Separation of variables The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) du 8² au for 0 0, (1) Ət əx² with the conditions ux(0, t) = ux (L, t) = 0 and u(x,0) = f(x), where L, a, DER are positive constants. 1. Seek a separation solution of the form u(x, t) = X(x)T(t) to show T' a X" + = =k, DT D X where k denotes the separation constant. 2. Use equation (2) to derive two ordinary differential equations (ODES), one in space x and one in time t. 3. Determine the boundary conditions for the ODE that depends on x. 4. Find the non-trivial solutions of X(x) = Xn(x) and corresponding values of the separation constant k = kn, for n = 0, 1, 2, 3.... Hint: you may need to consider each of the three cases k=-p², k = 0, k=p² (p = 0) to find all the non-trivial solutions 5. Find the solutions T(t) = Tn(t), for n = 0, 1, 2, 3.... 6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies the boundary conditions ur (0, t) = u₂(L, t) = 0. 7. Determine expressions for the unknown coefficients in your solution for u that satisfies the initial condition u(x,0) = f(x). 8. Find the solution when f(x) = 1 + cos лx, for values of L = 1, a = 1 and D = 1. 9. In a single figure, plot the solution found in Question 8 at times t = 0, 0.25, 0.5, 2. Briefly describe the behaviour as t increases. What is the constant value temperature of the environment that surrounds the bar? D
Separation of variables The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) du 8² au for 0 0, (1) Ət əx² with the conditions ux(0, t) = ux (L, t) = 0 and u(x,0) = f(x), where L, a, DER are positive constants. 1. Seek a separation solution of the form u(x, t) = X(x)T(t) to show T' a X" + = =k, DT D X where k denotes the separation constant. 2. Use equation (2) to derive two ordinary differential equations (ODES), one in space x and one in time t. 3. Determine the boundary conditions for the ODE that depends on x. 4. Find the non-trivial solutions of X(x) = Xn(x) and corresponding values of the separation constant k = kn, for n = 0, 1, 2, 3.... Hint: you may need to consider each of the three cases k=-p², k = 0, k=p² (p = 0) to find all the non-trivial solutions 5. Find the solutions T(t) = Tn(t), for n = 0, 1, 2, 3.... 6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies the boundary conditions ur (0, t) = u₂(L, t) = 0. 7. Determine expressions for the unknown coefficients in your solution for u that satisfies the initial condition u(x,0) = f(x). 8. Find the solution when f(x) = 1 + cos лx, for values of L = 1, a = 1 and D = 1. 9. In a single figure, plot the solution found in Question 8 at times t = 0, 0.25, 0.5, 2. Briefly describe the behaviour as t increases. What is the constant value temperature of the environment that surrounds the bar? D
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
Q solution
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 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,