Consider the following initial boundary value problem, with inhomogeneous Dirichlet boundary conditions where a, b ER are constants: Utt - Uxx = 0 u(0,t) =b, u(l,t) = a (x,t) = (0,1) x (0,∞) t€ (0,∞) u(x, 0) = (x) (x, 0) = (x) x = [0,1]. Find a solution to this problem by carrying out the steps below i) Using a formula from the notes, write down a solution v to the PDE Vit - Vxx = 0 under homogeneous Dirichlet boundary conditions v(0,t) = 0 = v(l,t) for all t > 0 and initial conditions v(x, 0) = (x) = ((a−b)x + b), vt(x, 0) x = [0,1]. - You should simplify the coefficients in your expansion where possible. Hint: You will need to check that -((ab)²+6) sin (77) 2 dx = ((-1)"a - b). пп = (x) for ii) Let u(x,t) = v(x,t)+(ab)x+b and show that u is the solution we are looking for. You do not need to use the infinite sum expression for v in order to check this!
Consider the following initial boundary value problem, with inhomogeneous Dirichlet boundary conditions where a, b ER are constants: Utt - Uxx = 0 u(0,t) =b, u(l,t) = a (x,t) = (0,1) x (0,∞) t€ (0,∞) u(x, 0) = (x) (x, 0) = (x) x = [0,1]. Find a solution to this problem by carrying out the steps below i) Using a formula from the notes, write down a solution v to the PDE Vit - Vxx = 0 under homogeneous Dirichlet boundary conditions v(0,t) = 0 = v(l,t) for all t > 0 and initial conditions v(x, 0) = (x) = ((a−b)x + b), vt(x, 0) x = [0,1]. - You should simplify the coefficients in your expansion where possible. Hint: You will need to check that -((ab)²+6) sin (77) 2 dx = ((-1)"a - b). пп = (x) for ii) Let u(x,t) = v(x,t)+(ab)x+b and show that u is the solution we are looking for. You do not need to use the infinite sum expression for v in order to check this!
Calculus: Early Transcendentals
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![Consider the following initial boundary value problem, with inhomogeneous Dirichlet
boundary conditions where a, b ER are constants:
Utt
-
Uxx = 0
u(0,t) =b, u(l,t) = a
(x,t) = (0,1) x (0,∞)
t€ (0,∞)
u(x, 0) = (x) (x, 0) = (x) x = [0,1].
Find a solution to this problem by carrying out the steps below
i) Using a formula from the notes, write down a solution v to the PDE Vit - Vxx = 0
under homogeneous Dirichlet boundary conditions v(0,t) = 0 = v(l,t) for all
t > 0 and initial conditions v(x, 0) = (x) = ((a−b)x + b), vt(x, 0)
x = [0,1].
-
You should simplify the coefficients in your expansion where possible.
Hint: You will need to check that
-((ab)²+6) sin (77)
2
dx =
((-1)"a - b).
пп
=
(x) for
ii) Let u(x,t) = v(x,t)+(ab)x+b and show that u is the solution we are looking for.
You do not need to use the infinite sum expression for v in order to check this!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea7bbfe4-f0d9-4343-839c-f9e5c129de8e%2F1d254ddd-f240-4a43-a7ad-f981b4012b5d%2Fhjct7xl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the following initial boundary value problem, with inhomogeneous Dirichlet
boundary conditions where a, b ER are constants:
Utt
-
Uxx = 0
u(0,t) =b, u(l,t) = a
(x,t) = (0,1) x (0,∞)
t€ (0,∞)
u(x, 0) = (x) (x, 0) = (x) x = [0,1].
Find a solution to this problem by carrying out the steps below
i) Using a formula from the notes, write down a solution v to the PDE Vit - Vxx = 0
under homogeneous Dirichlet boundary conditions v(0,t) = 0 = v(l,t) for all
t > 0 and initial conditions v(x, 0) = (x) = ((a−b)x + b), vt(x, 0)
x = [0,1].
-
You should simplify the coefficients in your expansion where possible.
Hint: You will need to check that
-((ab)²+6) sin (77)
2
dx =
((-1)"a - b).
пп
=
(x) for
ii) Let u(x,t) = v(x,t)+(ab)x+b and show that u is the solution we are looking for.
You do not need to use the infinite sum expression for v in order to check this!
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