4. The second order PDE ut = o²ur, where o is a real constant is called the (one-dimensional) heat equation (also referred to as the diffusion equation), whose solution is u = u(x, t). Here, as before, a is the space variable and t is the time variable. Show that u(x, t) = t-¹/²e-²/t, t> 0 is a solution to the heat equation, with o² = 1/4.

4. The second order PDE ut = o²ur, where o is a real constant is called the (one-dimensional) heat equation (also referred to as the diffusion equation), whose solution is u = u(x, t). Here, as before, a is the space variable and t is the time variable. Show that u(x, t) = t-¹/²e-²/t, t> 0 is a solution to the heat equation, with o² = 1/4.

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

please answer 19 through 24
Given yi (t) t² and y₂(t) = t¹ satisfy the corresponding homogeneous equation of t'y" - 2y = -2t¹-3, t> 0, the general solution to the nonhomogeneous equation can be written as y(t) = y(t) + C₁y₁(t) + C2y2(t). Use variation of parameters to find y, (t). Yp(t) Preview
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
(1-t)y +ty -y = (t² – 1)e*. (a) Show that yı = t and y2 = et are solutions of the associated homogeneous equation %3D (1-t)y +ty - y = 0. 3D0.
2. Find all the values of the constants a and b such that U(x, t) = eaz+bt satisfies the heat equation.
Which of the following statements is true for all invertible 3x3 matrices A and B? (3ABA-!)-|B = -3AB¬'A-!B 1 (3ABA-|)-'B = -AB-A-"B 1 (3ABA-!)-'B = 3 1 (3ABA¬!)-'B = =ABA-'B-!
A certain radioactive substance has a half-life of 3.6 years. Find how long it takes (in years) for the substance to decompose to 10% of its initial amount.
Given yı (t) = t and y2(t) = t satisfy the corresponding homogeneous equation of t'y" – 2y = 1 – 2t°, t > 0 Then the general solution to the nonhomogeneous equation can be written as y(t) = c1y1(t) + ©2Y2(t) + Yp(t). Use variation of parameters to find a particular solution yp(t). Y,(t) = Tip: Before you use the formula Y29(t) Yı9(t) JW(n, y2) Yp = - Y1 + y2 W (y1, Y2) the ODE should be in the form y'' + a(t)y' + b(t)y = g(t)
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).
1. For the following partial differential equations find the ordinary differential equations that are implied by the method of separation of variables. (DO NOT solve the ordinary differential equations.) du ди a) (xu), where k is a given costant at a2. b) ди = a at where 6 are given constants а. | du c) 2 (x2 Ou), where k is a given constant d) du? a2. = C where c is a given constant
b. Find the solution to the heat equation du Pu 4- t > 0, at u(0, t) = u(3, t) = 0, u(z,0) = x. !!