Problemi d'esame ed esercizi di Equazioni alle Derivate Parziali

630. Fourier equazione di LaplaceconInfine:γ 0n =γ 3n = 4γ 0n(2n+1) 2π +γ 2n, γ 4n =[22n+14γ 0n(2n+1) 2 +γ 1n4π(2n+1) , γ 1n = 8(−1)nπ(2n+1) 2 , γ 2n = 16(−1)n(2n+1) 2 − 32π(2n+1) 3 .].12. [20/4/2006 (ex)II] Risolvere con il metodo di FourierR.ove:econInfine:∆u = y +1, 0 < x < π,0 < y < π,u x (0,y) = 0, 0 < y < π,u(π,y) = 0, 0 < y < π,u y (x,0) = x 2 , 0 < x < π,u(x,π) = x, 0 < x < π.u(x,y) =∞∑n=0[α n (y)cos (2n+1) x ],2α n (y) = k 1n e 2n+12 y +k 2n e −2n+1 2 y − 4γ 0n(2n+1) 2(y +1),k 1n = γ 3n +γ 4n e −2n+1 2 πe 2n+1e 2n+12 π2 π +e , k 2n = γ 3n −γ 4n e 2n+1−2n+1 2 π 2 π +e −2n+1γ 3n = 4γ 0n(2n+1) 2(π +1)+γ 2n, γ 4n =22n+1[2 π ,]4γ 0n(2n+1) 2 +γ 1nγ 0n = 4(−1)nπ(2n+1) , γ 1n = 4π(−1)n2n+1 − 32(−1)nπ(2n+1) 3 , γ 2n = 4(−1)n2n+1 − 8π(2n+1) 2 ..13. [15/12/2006 (ex)I] Risolvere per serie di Fourier∆u = 0, 0 < x < π,0 < y < π,u(0,y) = 0, 0 < y < π,u x (π,y) = 1, 0 < y < π,u(x,0) = 1, 0 < x < π,u y (x,π) = 0, 0 < x < π.Soluzione220