area) under the curve. This method is popular and easy to use because the formula used to compute the plane area of trapezoids is well known to engineers.

Another popular numerical method for integration is by Simpson's rule, in particular, “Simpson's one-third rule.” This rule differs from the trapezoidal method by assuming that the function y(x) can be approximated in the range of interest by a parabolic function as shown in Figure 10.11c. The function y(x) = ax² + bx + c that describes the sector of the function in Figure 10.11c contains the unknown constant coefficients a, b, and c which can be determined with the following simultaneous equations relating to the function values at the discrete variable values x = −x , x = 0, and x = +x as follows:

from which we may solve for

10.9a

10.9b

The value of the integral of the function y(x) is equal to the plane area A in Figure 10.11c, or

By substituting the constant coefficients a and c in Equations 10.9a and 10.9b, together with Δx = x illustrated in Figure 10.11c, into the above expression, we get the following relation for Simpson's one-third rule for the integral I:

10.10

The analytical or “exact” result of the above integral is 191.45 from a table of definite integrals (Zwillinger 2003), which we may compare with the results I = 188.88 using three trapezoids in Example 10.5 and I = 194.70 with three function values using Simpson's one-third rule.

We will use the same illustration in Figure 10.13 and Equation 10.10 to derive the general expression for Simpson's one-third rule for numerical integration.

We derived Equation 10.10 with the first three function values y₀, y₁ and y₂ at x = x₀, x = x₁, and x = x₂, respectively, with an assumed parabolic function connecting these three points. The next parabolic function for the next adjoining three-point segment requires that the function value y₂ be evaluated twice. The same happens to the second last point at x_n−1 in the region for the integration at which the function value y_n−1 in Figure 10.13 being evaluated twice. The coefficients associated with the y_i for n points in Figure 10.13 are given in the following tabulation:

We may thus formulate the general expression of Simpson's one-third rule as follows:

10.11

We note from Equation 10.11 that the use of this relationship for Simpson's one-third rule requires even number of function values with odd number increments in the integration variable.

10.4.3 Numerical Integration by Gaussian Quadrature

Numerical integration of functions using the trapezoidal rule (Section 10.4.1) and Simpson's one-third rule (Section 10.4.2) enables us to determine approximate values of integrals of continuous functions f(x) over a range of (x_b–x_a) into equal parts with increment of the variable Δx (or h) as shown in Figures 10.7 and 10.9. This process allows us to select the sampling points and evaluate the integral in terms of the discrete values of the function at these points. These methods usually work well with well-behaved functions in integrals such as the ones used in Examples 10.5 to 10.8. However, neither the trapezoidal rule nor Simpson's one-third rule offers any guidance on the selection of the size of the increment of the variable in these numerical integration methods. There are times when engineers are expected to find numerical values of integrals involving functions that have drastic changes of shape over the range of the required integration. The two methods already discussed do not yield good approximations of the numerical values of these integrals because of improper selection of sampling points. Thus, it is desirable to have a numerical integration method that offers criteria for optimal sampling points in numerical integration.

The Gaussian integration method was established on the basis of strategically selected sampling points. The normal form of a Gaussian integral can be expressed as

10.12

in which n is the total number of sampling points, and H_i are the weighting coefficients corresponding to sampling points located at as given in Table 10.3.

The form of Gaussian integral shown in Equation 10.12 is rarely seen in practice. A transformation of coordinate is required to convert the general form of integration such as shown in Equation 10.6 to the form shown in Equation 10.12, as illustrated in Figure 10.15.

The transformation of coordinates from y(x) in the x-coordinate to the function F(ξ) in the coordinate ξ may be accomplished using the relationship

10.13

which leads to the following expression:

10.14

with

and

We will obtain the expression for the required evaluation of the integral in Equation 10.6 using Gaussian quadrature by substituting the relationship in Equation 10.12 into Equation 10.14.

10.15

Numerical values of the weight coefficients H_i with sampling points a_i are given in Table 10.3.

One needs to realize that the term F(a_i) on the right-hand-side of Equation 10.15 denotes the function y(x) evaluated at the sampling points a_i after it has been transformed to the function F(ξ) with integration limits of (−1 to +1) as in Figure 10.15b.

10.5.1 The Principle of Finite Difference

We have learned in Chapter 2 that differential equations are equations that involve derivatives. Physically, a derivative represents the rate of change of a physical quantity represented by a function with respect to the change of its variable(s).

