Boundary Element 1

Boundary Element 1 - Boundary Integral Equation for...

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Boundary Integral Equation for Laplace’s Equation We wish to solve the following boundary value problem 2 1 2 and 0 S q n u q S u u R u = = = = (1) This can be accomplished in a somewhat different scheme by using some standard methods of applied mathematics. Using the divergence theorem, we can write ∫∫ ∫∫∫ = S R dS dV n w w (2) Letting ) ( and 2 g f g f g f g f = + = = n n w w w and using these results in relation (2) gives ∫∫ ∫∫∫ = + S R dS n g f dV g f g f ) ( 2 (3) which is commonly called Green’s first formula or identity . Next, interchanging f and g in equation (3) and subtracting this result from (3) results in ∫∫ ∫∫∫ = S R dS n f g n g f dV f g g f ) ( 2 2 (4) Relation (4) is sometimes known as Green’s second formula or identity . Now getting back to our original problem (1), let
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Boundary Element 1 - Boundary Integral Equation for...

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