uestion 1. Consider the second order partial differential equation = c? dx² (1) r an unknown real-valued function u = u(t, x), where t represents time, x represents a point in space, and c> 0 is constant. 1. For any twice differentiable functions F = F(x) and G = G(x), show that u(t, x) = F(x + ct) + G(x – ct) satisfies (1). artial differential equations such as (1) are often solved as initial value problems, where the initial description of e unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are ven that u(0, x) = g(x) and (0, x) = h(x), for some given functions g and h. 2. Assume that u(t, x) = F(x + ct) + G(x - ct) for some functions F and G, as described in problem 1.1. If u = u(t, x) solves the initial value problem described above, show that g(x) F(x) + G(x) and h(x) = cF'(x) – cG'(x). %3D 3. By integrating the last equation for h(x), show that for any constant a E R, h(s) ds = cF(x) – CG(x) – cF(a)+ cG(a), and from here solve a linear system to show that F(x) = ; (9(2) +[ h(0) ds + F(a) – G(a)), - and (9(=) -[ ) . G(x) h(s) ds – F(a) + G(a) 4. Lastly, given that u(t, x) = F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial value problem for (1): 1 cr+ct u(t, x) = , [9(x + ct) + g(x – ct)] + h(s)ds. 2c

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
100%

Pls help ASAP. Show all work and calculations. 

Question 1. Consider the second order partial differential equation
и
(1)
dx² '
for an unknown real-valued function u =
u(t, x), wheret represents time, x represents a point in space,
and c > 0 is
a constant.
1. For any twice differentiable functions F
satisfies (1).
F(x) and G = G(x), show that u(t, x) = F(x + ct) + G(x – ct)
%3D
Partial differential equations such as (1) are often solved as initial value problems, where the initial description of
the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are
given that u(0, æ) = g(x) and " (0, x) = h(x), for some given functions
du
and h.
2. Assume that u(t, x)
u = u(t, x) solves the initial value problem described above, show that
F(x + ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If
g(x) = F(x)+ G(x)
and
h(x).
cF'(x) – cG' (x).
3. By integrating the last equation for h(x), show that for any constant a € R,
| h(s) ds = cF(æ) – cG(x) – cF(a)+ cG(a),
and from here solve a linear system to show that
1
F(x) = ; (g(x)
(o(e) + Mo) ds + F(a) – G(a)
and
G(e) = } (s(e) }
:| h(s) ds – F(a) + G(a)
2
4. Lastly, given that u(t, x) = F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial
value problem for (1):
%3D
1
1
px+ct
u(t, x) = ; [g(x + ct) + g(x – ct)] +
h(s)ds.
2c
x-ct
Transcribed Image Text:Question 1. Consider the second order partial differential equation и (1) dx² ' for an unknown real-valued function u = u(t, x), wheret represents time, x represents a point in space, and c > 0 is a constant. 1. For any twice differentiable functions F satisfies (1). F(x) and G = G(x), show that u(t, x) = F(x + ct) + G(x – ct) %3D Partial differential equations such as (1) are often solved as initial value problems, where the initial description of the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are given that u(0, æ) = g(x) and " (0, x) = h(x), for some given functions du and h. 2. Assume that u(t, x) u = u(t, x) solves the initial value problem described above, show that F(x + ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If g(x) = F(x)+ G(x) and h(x). cF'(x) – cG' (x). 3. By integrating the last equation for h(x), show that for any constant a € R, | h(s) ds = cF(æ) – cG(x) – cF(a)+ cG(a), and from here solve a linear system to show that 1 F(x) = ; (g(x) (o(e) + Mo) ds + F(a) – G(a) and G(e) = } (s(e) } :| h(s) ds – F(a) + G(a) 2 4. Lastly, given that u(t, x) = F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial value problem for (1): %3D 1 1 px+ct u(t, x) = ; [g(x + ct) + g(x – ct)] + h(s)ds. 2c x-ct
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,