Chapter 1 Introduction The solutions of engineering problems can be obtained using analytical methods or numerical methods. Analytical differentiation and integration provide a closed-form derivative and integral, respectively, only for simple functions. Very often analytical solutions cannot be obtained. Numerical methods can be used where analytical methods are not capable of or practical for providing solutions.

The procedure for solving a problem Identify the problem State the objectives Develop alternative solutions Evaluate the alternatives Implement the best alternative

Analytical VS Numerical Analysis To find minimum of Analytical: after differentiate 2x-3 the minimum solution occurs at x = 1.5 Numerical: define the interval from 1 to 2 increment of 0.2, min falls in 1.4 < x < 1.6 To improve the accuracy, we can search in 1.4 < x < 1.6 with a small increment, such as 0.04.

Analytical VS Numerical Analysis Analytical techniques provide a direct solution and will result in exact solution if one exists. Analytical methods are practical only for functions that have a simple, closed-form mathematical structure. Numerical methods can be used with any function. They often require many iterations to get the true solution. The numerical solution usually is not exact, and it is also necessary to provide initial estimates of the unknowns.

Taylor Series Taylor series where is the remainder. or First-order approximation Second-order approximation Third-order approximation

Taylor Series Expansion The importance of the individual terms of the Taylor series depends on the nature of the function and the distance h. The higher-order terms become more important as the nonlinearity of the function increases and the difference h = increases. If the function is linear, only the term with the first derivative is necessary. The error increases as fewer terms of the Taylor series are included.

Example: Nonlinear Polynomials 4th-order polynomial For a base point x0= 1.0 distance h = 0.5

Taylor Series Expansion of the Square Root The square-root function can be expressed as For a base point x0 = 1 and h = 0.001

Example Taylor Series

Example Taylor Series