A rod of lengthn coincides with the interval [0, T] on the x-axis. The rod has initial temperature f(x) = 5 for 0 < x < a with the ends at temperature zero for all t > 0. Assume the rod satisfies all the assumptions of the heat equation with a constant a² of three. Set up an initial-boundary value problem for the temperature u(x, t), then find the solution, the temperature u(x,t).

A rod of lengthn coincides with the interval [0, T] on the x-axis. The rod has initial temperature f(x) = 5 for 0 < x < a with the ends at temperature zero for all t > 0. Assume the rod satisfies all the assumptions of the heat equation with a constant a² of three. Set up an initial-boundary value problem for the temperature u(x, t), then find the solution, the temperature u(x,t).

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

Related questions

Q: Q6 A 1.0 m length copper bar with a constant cross section area of 0.0001 m² have the values of…

A: Maximum temperature occurs at middle point of the bar, initial temparature was 100° C at the middle…

Q: Consider the heat equation ƏT Ət J²T ər² 0≤x≤n. Solve the equation using separation of variables…

A: Ans- We are given a problem find a the solution for the given initial and boundary value problem…

Q: 3. Find the position function x(t) given the acceleration a(t), the initial velocity vo = %3D v(0),…

A: It is given that the acceleration function is at=6t+4. The initial velocity at t=0 is v0=v0 and the…

Q: A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the…

A: Given: Length of the bar L = 3 meters is situated along the x axis so that one end is at x = 0 and…

Q: the differential equation that describes the motion.

A: Spring constant k=8g/s2 and the mass stretched the spring by, l=245 cmThe damping coefficient…

Q: etermine whether the function u(x, t)=ecos() is a solution of the heat conduction equa- on u₁=a²uxx-

Q: 7. An outdoor swimming pool loses 0.05% of its water volume every day it is in use, due to losses…

A: Given : A outdoor swimming pool loses 0.05% of its water volume every day. The water is replaced at…

Q: Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at…

A: The heat equation given is as follows: k∂2u∂x2-hu=∂u∂t, 0<x<L,…

Q: A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the…

A: Given: Function, f(x)=150-15x Boundary conditions: u0,t=u3,t=0To find: ux,t, u42, 0.1

Q: A tank shaped like a vertical cylinder initially contains water to a depth of 9m. The bottom plug is…

Q: Consider the heat equation ƏT Ət ᏛᎢ . 7 əx² 0≤x≤n. Solve the equation using separation of variables…

A: The given problem is to find the solution for the given initial and boundary value problem of heat…

Q: H1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = 0. It…

A: a) Before the collision, the total momentum of the system is given by:P = m10 + m2(-v2)where v2 is…

Q: Q4: A vertical steel plate of dimensions 3cm x 3cmamd negligible thickness is in steady state…

Q: homogeneous solution to the second order constant coeﬀicient ODE’s. Solution must be real.

A: d3dx3y-3d2dx2y+3ddxy-y=0 A linear homogeneous ODE with constant coefficients has the form of…

Q: The temperature ne-dimensional distribution u(t) along a wire of length L is governed by th heat…

Q: Find the maximum possible temperature of an ideal gas undergoing the process p=p0e−βV, where β is a…

A: The given function is where p is the pressure ,β is a positive constant and V is the molar volume…

Q: Consider a lake of constant volume V (gallons) containing at time t (weeks) an amount Q(t) (grams)…

A: As per norms, we will be answering only first three subparts. If you need an answer to others kindly…

Q: A toy rocket is lauched from the top of a building 120 feet tall at an initial velocity of 200…

A: To find the time at which the rocket returns to a height of 120 feet, we set the position equation…

Q: Fund A accumulates at a force of interest of Sª = 0.03 1 + 0.05t at a constant force of interest of…

A: Given: For fund A, Force of interest: 0.051+0.05t at time t for t≥0, For fund B, Constant force of…

Q: A race Runner A begins at the point s(0) = 0 and runs on a straightand level road with velocity ν(t)…

A: Given: Runner A start at the point s0=0 and run with velocity vt=2t and runner B start at point s0=8…

Q: (4) Suppose the acceleration of an object is given by a(t) = t m/s. Its initial velocity is v(0) = 5…

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

A rod of length n coincides with the interval [0, 1] on the x-axis.
The rod has initial temperature f(x) = 5 for 0 < x <n with the ends at
temperature zero for all t > 0. Assume the rod satisfies all the
assumptions of the heat equation with a constant a? of three.
Set up an initial-boundary value problem for the temperature u(x,t), then find
the solution, the temperature u(x, t).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

2. Find all the values of the constants a and b such that U(x, t) = eaz+bt satisfies the heat equation.
A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the bar is k = 0.4. The bar's initial temperature f(x) = 50 degrees Celsius. The ends of the bar (x = 0 and x = 3) are then put in %3D an icy bath and kept at a constant 0 degrees C. Let u(x, t) be the temperature in the bar at x at time t, with t measured in seconds. Find u(x, t) and then U7 (2, 0.1). Put u (2, 0.1) calculated accurately to the nearest thousandth (3 decimal places) in the answer box.
A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the bar is k = 0.4. The bar's initial temperature f(x) = 90 degrees Celsius. The ends of the bar (x = 0 and x = 3) are then put in an icy bath and kept at a constant 0 degrees C. Let u(x, t) be the temperature in the bar at x at time t, with t measured in seconds. Find u(x, t) and then uz (2, 0.1). Put u7 (2, 0.1) calculated accurately to the nearest thousandth (3 decimal places) in the answer box.
a (x, v) a(x.t) For 0Sxsl with a step length h=0.2 and 0SIS0.2 with a step length k =0.02 where (x.0) = x (0,t)=0 and of (x, y) = 0.25 6. Use the Newton-Raphson iterative method to solve the following: Show that a root of the equation x+3x +5x+9=0 occurs between x = -2 and x -3. Evaluate the roots to 4 significant figures Show that the equation x+3sin x=2 has a root between x = 0.8 and x = 0.6. ii. Evaluate the root correct to 5 significant figures. ii. Given the table of values f(x) -2.63906 -2 -2.48491 -1.94591 -1.79176 Use Langrangian interpolation find the Value of (a) fl-0.8) (b) f(0.8) (c) f(5.5)
The handle of a newly manufactured skillet pan, which is of length 6 inches, has a perfectly insulated surface. To test its heat resistance, the temperature at both ends of the handle is kept consistently at zero. The initial temperature of the handle throughout follows the room temperature at 26°C. The 1-dimensional heat equation is given as follows: = 0.5 for all 0
Given the equation of state of a PVT system is bT² √P Where b is a constant. V = (a) Calculate the thermal expansivity B =) - = (3/P) ₁ 1 (c) Express the total differential of V in terms of ß and K. (b) Calculate the compressibility K =