Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation Ut + UU x = 0 (4) = on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt- plane.
Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation Ut + UU x = 0 (4) = on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt- plane.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Transcribed Image Text:2. Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation
= 0
ut + uux
-
on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a
quadratic equation, you need to pick the root that is consistent with the initial
conditions.) Make a sketch that illustrates some of the characteristics in the xt-
plane.
=
For the general initial-value problem u(x, 0) = A(x) use the strong solution to derive
an expression for u, in terms of A'(x – ut) (use the chain rule diligently). Use this
to show that shock formation is inevitable if the initial slope u(x, 0) < 0 at any x.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 28 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
