Let u be the solution to the initial boundary value problem for the Heat Equation,du(t, x) = 5 du(t, x),t € (0, ∞0),x € (0,5);with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initialcondition0,15u(0, x) = f(x) = 2,= [0, 2),= [5, ¹5).€ [15.5].0,XEThe solution u of the problem above, with the conventions given in class, has the formu(t, x) = Ï Cn Un(t) wn(x),n=1with the normalization conditions U,(0) = 1 and wn (5) = 1. Find the functionsUn, Wn, and the constants Cn.Un(t) =wn(x) =Cn =x EXEMM

Given that ut=5uxx, 0<x<5, t>0u0,t=0, u5,t=0, ux,o=fx=0,…

Answered: Let u be the solution to the initial…

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

Ex. The temperature u(x,t) in a strip of metal of width L is governed by the heat equation au =u, for 00. Use separation of variables method to find the unsteady state temperature distribution in the strip given that the initial condition is u(x,0) =x and the boundary conditions u (0,t)=u (L,t)=0 for t>0. The condition u, (0,t) = 0 expresses mathematically the constraint that no heat flows through the left boundary (insulated end condition). -H.W. Problems P.327 "wylie" Q.5,7,8
If use Taylor's method of order three to approximate an initial value problem with f(t, y) = et-y, we need h T(³) (t, y) = f(t, y) + − ƒ'(t, y) + + ½ ƒ'(t,y)+7ƒ"(t,y), 2 3! Write out the explicit expression of T(³) (t, y) (no ƒ‚ ƒ', or f" appears in the expression, should only involve t, h and y). (Do not carry out the actual calculations)
Verify that any function of the form Y (t) = c1vt + czt* satisfies the equation ty"(t) – ĮtY'(t) + 2Y (t) = 0. Determine the solution of the equation with the boundary conditions Y(1) = 1, Y(4) = 2.
(4) Suppose z = f(x, y) where f is differentiable (both f, and fy exist). In addition, suppose = g(t) and y=h(t), and I= dz Find when t-3. dt g(3) = 2, g'(3) = 5, fr(2,7)=6, h(3) = 7, h'(3) = -4, fy(2, 7) = −8.
Consider that the heat flow is described by the equation Pu + xet du %3D dt with BC: du(0.t) du(3.1) 0, and initial heat flow function u(x, 0) = r- %3D Find u(r, t) at any t> 0. 2.
Solve for the equilibrium temperature distribution using the 2D Laplace equation on an L x H sized rectangular domain with the following boundary conditions: 1. Left: u(0,y) = f(y) (fixed temperature) 2. Bottom: u₂(x,0) = 0 (insulating) 3. Top: u₂(x, H) = 0 (insulating) 4. Right: u(L, y) = 0 (zero temperature) Solve for a general boundary temperature f(y). Also solve for the particular temperature distribution f(y) = sin(4Ty/H). u(0,y)-f(y) U₂, (x,H) = 0 7² - 0 u(L, y) - 0 u₂(x,0) = 0 Without too much extra work, tell me how this solution would change if we also made the right boundary condition insulating?
Reduce the given wave equation for the vibration of a string streched between the points x=1 and x=m to the ordinary differential equations and separate the boundary conditions by using the method of separation of variables. U=4u 10 A X"-4AX=0, X(1)=0,X(m)=0. T"-AT=0, A0.
Let u be a solution of the heat equation u, -Uxx ; 00 u(0,t)=u(r,t)=0 u(x,0)= sin x+ sin 2x; 00 Then. 1: (a) u(x,t)→0 as t→o for all x e (0,7) (b) t'u(x,t)→0 as t→o for all xe (0, (c) - e'u(x,t) is a bounded function for xe (0,7),t >0 (d) e"u(x,t)→0 as t→o for all xe (0,7)
5. Solve the initial-boundary value problem u(x, t) = c²u(x, t) for 0 0; u(x, 0) = sin 2x; u₁(x,0) = re-²; u(0, t) = 0 for all t > 0.
Consider the differential equation dz dz +p- dt + kz = F(t) m dt2 which models an oscillator of mass m subject to a restoring force k, damping p and an external force F(t). The external force F(t) is given by F(t) = 18e-* and also m = 1, p = 4 and dz k = 13. Initial conditions are that z = 1 and = 4 when t = 0. dt (i) By taking the Laplace Transform of the differential equation show that 7, (the Laplace Transform of the solution) satisfies s? + 10s + 34 (s + 2)(s² + 4s + 13) (ii) Hence find z(t). post (iii) Explain what happens to z(t) for large t.

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

Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 5 du(t, x), te (0, 00), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, € [0, € [$., 15). u(0, x) = f(x) = {2, 0, XE XE x € [15,5]. XE