chap10 - Boundary Value Problems Numerical Methods for BVPs...

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Unformatted text preview: Boundary Value Problems Numerical Methods for BVPs Scientific Computing: An Introductory Survey Chapter 10 – Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 45 Boundary Value Problems Numerical Methods for BVPs Outline 1 Boundary Value Problems 2 Numerical Methods for BVPs Michael T. Heath Scientific Computing 2 / 45 Boundary Value Problems Numerical Methods for BVPs Boundary Values Existence and Uniqueness Conditioning and Stability Boundary Value Problems Side conditions prescribing solution or derivative values at specified points are required to make solution of ODE unique For initial value problem, all side conditions are specified at single point, say t For boundary value problem (BVP), side conditions are specified at more than one point k th order ODE, or equivalent first-order system, requires k side conditions For ODEs, side conditions are typically specified at endpoints of interval [ a, b ] , so we have two-point boundary value problem with boundary conditions (BC) at a and b . Michael T. Heath Scientific Computing 3 / 45 Boundary Value Problems Numerical Methods for BVPs Boundary Values Existence and Uniqueness Conditioning and Stability Boundary Value Problems, continued General first-order two-point BVP has form y = f ( t, y ) , a < t < b with BC g ( y ( a ) , y ( b )) = where f : R n +1 → R n and g : R 2 n → R n Boundary conditions are separated if any given component of g involves solution values only at a or at b , but not both Boundary conditions are linear if they are of form B a y ( a ) + B b y ( b ) = c where B a , B b ∈ R n × n and c ∈ R n BVP is linear if ODE and BC are both linear Michael T. Heath Scientific Computing 4 / 45 Boundary Value Problems Numerical Methods for BVPs Boundary Values Existence and Uniqueness Conditioning and Stability Example: Separated Linear Boundary Conditions Two-point BVP for second-order scalar ODE u 00 = f ( t, u, u ) , a < t < b with BC u ( a ) = α, u ( b ) = β is equivalent to first-order system of ODEs y 1 y 2 = y 2 f ( t, y 1 , y 2 ) , a < t < b with separated linear BC 1 y 1 ( a ) y 2 ( a ) + 1 y 1 ( b ) y 2 ( b ) = α β Michael T. Heath Scientific Computing 5 / 45 Boundary Value Problems Numerical Methods for BVPs Boundary Values Existence and Uniqueness Conditioning and Stability Existence and Uniqueness Unlike IVP, with BVP we cannot begin at initial point and continue solution step by step to nearby points Instead, solution is determined everywhere simultaneously, so existence and/or uniqueness may not hold For example, u 00 =- u, < t < b with BC u (0) = 0 , u ( b ) = β with b integer multiple of π , has infinitely many solutions if β = 0 , but no solution if β 6 = 0 Michael T. Heath Scientific Computing 6 / 45 Boundary Value Problems Numerical Methods for BVPs...
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chap10 - Boundary Value Problems Numerical Methods for BVPs...

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