Unformatted text preview: H ( x ) is not di±erentiable) • Show that the distributional integral i x −∞ g ( y ) δ ( y-x ) dy is given by g ( x ) H ( x-x ). 5. (parts of of # 9.3.6) Solve for the Green’s function directly: d 2 G dx 2 = δ ( x-x ) , x ∈ (0 ,L ) , G (0 ,x ) = dG dx ( L,x ) = 0 , and compare your result to that of problem 2 above. 6. (parts of # 9.3.11) Considering the Helmholtz equation and assuming that L is no multiple of π , solve for the Green’s function directly: d 2 G dx 2 + G = δ ( x-x ) , x ∈ (0 ,L ) , G (0 ,x ) = G ( L,x ) = 0 . Show that G is symmetric: G ( x,x ) = G ( x ,x ). (Note: use the variation of parameters method to solve for G directly, and use the rules of distributional integration; for instance, you might ²nd that the results of problem 4 come in handy.) * MAP 4341; Instructor: Patrick De Leenheer. 1...
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- Summer '06
- Derivative, Dirac delta function, Patrick De Leenheer