Referring to Figure 10.16, we have a continuous function f(x) that has values f_i−1, f_i, and f_i+1 corresponding to the three values of its variable x at x_i−1, x_i, and x_i+1, respectively. We may also write the three function values at the three x-values as

10.16a

10.16b

10.16c

The derivative of the function f(x) at point A with x = x_i in Figure 10.16 is graphically represented by the tangent line A″–A′ to the curve representing function f(x) at point A. Mathematically, we may express the derivative as given in Equation (2.9), or in the form

10.17

where Δx is the increment used to change the values of the variable x, and dx is the increments of variable x with infinitesimally small sizes.

One may observe from Equation 10.17 the important relation that the derivative may be approximated by finite increments of Δf and Δx as indicated in Equation 10.18:

10.18

if the condition on the increment Δx→0 is removed in the evaluation of the rate of change of the function f(x) in Equation 10.17.

Thus from Equation 10.17 we see that derivatives of functions may be approximated by adopting finite—but not infinitesimally small—increments of the variable x. A formulation with such an approximation is called “finite difference”.

10.5.2 The Three Basic Finite-difference Schemes

There are three basic schemes that one may use to approximate a derivative: (1) the “forward difference” scheme; (2) the “backward difference” scheme; and (3) the “central difference” scheme. Mathematical expressions of these difference schemes are given below.

1. The forward difference scheme

In the “forward difference scheme,” the rate of change of the function f(x) with respect to the variable x is accounted for between the function value at the current value x = x_i and the value of the same function at the next step, that is, x_i+1 = x + Δx in the triangle ΔA′Aa′ in Figure 10.16. The mathematical expression of this scheme is given in Equation 10.19:

10.19

in which h = Δx is the “step size.”

The derivative of the function f(x) at other values of the variable x in the positive direction can be expressed following Equation 10.19 as

10.20

The second-order derivative of the function f(x) at x can be obtained according to the following procedure:

10.21

The backward difference scheme

In this difference scheme, the rate of the change of the function with respect to the variable x is accounted for between the current value at x = x_i and the step backward, that is, x_i−1 = x − Δx in the triangle ΔAA″a″ in Figure 10.16. The mathematical expression of this scheme is given in Equation 10.22:

10.22

Following a similar procedure as in the forward difference scheme, we may express the second-order derivative in the following form:

10.23

The central difference scheme:

The rate of change of function f(x) in this finite-difference scheme includes the function values between the preceding step at (x − Δx) and the step ahead, that is, (x + Δx). The triangle involved in this difference scheme is ΔA′A″a‴ in Figure 10.16. We have the first-order derivative as in Equation 10.24:

10.24

Note that in this finite-difference scheme a much larger step of size 2h is used in the first-order derivative as given in Equation 10.24. These “coarse” steps will compromise the accuracy of the values of the derivatives. A better central difference scheme is to employ for “half” steps in both directions. In other words, if we define

10.25

we will then have the derivative of the function f(x) using this modified central difference scheme as

10.26

One will observe from the tabulated values that the percentage error of the results obtained from the finite-difference method increases with the increase of the variable t. One may also show that the accuracy of the finite-difference method improves with smaller increments of the variables—the Δt in the present example.

10.5.3 Finite-difference Formulation for Partial Derivatives

Partial derivatives along a single dimension are computed in the same fashion as for ordinary derivatives (Chapra 2012) illustrated in Section 10.5.2. For example, the central difference scheme for function f(x,y) can be shown to have the following expressions:

10.27a

10.27b

The error of the above approximation is of order Δx = h.

For higher-order partial derivatives, such as :

10.28

Evaluating each of the partial derivatives in Equation 10.28 will lead to the following final expression:

10.29

The error of the approximation of second-order differentiation is of order (Δx)².

10.5.4 Numerical Solution of Differential Equations

There are a number of numerical solution methods available for solving differential equations relating to both types of initial value and boundary value problems (Hoffman, 1992; Bronson, 1994; Ferziger, 1998; Chapra, 2012). We will present only the expressions used in the classic fourth-order Runge–Kutta method to illustrate the power of numerical methods for solving differential equations for initial value problems.

The simplest method of solving differential equations is to convert the derivatives in differential equations to the forms of “finite differences” as presented in Section 10.5.2 and Example 10.12. This method is straightforward but usually has significant accumulation of errors in the solution, as indicated in the numerical illustration in Example 10.12. There are several alternative versions available for solutions with better accuracies in the references cited above. We will include the Runge–Kutta method in this section for numerical solution of differential equations using initial value processes.

