Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Suppose that u(x, t) satisfies the heat equation Ut = α-Uxx, a²uxx, 0 0, with a² being the thermal diffusivity. The boundary conditions u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0, are imposed. (a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential equations X"(x) + AX(x) = 0, İ(t) + Aa²T (t) = 0, where A is the separation constant; and express the boundary conditions in terms of X(x). (b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would suffice for A< 0). (c) Show that non-trivial solutions exist for λ = µ²> 0 provided that tan μ == 2 By considering the graphs of tan µ and -µ/2, explain why there are an infinite num- ber of positive eigenvalues A = An, (n = 1,2,3,...).

Suppose that u(x, t) satisfies the heat equation Ut = α-Uxx, a²uxx, 0 0, with a² being the thermal diffusivity. The boundary conditions u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0, are imposed. (a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential equations X"(x) + AX(x) = 0, İ(t) + Aa²T (t) = 0, where A is the separation constant; and express the boundary conditions in terms of X(x). (b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would suffice for A< 0). (c) Show that non-trivial solutions exist for λ = µ²> 0 provided that tan μ == 2 By considering the graphs of tan µ and -µ/2, explain why there are an infinite num- ber of positive eigenvalues A = An, (n = 1,2,3,...).

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
2. Suppose that u(x, t) satisfies the heat equation Ut = α² a²uxx, 0<x<1, t>0, with a² being the thermal diffusivity. The boundary conditions u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0, are imposed. (a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential equations X"(x) + AX(x) = 0, İ(t) + Aa²T (t) = 0, where A is the separation constant; and express the boundary conditions in terms of X(x). (b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would suffice for A < 0). (c) Show that non-trivial solutions exist for A = tan u = µ² > 0 provided that μ 2 By considering the graphs of tan µ and —µ/2, explain why there are an infinite num- ber of positive eigenvalues λ = An, (n = 1,2,3,...). (d) Determine the general solution u(x, t) in terms of the (unknown) eigenvalues An.
Transcribed Image Text:2. Suppose that u(x, t) satisfies the heat equation Ut = α² a²uxx, 0<x<1, t>0, with a² being the thermal diffusivity. The boundary conditions u(0, t) = 0, ux(1, t) +2u(1, t) = 0, t> 0, are imposed. (a) Using the separation of variables u(x, t) = X(x)T(t), derive the ordinary differential equations X"(x) + AX(x) = 0, İ(t) + Aa²T (t) = 0, where A is the separation constant; and express the boundary conditions in terms of X(x). (b) Show that there are no non-trivial solutions for A ≤ 0. (A graphical explanation would suffice for A < 0). (c) Show that non-trivial solutions exist for A = tan u = µ² > 0 provided that μ 2 By considering the graphs of tan µ and —µ/2, explain why there are an infinite num- ber of positive eigenvalues λ = An, (n = 1,2,3,...). (d) Determine the general solution u(x, t) in terms of the (unknown) eigenvalues An.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
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