Differential Equations Lecture Work Solutions 308

Differential Equations Lecture Work Solutions 308 - 2. The...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
2. The DuFort Frankel method for solving the heat equation requires solution of the di±er- ence equation u n +1 j u n 1 j 2∆ t = α (∆ x ) 2 u n j +1 u n +1 j u n 1 j + u n j 1 Develop the stability requirements necessary for the solution of this equation. Let r = α t (∆ x ) 2 then DuFort Frankel method can be written as u n +1 j u n 1 j =2 r u n j +1 u n +1 j u n 1 j + u n j 1 or (1 + 2 r ) u n +1 j r u n j +1 + u n j 1 +(1 2 r ) u n 1 j Use λ n e ik m j x λ n +1 (1 + 2 r ) 2 r e ik m x e ik m x | {z } 2cos β λ n (1 2 r ) λ n 1 =0 where β = k m x . This leads to a quadratic equation for λ (1 + 2 r ) λ 2 4 r cos βλ (1 2 r )=0 and the solutions are
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

Ask a homework question - tutors are online