*Note: Please answer all parts, and explain all workings. Thank you! 3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of m and n, the corresponding function of time is Tmn (ț)-A, , cos ω'nnt + Bnn sina, nt, find the natural frequencies a,nn. The solution of u is the summation of all the terms like Xm . Y, . Tmn . Apply the initial conditions, and use the orthogonality of the eigen-functions to obtain Amn and Bmn. c) 3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of m and n, the corresponding function of time is Tmn (ț)-A, , cos ω'nnt + Bnn sina, nt, find the natural frequencies a,nn. The solution of u is the summation of all the terms like Xm . Y, . Tmn . Apply the initial conditions, and use the orthogonality of the eigen-functions to obtain Amn and Bmn. c)


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