unj fmipa-fisika

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

6. Penyelesaian: Sempadan atas : u(x, 1) = x 3 -3x+1, u 0,2 = u(0, 1) = 1, u 1,2 = u(1/2, 1) = -0.375 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=-0.375 U=-1

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 |