Please help with explanation Exercise 19.7. Solve the heat equationдиƏtwith Neumann boundary conditionsJ²uმ2duəxfor t≥ 0, 0≤x≤ 1,(t,0) = 1,диəx(t, 1) = 1.(Hint: The function x that is independent of t has constant x-partial 1 that “linearly interpolates" betweenthe boundary values 1 and 1. What BVP does v(t, x) = u(t, x) − x satisfy?)

Using the hints, transform the nonhomogeneous boundary conditions to homogeneous boundary…

Answered: Exercise 19.7. Solve the heat equation…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider a 6-meter metal bar with a uniform initial temperature (across the bar) of 35°C . Suppose it is in thermal contact with an external source of heat given by h(x)= 3−x, 0 ≤ x ≤ 6. So the temperature u(x,t) had the ut=uxx+h(x) permission. Suppose further that the temperature of the ends are kept constant, being at x=0 of 5°C , while at x=6 of 30°C . Under such conditions: Find the steady-state temperature distribution of the bar and the boundary value problem that determines the transient distribution. (no need to solve the problem).
Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if: u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.
solve the following elliptic neumann partial differential equation for the steady state temperature distribution over a thin rod of length L. The boundary conditions are:
Find the steady-state temperature u(r,θ) in a circular hole in an infinite plate of radius r=1 subject to the condition u(1,θ) = {π-θ, 0<θ<π and θ-π, π<θ<2π}
B) Let dP/dt =.5P - 50. Find the equilibrium solution for P. Furthermore, determine whether P is intially increasing faster if the initial population is 120 or 200.
A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).
Show that y=y1+y2where y1=c1cos(x) & y2=c2sin(x) is a solution to y"+y=0 and show that the Wronskian of S={c1cos(x), c2sin(x)} is nonzero & equal to (c1c2).
Determine the largest interval (a,b) for which Theorem 1 guarantees the existence of a unique solution on (a,b) to the initial value problem below. x(x+7)y′′′−6xy′+y=0, y− 1 6=1, y′− 1 6=0, y′′− 1 6=0

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