31 Boundary Value Problems

31 Boundary Value Problems - BVPs 1D Poisson 2D Poisson...

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BVPs 1D Poisson 2D Poisson Boundary-Value Problems Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) Boundary-Value Problems MATH 2070U 1 / 19 BVPs 1D Poisson 2D Poisson Boundary-Value Problems 1 Boundary-value problems 2 Poisson equation in one dimension 3 Poisson equation in two dimensions c ± D. Aruliah (UOIT) Boundary-Value Problems MATH 2070U 2 / 19 BVPs 1D Poisson 2D Poisson Boundary-value problems Boundary-value problem (BVP) consists of differential equation together with boundary conditions (BCs) Independent variable(s) spatial coordinates rather than time ODEs in one dimension, PDEs in two or more dimensions Underlying theory differs from IVPs BVPs ubiquitous in science/engineering applications c ± D. Aruliah (UOIT) Boundary-Value Problems MATH 2070U 4 / 19 BVPs 1D Poisson 2D Poisson Examples Poisson equation in one dimension (Dirichlet BC) ± - u 00 ( x ) = f ( x ) , x ( a , b ) u ( a ) = α , u ( b ) = β Poisson equation in two dimensions (Dirichlet BC) ± -4 u = f ( x , y ) , ( x , y ) Ω u ( x , y ) = g ( x , y ) , ( x , y ) Ω I 4 u : = 2 u x 2 + 2 u y 2 is the Laplacian operator (alternatively 2 u ) I Ω R 2 is some 2D domain I Ω is boundary of Ω I (Laplacian 4 u also written as 2 u ) c ± D. Aruliah (UOIT) Boundary-Value Problems MATH 2070U 5 / 19

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BVPs 1D Poisson 2D Poisson Construction of grid Given interval [ a , b ] , choose integer N > 2 and set h : = b - a N + 1 , x j = a + jh ( j = 0: N + 1 ) Introduce unknown values { u j } N j = 1 such that u j ' u ( x j ) ( j = 1: N ) Use boundary data to set u 0 : = α and u N + 1 = β Numerical solution consists of mesh { x j } N + 1 j = 0 and values
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31 Boundary Value Problems - BVPs 1D Poisson 2D Poisson...

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