Differential Equations Lecture Work Solutions 102

Differential Equations Lecture Work Solutions 102 - 3. See...

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3. See the solution of Helmholtz equation (problem 2 in section 7.4) ψ nm ( x, y )=s in π a ( m + n ) x sin π a ny ( 1) m sin π a ( m + n ) y sin π a nx λ nm = π a ± ( m + n ) 2 + n 2 n, m =1 , 2 , ··· The solution is similar to 1b u ( x, y, t )= X n =1 X m =1 ² a nm cos c ± λ nm t + b nm sin c ± λ nm t ³ ψ nm ( x, y ) f ( x, y )= X n =1 X m =1 a nm ψ nm ( x, y ) g ( x, y )= X n =1 X m =1 c ± λ nm b nm ψ nm ( x, y ) a nm ,b nm
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