The Runge–Kutta methods are integrative methods for approximation of solutions of differential equations. These method, with several versions of the technique, were developed around 1900s by German mathematicians C. Runge and M.W. Kutta. The essence of the Runge–Kutta methods involves numerical integration of the function in a differential equation by using a trial step at mid-point of an interval—within a step Δx or h—using numerical integration techniques such as the trapezoidal or Simpson's rules as presented in Section 10.4. The numerical integrations will allow the cancellation of low-order error terms for more accurate solutions. Several versions of Runge–Kutta methods with different orders for solving differential equations have been developed over the years.

10.5.4.1 The Second-order Runge–Kutta Method

This is the simplest form of the Runge–Kutta method, with the formulation for the solution of first-order differential equation in the following form:

10.30

with a specified solution point corresponding to one specific condition for Equation 10.30. The solution points of this differential equation can be expressed as

10.31

where O(h³) is the order of error of the step h³, and

10.32a

10.32b

10.5.4.2 The Fourth-order Runge–Kutta Method

This is the most popular version of the Runge–Kutta method for solving differential equations in initial value problems. Formulation of this solution method is similar to that of the second-order method.

The differential equation is similar to that shown in Equation 10.30:

with the solution point given by the following formula:

10.33

where

10.34a

10.34b

10.34c

10.34d

10.5.4.3 Runge–Kutta Method for Higher-order Differential Equations

We have seen that Runge–Kutta method can solve differential equations often with remarkable accuracy as demonstrated in Example 10.14. Unfortunately, most textbooks offer the application of this valuable method only for solving first-order differential equations. Its use for solving higher- order differential equations requires the conversion of higher-order differential equations to the first-order-equivalent forms such as that shown in Equation 10.30. The solution of the converted higher-order differential equations can be obtained using expressions such as that given in Equation 10.33 for the fourth-order Runge–Kutta formulation. We will present the following formulation to illustrate how the fourth-order Runge–Kutta method can be used to solve second-order ordinary differential equations (the treatment is derived from an online tutorial http://www.eng.colostate.edu/∼thompson/Page/CourseMat/Tutorials/CompMethods/Rungekutta.pdf).

We will use the Runge–Kutta method to solve a second-order ordinary differential equation of the form

10.35

The left-hand side of Equation 10.35 may be expressed as dy′(x)/dx, which thus converts the second-order differential equation in (10.35) into a first-order differential equation in the form

10.36a

with

10.36b

Solution of the second order differential equation in Equation 10.35 may be obtained by the solutions of Equations 10.36a and 10.36b using the fourth-order Runge–Kutta formulations given in Equations 10.37 and Equation 10.38:

10.37

and

10.38

We note that the expression in Equation 10.38 is similar to that in Equation 10.33 for the converted first-order differential equation in Equation 10.36a.

The coefficients f₁, f₂, f₃, and f₄ in Equations 10.37 and 10.38 can be obtained from the expressions given in Table 10.4

The three solution points at x = 0.2, 0.1, and 0.4 obtained by using the Runge–Kutta method appeared tedious and time-consuming. The same differential equation with the same specified conditions was solved using the software package MatLAB, with the input/output information included in Case 3 of Appendix 4. The results so obtained were remarkably accurate, with solution at the same three points being the exact values as shown tabulated above. MatLAB also offers graphical output such as that shown in Figure 10.17. An overview description of MatLAB software will be presented in Section 10.6.2.

This numerical solution has an error of 0.48% from the exact solution, and it is more accurate than that obtained from the simple forward difference scheme in Example 10.12.

10.6.1 Introduction to Mathematica

Mathematica is a computational software program based on symbolic mathematics. It is used in many scientific, engineering, mathematical, and computing fields. The programming languages used in Mathematica are the Wolfram Language by Stephen Wolfram, together with C, C++, and Java. This software package has been in the marketplace since June 1988. It is capable of handling the following major functions and features in engineering analysis:

1. 1. Determining roots of polynomial equations of cubic or higher orders.
2. 2. Integrating and differentiating complicated expressions. For instance, the integral of the function sin(x³), for which ; in contrast, similar integrals with the simple function sin(x) but with higher power of the function in the integral, such as , are available from integration tables in mathematical handbooks (Zwillinger, 2003).
3. 3. Solving linear and nonlinear differential equations.
4. 4. Elementary and special mathematical function libraries.
5. 5. Matrix and data manipulation tools.
6. 6. Numeric and symbolic tools for discrete and continuous calculus.

