Use the method of separation of variables and the superposition principle to solve the heat equation192subject to the boundary conditions0 < x < 4, t > 0.(0, t) 0, u(4, t) = 0, t > 0,u(x, 0)and the initial condition=3 cos(³x) + cos(³), 0

Answered: Image /qna-images/answer/3b33166c-0662-4169-9482-337d1f5a7777.jpg

Answered: Use the method of separation of…

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

Similar questions

Use Euler's method to approximate the solutions for the initial value problem: y' : = cos 2t + sin 3t, 0 ≤ t ≤ 1, y(0) = 1, with h = 0.5.
Consider the following initial value problem: / = sin(y + t) +e - sin(e' +t), y(0) = 2 (a) Show that there exists an intervalICR such that 0 E I, and a differentiable function y:1R which is a solution to this initial value problem. (b) Is the solution in (a) unique? Explain why or why not. (c) One of the following three things is true: 1. y(t) >e for all teI 2. y(t)
1. A string has a partial differential equation with its initial conditions and boundar equation of 8² u Ət² = 9 J²u მ2 u (x,0) = 2x(x-1) du Ət u (0, t) = 0 u (L, t) = 0 Find u(t). -(x,0) = 0
Determine the longest interval in which the initial value problem (8t + 6)y" + 3ty - 4y = 3 cos(t), y(-6) = 2, y(-6) = 0 is certain to have a unique twice-differentiable solution. Do not attempt to find the solution. Answer:
Find the solution of the given initial value problem: y(t): = y"+y' = sec(t), y(0) = 11, y'(0) = 5, y"(0) = -6. -(t cos(t))+ 6 cos(t) +5 sin(t) + ln(tan(t) + sec(t))+ ln(cos(t)) sin(t) + 5 X
Determine the longest interval in which the initial value problem is certain to have a unique twice differentiable solution. (Do not attempt to find the solution.) (t− 3)y" - 3ty' + 5y = sint, y( − 7) = 7, y'(-7) = 3 ◄►
Example 5: Solve the DE (3 + y + 2y^2sin^2 x) dx + (x+ 2xy − y sin 2x) dy = 0i. f(x,y) = M(x,y) dx = F(x,y) + Q(x) + T(y) where y is held constant.
The function Ур is a particular solution to the specified nonhomogeneous equation. Find the unique solution satisfying the equation and the given initial conditions. y" - 2y + y = 4ex, yp = 2x² ex, y(0) = 1, y'(0) = -1
The temperature u(x, t) of a metal bar of length e = 2 at a distance r from one end and at time t is modelled by the partial differential equation Ut = aurr (0 0) It is given that the metal has diffusivity a = 2.25, that the two ends of the bar are kept at temperature u = 0 and that the initial temperature distribution is u(x,0) = f(x) = sin(Tx/0) Use the explicit difference parabolic method with Ar = 0.5 and At 0.05 to approximate u(x, t) at t= 0.05 and t= 0.10.
Find a general solution to (x'+xy’)y' = y' Give an implict form F(x, y) =C :
7.(This one is a gift) Consider a homogeneous differential equation, p(x, y) dx + q(x, y) dy = 0, where the coefficients p(x, y) and q(x, y) satisfy the special homogeneity condition: p(Ax, Xy) = X®p(x, y); q(Ax, Ay) = X°q(x, y). Explain why it is always possible to express any such homogeneous differential equation in the form dy h. F dx
Consider the following example of the heat equation: Uț = 3 · uz, u (x, 0) = 4 · (2 – x) · x², u (0, t) = 0, u (2, t) = 0. Assume we discretize with five nodes and that we choose At = 1.0. Use the Crank-Nicholson method (Lecture 28) to find the solution on the interior nodes after 1 step (enter your answer to 3dps).

SEE MORE QUESTIONS

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,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

Use the method of separation of variables and the superposition principle to solve the heat equation - 1924, 0 < x < 4, t > 0. subject to the boundary conditions (0, t) 0, u(4, t) = 0, t > 0, and the initial condition u(x, 0) = 3 cos (³x) + cos(5), 0 < x < 4. Enter u(x, t).