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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 31 Ordinary Differential Equations

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Fig 23.1 FORWARD FINITE DIFFERENCE

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Fig 23.2 BACKWARD FINITE DIFFERENCE

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Fig 23.3 CENTERED FINITE DIFFERENCE

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Data with Errors

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Pendulum W=mg Ordinary Differential Equation

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ODEs Non Linear Linearization Assume is small

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ODEs Second Order Systems of ODEs

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Application of ODEs in Engineering Problem SOlving

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ODE

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ODE - OBJECTIVES Undetermined

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ODE- Objectives Initial Conditions

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ODE-Objectives Given Calculate

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Runge-Kutta Methods New Value = Old Value + Slope X Step Size

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Runge Kutta Methods Definition of yields different Runge-Kutta Methods

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Euler’s Method Let

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Example

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Euler h=0.5

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Sources of Error Truncation: Caused by discretization Local Truncation Propagated Truncation Roundoff: Limited number of significant digits

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Sources of Error Propagated Local

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Euler’s Method

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Heun’s Method PredictorCorrector 2-Steps

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Heun’s Method Predict Predictor-Corrector Solution in 2 steps Let

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Heun’s Method Correct Corrector Estimate Let

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Error in Heun’s Method

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The Mid-Point Method Remember: Definition of yields different Runge-Kutta Methods

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Mid-Point Method Predictor Corrector 2-Steps

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Mid-Point Method Predictor Predict Let

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Mid-Point Method Corrector Correct Estimate Let

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