Chapter 1 Introduction The solutions of engineering problems can be obtained using analytical methods or numerical methods. Analytical differentiation.
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1 Chapter 1 IntroductionThe 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.
2 The procedure for solving a problem Identify the problemState the objectivesDevelop alternative solutionsEvaluate the alternativesImplement the best alternative
3 Analytical VS Numerical Analysis To find minimum ofAnalytical: after differentiate 2x-3the minimum solution occurs at x = 1.5Numerical: define the interval from 1 to 2 increment of 0.2, min falls in 1.4 < x < 1.6To improve the accuracy, we can search in 1.4 < x < 1.6 with a small increment, such as 0.04.
4 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.
5 Taylor Series Taylor series where is the remainder. or First-order approximationSecond-order approximationThird-order approximation
6 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.
7 Example: Nonlinear Polynomials 4th-order polynomialFor a base point x0= 1.0distance h = 0.5