Show that V(x, t) = π – S√4xt - e ds is a solution to the heat equation V₁ = xVxx, x = R.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 5T
icon
Related questions
Question
100%
Show that V(x, t) =
- Sov
√4xt e-s² ds is a solution to the heat equation
V₁ = xV₁, x € R.
with initial value
=T-
Suppose U solves the heat equation on the real line
4Uxx, x ER
Ut=
=
U(x,0) =
4, x ≤0
2, x > 0.
(i) Use the Fourier-Poisson formula to give an explicit expression for the solution
U.
(ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot out
the solution at several instants of time to explain your answer. What is the limit
of U as t→∞?
Transcribed Image Text:Show that V(x, t) = - Sov √4xt e-s² ds is a solution to the heat equation V₁ = xV₁, x € R. with initial value =T- Suppose U solves the heat equation on the real line 4Uxx, x ER Ut= = U(x,0) = 4, x ≤0 2, x > 0. (i) Use the Fourier-Poisson formula to give an explicit expression for the solution U. (ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot out the solution at several instants of time to explain your answer. What is the limit of U as t→∞?
Expert Solution
steps

Step by step

Solved in 3 steps with 1 images

Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage