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1. The transport PDE, du has a family of solutions u = f(z – vt) where f can be any function. (a) Show that the proposed solution obeys the wave oquation (b) Find f when u(r,0) = e=='/2; (c) Write down the solution u(z,t) when u(z,0) = e~x*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v =1

1. The transport PDE, du has a family of solutions u = f(z – vt) where f can be any function. (a) Show that the proposed solution obeys the wave oquation (b) Find f when u(r,0) = e=='/2; (c) Write down the solution u(z,t) when u(z,0) = e~x*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v =1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. The transport PDE, du du has a family of solutions u = f(z – vt) where f can be any function. (a) Show that the proposed solution obeys the wave oquation (b) Find f when u(r,0) = e==*/2; (c) Write down the solution u(z, t) when u(z,0) = e-*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1 2. The wave equation, has a family of solutions u = f(x – vt) + g(z+ et) where ƒ can be any function EDIT: provided v = 1. (a) Show that the proposed solution obeys the wave equation (b) Find f and g when u(r,0) = e-*/² and it has zero initial velocity, dule = 0. i.e., (c) Write down the solution u(z,t) when u(z,0) = e~x*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1
Transcribed Image Text:1. The transport PDE, du du has a family of solutions u = f(z – vt) where f can be any function. (a) Show that the proposed solution obeys the wave oquation (b) Find f when u(r,0) = e==*/2; (c) Write down the solution u(z, t) when u(z,0) = e-*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1 2. The wave equation, has a family of solutions u = f(x – vt) + g(z+ et) where ƒ can be any function EDIT: provided v = 1. (a) Show that the proposed solution obeys the wave equation (b) Find f and g when u(r,0) = e-*/² and it has zero initial velocity, dule = 0. i.e., (c) Write down the solution u(z,t) when u(z,0) = e~x*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1
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