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Unformatted text preview: 1 Introduction Numerical methods are mathematical techniques for solving mathematical prob- lems which are either difficult or impossible to solve analytically . An analytical solution is an exact answer which is given in a form of a mathematical expression which is in terms of the variables associated with the given problem. For exam- ple in your algebra and calculus classes, typically solving a quadratic equation or integrating a function involved methods which resulted in analytical solutions. On the other hand, a numerical solution is an approximate numerical value (number) for the solution. While numerical solutions are approximate, they can be very accurate. Often times the accuracy of the solution is dictated by the analyst (you in this course). For many problems numerical solutions are found in an iterative manner until a desired solution accuracy is obtained. You will encounter this concept often in this course, particulary when dealing with prob- lems which are nonlinear . The techniques for numerical methods were, most often, developed hundreds of years ago. However, as the solutions often involve many thousands or millions of calculations, only within the last 50 or so years since the advent of the digital computer have such methods been extensively used in engineering and science. 1.1 Solving a problem in science and engineering The process of solving problems in science and engineering can be split into four parts Problem Statement- Defines the problem. Gives a description of the prob- lem, lists the variables that are involved and identifies the constraints. When dealing with problems which involve differential equations, these contraints take on the form of boundary and/or initial conditions. Often times this part of the solution process (along with the next part) is the most difficult and requires the insight of a trained engineer or scientist ( YOU!! ) to decide what sort of assumptions can be made to the problem to simplify its numerical solution while still giving adequate insight into the problem. Formulation of solution- Consists of the model (physical law or laws) that is used to represent the problem and derivation of the governing equations that need to be solved. Newtons laws, conservation laws and the laws of thermodynamics are all examples of such laws. If analytical methods are to be used to solve the problem, the governing equations must be such that they admit analytical solutions and if they dont then they will have to be simplified such that the equations can be solved analytically. If, instead, numerical methods are to be used, the equations can be more complicated. However, simplification is still often necessary if one wants to obtain a solution in a reasonable amount of time....
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- Spring '09