Question 1. Solve the following IVPS for the 2D wave on the rectangle0, 1] x [0, 1] (where we assume that the solution u is always zero on theboundary):1.UttΔυ= –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 theintegrals)Utt =Δυu(r, y, 0) = x(x – 1)°y(y – 1)*uz(r, y,0)-= sin(Tx) sin(TY)

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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Expert Solution

Step 1

Given

$u_{t t} = Δ u$

$u (x, y, 0) = - 10 \sin (3 πx) \sin (ππy)$

$u_{t} (x, y, 0) = \sin (πx) \sin (πy)$

Step 2

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

$u (x, y, +) = X (x) Y (y) T (t) \frac{T^{''}}{T} = \frac{X^{''}}{X} + \frac{Y^{''}}{Y}$

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

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

$X_{n} (x) = \sin (n πx) Y_{m} (y) = \sin (m πy)$

$\frac{T^{''}}{T} = - {(n π)}^{2} - {(m π)}^{2}$

$T (+) = A_{n, m} \cos (\sqrt{n^{2} + m 2} π^{+}) + B_{n, m} \sin (\sqrt{n^{2} + m 2} π^{+})$

Step 3

$u (x, y, +) = \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} [A_{n, m} \cos (\sqrt{n^{2} + m 2} π^{+}) + B_{n, m} \sin (\sqrt{n^{2} + m 2} π^{+})]$ $= \sin (n x π) \sin (m y π)$

Substitute n=3 and m=1,

$u (x, y, 0) = \sin (3 x π) \sin (πy)$ $u_{+} (x, y, 0) = \sin (πx) \sin (πy)$

$A_{3, 1} = - 10 B_{1, 1} = \frac{1}{\sqrt{2} π}$

$u (x, y, +) = - 10 \cos (\sqrt{10} π^{+}) \sin (3 x π) \sin (y π) + \frac{1}{\sqrt{2} π} \sin (\sqrt{2} π^{+}) \sin (πx) \sin (πy)$

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