Lec14 - Numerical Solution of Boundary Value ODEs...

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Numerical Solution of Boundary Value ODEs Elementary Methods: Shooting Method and Finite Differences
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Lecture 14 2 Boundary Value ODEs z Boundary Value Problems ODE + Boundary Conditions = BV Problem z Many steady state problems in transport phenomena and reaction engineering can be reduced to BV problems z A generic 2 nd Order BV problem: Where f(x,y) is in general a nonlinear function of both x and y(x) One boundary condition is specified at both x=a and x=b Common examples: “Dirichlet” BCs “Neumann” BCs “Robbins” or “Mixed” BCs
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Lecture 14 3 A Steady-State Heat Transport Example z Consider a non-insulated metal bar suspended in air and in contact with two thermal reservoirs at each end (T 1 < T 2 ): T 1 T 2 x=0 x=L T a z A Dirichlet BV problem for temperature distribution along the rod (Fourier’s law + Newton’s law of cooling): x
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Lecture 14 4 The Shooting Method z One approach that can be taken to solve such problems is the Shooting Method z The basic idea is: BV Problem = IV Problem + Root Finding Problem z We start by replacing our 2 nd order ODE with two 1 st order ODEs: z Approach: guess v(a)=v 0 then integrate IV system from x=a to x=b . If y(b)=y 1 , stop; otherwise iterate v 0 to solve for the root of:
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Lecture 14 5 Heat Transfer Example: The Equations z Lets try this out for our heat transfer problem with L=10, h/k=0.01, T(0)=40, T(10)=200, T a =20 z Call y 1 (x) =T(x), y 2 (x) = dT(x)/dx
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Lecture 14 6 Heat Transfer Example: Shooting to Bracket Solution z Call y 1 (x) =T(x), y 2 (x) = dT(x)/dx z We need some initial guesses of y 2 (0) that when integrated forward in x will lead to y 1 (10) values that bracket the target 200 z Lets try: y 2 (0) = 10 y 2 (0) = 20 z See scriptL14.m target: 10 20
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Lecture 14 7 Heat Transfer Example: Root-finding + shooting to find solution z Now that we have bracketed the initial slope y 2 (0)
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Lec14 - Numerical Solution of Boundary Value ODEs...

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