Consider the following initial boundary value problem for the wave equation under homogeneous Neumann boundary conditions. Utt (x,t) = Uxx(x, t) for (x,t) = (0,π) × (0,∞) ux (0,t) = 0 = ux(π,t) when t> 0 u(x, 0) = (x) ut(x, 0) = v(x) x = [0, π]. We will assume that and are differentiable on [0, 7] and also satisfy the homogenous Neumann boundary condition themselves. i) Using separation of variables derive an expression for a solution u to the above as an infinite sum. You will need to show that u takes the form u(x,t) = A0 +tBo + Σ (An cos(nt) + B₂ sin(nt)) cos(nx). n=1 Describe the steps you carry out and give explicit expressions for An and Bn in terms of and . Hint: You can use the following without proof: If a differentiable function g : [0,7] → R satisfies the homogeneous Neumann condition 02 (0) = 0 = 02 (7) then it can be expressed via: ½ П მ მ S₁g(s) cos (ns) ds if n ≥ 1 g(x) = = Σ In cos (nx) where In = n=0 חי √₁₂ g(s) ds if n = = 0
Consider the following initial boundary value problem for the wave equation under homogeneous Neumann boundary conditions. Utt (x,t) = Uxx(x, t) for (x,t) = (0,π) × (0,∞) ux (0,t) = 0 = ux(π,t) when t> 0 u(x, 0) = (x) ut(x, 0) = v(x) x = [0, π]. We will assume that and are differentiable on [0, 7] and also satisfy the homogenous Neumann boundary condition themselves. i) Using separation of variables derive an expression for a solution u to the above as an infinite sum. You will need to show that u takes the form u(x,t) = A0 +tBo + Σ (An cos(nt) + B₂ sin(nt)) cos(nx). n=1 Describe the steps you carry out and give explicit expressions for An and Bn in terms of and . Hint: You can use the following without proof: If a differentiable function g : [0,7] → R satisfies the homogeneous Neumann condition 02 (0) = 0 = 02 (7) then it can be expressed via: ½ П მ მ S₁g(s) cos (ns) ds if n ≥ 1 g(x) = = Σ In cos (nx) where In = n=0 חי √₁₂ g(s) ds if n = = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Question
![Consider the following initial boundary value problem for the wave equation under
homogeneous Neumann boundary conditions.
Utt (x,t) = Uxx(x, t)
for (x,t) = (0,π) × (0,∞)
ux (0,t) = 0 = ux(π,t)
when t> 0
u(x, 0) = (x)
ut(x, 0) = v(x) x = [0, π].
We will assume that and are differentiable on [0, 7] and also satisfy the homogenous
Neumann boundary condition themselves.
i) Using separation of variables derive an expression for a solution u to the above as
an infinite sum. You will need to show that u takes the form
u(x,t) = A0 +tBo + Σ (An cos(nt) + B₂ sin(nt)) cos(nx).
n=1
Describe the steps you carry out and give explicit expressions for An and Bn in
terms of and .
Hint: You can use the following without proof: If a differentiable function g :
[0,7] → R satisfies the homogeneous Neumann condition 02 (0) = 0 = 02 (7) then
it can be expressed via:
½
П
მ
მ
S₁g(s) cos (ns) ds if n ≥ 1
g(x) =
=
Σ In cos (nx) where
In
=
n=0
חי
√₁₂
g(s) ds
if n =
= 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea7bbfe4-f0d9-4343-839c-f9e5c129de8e%2Ff9b67086-09fe-42ba-900b-6492be53dd79%2F3ccyxcl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the following initial boundary value problem for the wave equation under
homogeneous Neumann boundary conditions.
Utt (x,t) = Uxx(x, t)
for (x,t) = (0,π) × (0,∞)
ux (0,t) = 0 = ux(π,t)
when t> 0
u(x, 0) = (x)
ut(x, 0) = v(x) x = [0, π].
We will assume that and are differentiable on [0, 7] and also satisfy the homogenous
Neumann boundary condition themselves.
i) Using separation of variables derive an expression for a solution u to the above as
an infinite sum. You will need to show that u takes the form
u(x,t) = A0 +tBo + Σ (An cos(nt) + B₂ sin(nt)) cos(nx).
n=1
Describe the steps you carry out and give explicit expressions for An and Bn in
terms of and .
Hint: You can use the following without proof: If a differentiable function g :
[0,7] → R satisfies the homogeneous Neumann condition 02 (0) = 0 = 02 (7) then
it can be expressed via:
½
П
მ
მ
S₁g(s) cos (ns) ds if n ≥ 1
g(x) =
=
Σ In cos (nx) where
In
=
n=0
חי
√₁₂
g(s) ds
if n =
= 0
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