Note: Use the prime notation for derivatives, so the derivative of X is written as X' Do NOT use X'(r). The variable A i

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Note Use The Prime Notation For Derivatives So The Derivative Of X Is Written As X Do Not Use X R The Variable A I 1 (16.29 KiB) Viewed 49 times

Note Use The Prime Notation For Derivatives So The Derivative Of X Is Written As X Do Not Use X R The Variable A I 2 (22.3 KiB) Viewed 49 times

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Note: Use the prime notation for derivatives, so the derivative of X is written as X' Do NOT use X'(r). The variable A is typed lambda, jais typed as mu Solve the two-dimensional wave equation with clamped edges and the given initial conditions 20<2<b,0<y<at>0 D (0,y,t)-0, u (b, y,t) = 0, 0<g<ct>0 0<r<b,t>0 u (z,0,1)=0,u (z,c,t) 0, u(x, y,0)=f(x,y). 0<x<b,0<y<e, Du = g(x, y), 0<z<b,0<y<e, Ot -0

Part 1 Separating the variables We try for a separable solution u(z,y)= X(z)Y(y)T(t), plugging XYT into the PDE for u we get Therefore X"YT = Divide both sides by XYT leads to the separated equations For now we will work with the middle of this equation, the equation in Y and T. This equation also separates Y" =-P Y We now have three separated differential equations on the three variables: ODE in X = 0 (use m for the multiple of #) ODE in Y. = 0 (use n for the multiple of x) ODE in T =0

Part 2 The Sturm-Liouville Problems From the boundary conditions on the plate we see that X(0) = X(b) = Y(0) = Y(c) = That makes the differential equations in X and Y Sturm-Liouville problems. We can use them to find A and µ, and then X,Y, T X(X) = (use m as the multiple of ) Y(y) = (use n as the multiple of ) T(t) = (use Amn as the coefficient on cos, Bmn as the coefficient on sin)

Part 3 The series solution 00 00 Ma(iF,9,1) = Σ Σ ΧΥΤ' = m-In-1 mil We now need to find the coefficients Amn and Ben Using the first initial condition 00 00 u(z,y,0) = Σ E m-In-1 Therefore Amp Now we use the second initial condition: thu = ΣΕ 20 t-0 m-ln-1 Therefore Bm = f(x,y) f(x,y) = g(x,y) 1 / g(z,y) dydx dydx