mase201sp09_Apr07_PRINTABLE

mase201sp09_Apr07_PRINTABLE - Numerical Solution of...

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Numerical Solution of Ordinary Diff Eqns • Intro to numerical solution of Ordinary Differential Equations (ODEs): – Approaches • Problem statement • Explicit methods plicit methods • Implicit methods – Euler methods xplicit ethod Explicit method • Error propagation in explicit method • Implicit method • Runge-Kutta methods • MATLAB ODE solvers (sec 10.4) 4/07/2009 1 • Solving systems of first-order ODEs w/ MATLAB MASE201 Spring 2009 Ordinary Differential Equations: Terminology rdinary one independent variable • Ordinary : one independent variable • Order : determined by highest derivative that ppears in the equation e g y appears in the equation, e.g. d y /d x = f ( x,y ) is a first-order differential equation inear linear dependence on nd its • Linear : linear dependence on y and its derivatives. The “canonical form” for a linear 1 st order ODE is : y omogeneous right- and side, = 0     ) ( 1 2 x r y x a dx dy x a Homogeneous : right hand side, r ( x ) 0 4/07/2009 2 MASE201 Spring 2009 Boundary Conditions and Initial Conditions hen solving an ODE arbitrary constants • When solving an ODE, arbitrary constants appear in the solution. o find the values of these constants the • To find the values of these constants, the value of the solution of f ( x ) and its derivatives p to the order of the ODE must be fixed up to the order of the ODE 1 must be fixed • When the independent variable is time, the ODE is called an initial value problem and we p need to specify y (0), d y /d t (0) etc. • When the independent variable is position (e.g. x, y, or z ) the ODE is called a boundary value problem 4/07/2009 3 MASE201 Spring 2009 Example of a 1 st order, non-linear ODE tank is filled at a time arying • A tank is filled at a time-varying mass flow rate: m     ole out ut in in gh A dm t Q dt dm t Q 2  hole in out in hole out gh A t Q dt dm dt dm dt dm dt 2 hole in tank gh A t Q t dh h A m 2 1 tank in tank hole tank A t Q gh A A dt dh A dt 2 4/07/2009 4 MASE201 Spring 2009 Figure 8-1 from Gilat, “Numerical Methods for Engineers and Scientists”
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Analytic Solution to Example ODE omogeneous solution i e 0 • Homogeneous solution, i.e. Q in ( t ) = 0 gh A t dh hole 0 2 dt g A A dh h A dt tank hole tank 2 2 / 1 t A h h d g A A d hole t tank hole h h 2 0 2 2 2 2 / 1 2 / 1 0 ) 0 ( 2 / 1 h t g A A h g A nk hole tank ) 0 ( 2 2 1 ) ( 2 2 / 1 h g A A g A A h t g A A dt dh tank hole tank hole tank hole tank 2 2 2 1 ) 0 ( 2 2 1 2 2 / 1 4/07/2009 5 MASE201 Spring 2009 Graph of Homogeneous Solution lot for different ratios of the area of the hole • Plot for different ratios of the area of the hole to the area of the tank:
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mase201sp09_Apr07_PRINTABLE - Numerical Solution of...

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