Fisika Matematika II (9 - 11) persamaan-diferensialparsial
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The document discusses partial differential equations, including the Laplace equation, diffusion equation, wave equation, and Poisson equation. It provides an example of solving the Poisson equation on the domain 0<x<1, 0<y<1 using the finite difference method. The example calculates the value at various interior grid points by taking the average of the neighboring values, based on the Poisson equation and specified boundary conditions.
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- 1. Pertemuan 9 - 11 PERSAMAAN DIFERENSIAL PARSIAL Dr. I Made Astra, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam
- 3. u i+1,j - 2u i,j + u i-1,j + u i,j +1 - 2u i,j + u i,j -1 = f i, j h 2 k 2 Utk memudahkan pengiraan ambil h = k, diperolehi u i,j = ¼[u i-1,j + u i+1,j + u i,j-1 + u i,j+1 – h 2 f i, j ] 1 1 1 1 4 u i,j u i-1,j u i+1,j u i,j-1 u i,j+1 -h 2 f i, j 02/02/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
- 5. Penyelesaian: h = a/M = 1/2 k = b/N = 1/2 1 1 1/2 0,0 1/2 Sempadan bawah : u(x, 0) = x 3 u 0,0 = u(0, 0) = 0, u 1,0 = u(1/2, 0) = 0.125 u 2,0 = u(1, 0) = 1 x i = (1/2) i y j = (1/2) j i= 0 i=1 i=2 j=2 j=1 j=0 02/02/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | U=0.125 U=1
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- 7. Penyelesaian: Sempadan kiri : u(0, y) = y, u 0,0 = u(0, 0) = 0, u 0,1 = u(0, 1/2) = 1/2 u 0,2 = u(0, 1) = 1 1 1 1/2 0,0 1/2 i= 0 i=1 i=2 j=2 j=1 j=0 02/02/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
- 8. Penyelesaian: Sempadan kanan : u(1, y) = -2 y+1, u 2,0 = u(1, 0) = 1, u 2,1 = u(1, 1/2) = 0 u 2,2 = u(1, 1) = -1 1 1 1/2 0,0 1/2 i= 0 i=1 i=2 j=2 j=1 j=0 02/02/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | U=0 U=-1
- 9. Penyelesaian: Nak cari u 1,1 u i,j = ¼[u i-1,j + u i+1,j + u i,j-1 + u i,j+1 – h 2 f i, j ] u 1,1 = ¼[ u 0,1 + u 2,1 + u 1,0 + u 1,2 – (1/2) 2 6x 1 ] = ¼[0.5+0+0.125-0.375-0.25(6)(0.5)] = -0.125 U=1 U=1 U=0.125 U=0 U=1/2 i= 0 i=1 i=2 j=2 j=1 j=0 U=-0.375 02/02/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | U=0 U=-1
- 10. TERIMA KASIH 02/02/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |