Consider the following example of the heat equation: Uf = 3 . · Uzz, u (x, 0) = 4 · (2 – x) · æ², 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).

Consider the following example of the heat equation: Uf = 3 . · Uzz, u (x, 0) = 4 · (2 – x) · æ², 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).

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: 4. The equation of motion for a damped system with periodic external force is given below. * + 8x +…

Q: Consider the autonomous first order equation y' = y° + 3y³ – 10y*.

Q: Q.3 Solve the heat equation: ************ U,=U.. U_(0,t)=0, U,(7,1)=0, U (x,0)=3-3cos2.x, *********…

A: The given problem is to find the solution for the given heat equation partial differential equation…

Q: 00 ди (0,t) 3 0%; ди ди (T, t) = 0; t> 0 at ax2 at дх и(х, 0) — seпх; 0 < x < T %3D

A: Given: The given heat transmission equation ∂u∂t=∂2u∂x2

Q: Ut = Uxx e-3t sin(5x) + e−²t sin x, -2t u(0,t) = u(T,t) =0, u(x,0) = sin(2x).

A: Introduction: Laplace transformation and inverse Laplace transformation is very significant in…

Q: How long will it take to drive 700 km at 100 km per hour, if you take a 1-hour break after every 5…

A: We have to find time taken to drive 700km using given data. We will use formulaes,…

Q: 3. Solve Bernoulli equation: xy' = y + y³

A: The differential equation is given as xy'=y+y3 It can be rewritten as…

Q: 13. (x? — ху + у?)dх — хуdy %3D 0

Q: Solve the IBVP for the heat equation = 2uxr, 00 e(0, 1) %3D и(4, ) 3 0, 1>0 u(x, 0) = 14sin(Tx) -…

A: Given, (1)…

Q: Q.3 Solve the heat equation: U, =U U (0,1) = 0, U(2,1) = 0, U(x,0) = 3sin 2x - 3sin 3x

A: The given problem is to solve the given heat equation with given initial conditions and boundary…

Q: Let Y(t)= X(t) be the solution of , ly(t). 6 - 1 Y' = -2 Y, Y(0) = 8 5 4 Write y(T/10), round your…

Q: 3. Solve the Euler-Cauchy equation. (xD² +4D) y=0, y(1)=12, y'(1)=-6

Q: Solve PDE : 1 Urr + -Ur + 2u 1 zU00 = 0, 1<r< 3; ВС: u(1,0) = 3, u(3, 0) = sin(30). %3D

A: Given, PDF is urr+1rur+1r2uθθ=0, 1<r<3; (1) and BC:…

Q: An insulated composite rod is formed of two parts arranged end te equal length. Part a has thermal…

A: Given, the differential equation ∂2u∂x2=∂u∂t, 0≤x≤12 (1)…

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: y'+y = cos(e*

A: Given equation is: y'+y=cosex

Q: b) d² y dx² +4y= cos 2x, y(0) = 2, y'(0) = 4

Q: 3. (4r'y-15r-y)dt + (r +3y-1)dy = 0 CS Scanned with CamScarner

A: We can solve this using exact equation of differential equation

Q: Solve the heat problem Ut = Uzz, 0 < I < 7, uz(0, t) = 0, Uz(7, t) = 0 u(x, 0) = 1 – cos(x) u(x, t)…

A: The given problem is to solve the given heat equation partial differential equation with given…

Q: An isolated bar is kept at zero temperature at its ends. If it's given initially temperature, f(x) =…

A: First of all the question has a small error in place of (dt) (dx) was printed. Which I had corrected…

Q: Solve the heat equation in the glass bar U; = 0,0034ux u(0, t) = 0 %3D u(24r , t) = 0 u(x, 0) = 16…

A: In this question, we use the separation of variable method. and then put x=8π

Q: Solve the given IVP: y" + 7y" + 33y' – 41y = 0; y(0) = 1, y'(0) = 2, y"(0) = 4.

A: The given initial value problem is y'''+7y''+33y'-41y=0 ;…

Q: 8. (y – In æ) dæ + (x +1) dy = 0, y(1) = 10 Ghenoral Form : y' + Pixly= Q(x) I (x): PLk) du : Final…

Q: Solve the following heat equation = ut (x, t) u(0, t) = t, u(1,t) = 0, u(x,0) = x². Uxx (x, t) + x…

Q: 9. Ut = 4uxx, (xo, to) = (5,4)

Q: 0 0 и(0,t) 3D 0; и(1,t) 3D 0; t>0 и(х,0) 3D 5seппх ди a2u = 2 at 1 E sen3пx; 0 <x<1 - 5

A: Given ∂u∂t=2∂2u∂x2 0<x<1 ;t>0u0,t=0; u1,t=0; t>0ux,0=5secπx-15sec3πx ; 0<x<1

Q: Y, (x) = x y, (x) = Xe %3D

Q: 13. Use Euler's method to approximate y(2) given the following information. Show all of your steps.…

Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

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).

Expert Solution

This question has been solved!

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

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Equations$

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

Solve the heat equation U; = 4u xx, - 00, %3D u(x,0)=6(x-2). 8. O A Ux,1) == Je-dax cos[@x-2)]dm. -88 C8 OB U(x,f) = Se4at cos[@x-2)]d o. 1 Ocu(x,t) = Je-4 cos[@(x+2)]d o. -8 O D. None of these. 8. OE U(x,t) = 1 Te-Aat sin[ @x-2)]dw. %3D -48
The motion of a mass-spring system with damping is governed by y''(t) + by' (t) +9y(t) = 0; y(0) = 3, and y'(0) = 0. Find the equation of motion and sketch its graph for b = 0, 2, 6, and 10. What is the equation of motion for b = 0? y(t) = 3 cos 3t (Type an exact answer, using radicals as needed.) What is the equation of motion for b = 2? y(t) = (Type an exact answer, using radicals as needed.) ( ・・・ )
Apply the improved Euler method to approximate the solution on the interval [0, 0.5] with step size h = 0.1. Construct a table showing values of the approximate solution and the actual solution at the points x = 0.1, 0.2, 0.3, 0.4, 0.5. y' y 3x-3, y(0) = 3; y(x) = 6+3x-3e* Complete the table below. (Round to four decimal places as needed.) ✗n Actual, y (xn) Improved Euler, yn 0.1 0.2 0.3 0.4 0.5
Find Jeneral Solution- S+y = Cos (e*)
• Sheet no.2: Exercises/solve the following D.E using L.T. 1. y"+3y'+2y = c', y(0) = 1, y'(0) = 0,
Consider the second order DE 2y" + 3y' + y = 0. Find two different solutions y1 and y2 of the form y = e"t.
Solve for homogeneous de
Find the value of a, if y(x) = (₁ +₂ In x) is solution of the Cauchy Euler equation Select one: O A. 2 O B. 2/3 O C. 1 O D. O E. 3 2 a²x²y" + a(a + 2)xy' + y = 0 x > 0(a = 0).
Show, expand and explain the solution in the first image. Show the complete solution. And get as well the constants using the conditions in second image.