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Question 1. Solve the following IVPS for the 2D wave on the rectangle [0, 1] x [0, 1] (where we assume that the solution u is always zero on the boundary): 1. Utt :Au u(x, y,0) = – 10 sin(37x) sin(7y) Uz (x, y, 0) = sin(7x) sin(Ty) 2. (for this one, you can use a computer algebra software to compute the integrals) Utt = Δυ u(r, y, 0) = x(x – 1)°y(y – 1)³ u(r, y, 0) = sin(rr) sin(ry)

Question 1. Solve the following IVPS for the 2D wave on the rectangle [0, 1] x [0, 1] (where we assume that the solution u is always zero on the boundary): 1. Utt :Au u(x, y,0) = – 10 sin(37x) sin(7y) Uz (x, y, 0) = sin(7x) sin(Ty) 2. (for this one, you can use a computer algebra software to compute the integrals) Utt = Δυ u(r, y, 0) = x(x – 1)°y(y – 1)³ u(r, y, 0) = sin(rr) sin(ry)

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Question 1. Solve the following IVPS for the 2D wave on the rectangle 0, 1] x [0, 1] (where we assume that the solution u is always zero on the boundary): 1. Utt Δυ = –10 sin(37x) sin(7y) Ut(r, Y, 0) = sin(r 2) sin(ry) u(x, y, 0) 2. (for this one, you can use a computer algebra software to compute the integrals) Utt = Δυ u(r, y, 0) = x(x – 1)°y(y – 1)* uz(r, y,0) - = sin(Tx) sin(TY)
Transcribed Image Text:Question 1. Solve the following IVPS for the 2D wave on the rectangle 0, 1] x [0, 1] (where we assume that the solution u is always zero on the boundary): 1. Utt Δυ = –10 sin(37x) sin(7y) Ut(r, Y, 0) = sin(r 2) sin(ry) u(x, y, 0) 2. (for this one, you can use a computer algebra software to compute the integrals) Utt = Δυ u(r, y, 0) = x(x – 1)°y(y – 1)* uz(r, y,0) - = sin(Tx) sin(TY)
Expert Solution
Step 1

Given

utt=Δu

u(x,y,0)=-10sin(3πx)sin(ππy)

ut(x,y,0)=sin(πx)sin(πy)

Step 2

u(0,y,+)=u(1,y,+)=u(x,0,+)=u(x,1,+)=0

u(x,y,+)=X(x)Y(y)T(t)T''T=X''X+Y''Y

 

X(0)=0 and X(1)=0

Y(0)=0 and Y(1)=0

Xn(x)=sin(nπx)Ym(y)=sin(mπy)

 

T''T=-(nπ)2-(mπ)2

T(+)=An,mcos(n2+m2π+)+Bn,msin(n2+m2π+)

Step 3

ux,y,+=∑n=1∞∑m=1∞[An,mcos(n2+m2π+)+Bn,msin(n2+m2π+)]=sin(nxπ)sin(myπ)

 

Substitute n=3 and m=1,

u(x,y,0)=sin(3xπ)sin(πy)u+(x,y,0)=sin(πx)sin(πy)

A3,1=-10B1,1=12π

 

u(x,y,+)=-10cos(10π+)sin(3xπ)sin(yπ)+12πsin(2π+)sin(πx)sin(πy)

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