It can also solve the following common analytical engineering problems:

1. 1. Determining Laplace and Fourier transforms of functions.
2. 2. Generating graphics in two and three dimensions.
3. 3. Simplifying trigonometric and algebraic expressions.

According to the Wolfram Language and Systems Documentation Center, Mathematica has the following features and capabilities that are of great value in advanced engineering analyses:

• Support for complex number, arbitrary precision, interval arithmetic, and symbolic computation.
• Solvers for systems of equations, Diophantine equations, ODEs, PDEs, and so on.
• Multivariate statistics libraries including fitting, hypothesis testing, and probability and expectation calculations on over 140 distributions.
• Calculations and simulations on random processes and queues.
• Computational geometry in 2D, 3D, and higher dimensions.
• Finite-element analysis including 2D and 3D adaptive mesh generation.
• Constrained and unconstrained local and global optimization.
• Toolkit for adding user interfaces to calculations and applications.
• Tools for 2D and 3D image processing and morphological image processing including image recognition.
• Tools for visualizing and analyzing directed and undirected graphs.
• Tools for combinatorics problems.
• Data mining tools such as cluster analysis, sequence alignment, and pattern matching.
• Group theory and symbolic tensor functions.
• Libraries for signal processing including wavelet analysis on sounds, images, and data.
• Linear and nonlinear control systems libraries.
• Continuous and discrete integral transforms.
• Import and export filters for data, images, video, sound, CAD, GIS, documents, and biomedical formats.
• Database collection for mathematical, scientific, and socioeconomic information and access to Wolfram alpha data and computations.
• Technical word processing including formula editing and automated report generation.
• Tools for connecting to DLL, SQL, Java, .NET, C++, Fortran, CUDA, OpenCL, and http based systems.
• Tools for parallel programming.
• Mathematica language in notebook computers when connected to the Internet.

The last of the features listed is of particular value to engineers. For example, in Example 10.3 we were required to find the root of the cubic equation . A meaningful root of this equation found by the Newton–Raphson method was as shown in Example 10.3. A similar solution of was obtained by the solution method offered via the internet at the Wolfram/Alpha Widgets website (www.wolframalpha.com/widgets/ ) with user input of the coefficients of this equation. It offered an instant solution and with an excellent user interface feature.

10.6.2 Introduction to MATLAB

MATLAB is an acronym of “matrix laboratory.” This numerical analysis package was designed by Cleve Moler in the late 1970s with an initial release to the public in 1984. The latest version, Version 8.6 was released in September 2015.

MATLAB provides a multiparadigm numerical computing environment and fourth-generation programming language, a proprietary programming language developed by MathWorks. It allows matrix manipulations, plotting of functions and data, implementation of algorithms, and creation of user interfaces that include interfacing with programs written in other languages, including C, C++, Java, Fortran, and Python. It is a popular numerical analysis package mainly because of it has graphics and graphical user interfacing programming capability.

Like Mathematica, MATLAB is capable of handling the following common problems in engineering analysis (Malek-Madani, 1998):

1. 1. Finding roots of polynomials, summing series, and determining limits of sequences.
2. 2. Symbolically integrating and differentiating complicated expressions.
3. 3. Plotting graphics in two and three dimensions.
4. 4. Simplify trigonometric and algebraic expressions,
5. 5. Solving linear and nonlinear differential equations.
6. 6. Determining the Laplace transforms of functions.

Additionally it can handle a variety of other mathematical operations.

Operation of MATLAB requires the user to input simple programs for the solution of the problems. These programs usually consists of three “windows:” (1) the “command window” for the user to enter commands and data; (2) the “graphics window” to display the results in plots and graphs; and (3) the “edit window” to create and edit the M-files, which provide alternative ways of performing operations that can expand MATLAB's problem-solving capabilities.

Detailed instructions for using MATLAB for solving a variety of mathematical problems are available in MATLAB Primer published by MathWorks, Inc.(see www.mathworks.com) and two excellent references (Malek-Madani, 1998 Chapra, 2012). Appendix 4 will present the procedure for the input/output (I/O) of three cases of engineering analysis using the MatLAB package:

1. Case 1: Graphic solution of the amplitudes with “beats” offered by the solution in Equation (8.40) for the near-resonant vibration of a metal stamping machine.
2. Case 2: The numerical solution with graphic representations of the amplitudes and mode shapes of a flexible rectangular pad subjected to transverse vibration as presented in Equation (9.76).
3. Case 3: The solution with graphic representation of a nonhomogeneous second-order differential equation.

These cases will demonstrate the value of the MatLAB software package in solving complicated engineering analysis problems.