Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on -l < x

we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on -l < x

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
See similar textbooks
icon
Related questions
Question

[Second Order Equations] How do you solve 3? ty

we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on -l < x <l utt - c²uxx = 0, with periodic boundary conditions and initial conditions u (-l, t) = u(l, t), ux (−l, t) = ux (l, t), (1) u (x,0) = (x), ut (x,0) = (x). 1. Suppose we have a "separated" solution u (x, t) = X(x)T(t). By following the method in class, show that there must exist a real number number A so that X" (x) + AX (x) = 0, T" (t) + c²AT (t) = 0. 2. Show that in order for X to satisfy the correct boundary conditions, we must have λ = (¹7) ² for n = 0,1,2,... and that we can take X to be any linear combination of cos (¹) or sin (¹7x). ηπχ 3. Solve the equation for T(t) (as we did in lecture) and then (just as in lecture) sum over all possible separated solutions to obtain a guess for the general form of the solution u. 4. By evaluating the formula (and the time derivative of the formula) at t = 0, relate this general formula to the initial data & and 4. Using the formulas for Fourier series on -l < x < l, show that you can arrange the constants in your formula so that the function u determined by the formula obtains the initial data correctly.
Transcribed Image Text:we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on -l < x <l utt - c²uxx = 0, with periodic boundary conditions and initial conditions u (-l, t) = u(l, t), ux (−l, t) = ux (l, t), (1) u (x,0) = (x), ut (x,0) = (x). 1. Suppose we have a "separated" solution u (x, t) = X(x)T(t). By following the method in class, show that there must exist a real number number A so that X" (x) + AX (x) = 0, T" (t) + c²AT (t) = 0. 2. Show that in order for X to satisfy the correct boundary conditions, we must have λ = (¹7) ² for n = 0,1,2,... and that we can take X to be any linear combination of cos (¹) or sin (¹7x). ηπχ 3. Solve the equation for T(t) (as we did in lecture) and then (just as in lecture) sum over all possible separated solutions to obtain a guess for the general form of the solution u. 4. By evaluating the formula (and the time derivative of the formula) at t = 0, relate this general formula to the initial data & and 4. Using the formulas for Fourier series on -l < x < l, show that you can arrange the constants in your formula so that the function u determined by the formula obtains the initial data correctly.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
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,
SEE MORE TEXTBOOKS