lecture14 - Numerical Methods in Engineering ENGR 391...

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Unformatted text preview: Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Ordinary differential equations Chapter 8 and 9 Ordinary Differential Equations • Equations which are composed of an unknown function and its derivatives are called differential equations . • Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change. v- dependent variable t- independent variable v m c g dt dv − = • When a function involves one dependent variable, the equation is called an ordinary differential equation (or ODE). A partial differential equation (or PDE) involves two or more independent variables. • Differential equations are also classified as to their order. – A first order equation includes a first derivative as its highest derivative. – A second order equation includes a second derivative. • Higher order equations can be reduced to a system of first order equations, by redefining a variable. Ordinary Differential Equations Ordinary Differential Equations • Ordinary (O.D.E.) = all derivatives are with respect to single independent variable (often representing time) • An O.D.E. by itself does not determine an unique solution function. The solution can be represented by a family of curve . y’ = y Solution: y = α e t Initial value problem • Initial value problem = solution to a differential equation that satisfies a given initial condition y ( t o ) = y o • In this chapter, we will study numerical method to approximate the solution for: ) , ( y t f dt dy = for a ≤ t ≤ b Subject to an initial condition y ( t o ) = y o Euler method (tangent line method) Derive from Taylor’s theorem Expand the solution y ( t ) about t i ) , ( y t f y = ′ ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 2 1 1 2 1 1 1...
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This note was uploaded on 07/11/2011 for the course ENGR 391 taught by Professor Hoidick during the Winter '09 term at Concordia Canada.

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lecture14 - Numerical Methods in Engineering ENGR 391...

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