Question: (a) Show that u(t, x) = e-k2t+ikx is a complex solution to the heat equation u/t = 2u/x2 for any real constant k. (b) Write

(a) Show that u(t, x) = e-k2t+ikx is a complex solution to the heat equation
∂u/∂t = ∂2u/∂x2
for any real constant k.
(b) Write down another complex solution by using complex conjugation.
(c) Find two independent real solutions to the heat equation.
(d) Can k be complex? If so, what real solutions are produced?
(e) Which of your solutions decay to zero as t → ∞?

Step by Step Solution

★★★★★

3.38 Rating (151 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a ut k 2 e k2 ti kx 2 ux 2 b e k2 t i kx c e k2t cos kx e k2t sin kx d Yes Wh... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

952-M-L-A-E (2549).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Find all complex exponential solutions u(t, x) = ewt+kx of the beam equation 2u/t2 = 4u/x4. How many different real solutions can you produce?

Write down all solutions to the following boundary value problems. Label your answer as (i) Unique solution, (ii) No solution, (iii) Infinitely many solutions. (a) u" + 2u = 2a, u(0) = 0, u() = 0 (b)...

The wave equation c2 (2u/(x2 = (2u/(t2 and the heat equation c (2u/(x2 = (u/(t are two of the most important equations in physics (c is a constant). These are called partial differential equations....

5, MATH 525 FALL 2015 This homework is due in class Nov 18th. Please write up your solutions on your own and give complete arguments. I. Let u(x, t) be the solution to the heat equation with diusion...

\fCOMPLEX VARIABLES AND APPLICATIONS Eighth Edition James Ward Brown Professor of Mathematics The University of Michigan-Dearborn Ruel V. Churchill Late Professor of Mathematics The University of...

MAT 402 - EBM 607 Mini Project In this project we will go over all steps of scientific computing. We will first derive the differential equation of a physical phenomenon, namely, unsteady heat...

SOLVE Questions 1-5: PLEASE AND THANK YOU 1: What is the difference between homogenous and non-homogenous differential equation? Write an example for each and solve it. 2: What is the main formula to...

9/14/10 12:39 PM Page 381 Chapter dug84356_ch06a.qxd 6 Rational Expressions Advanced technical developments have made sports equipment faster, lighter, and more responsive to the human body. Behind...

Can you please help me with number 31. I have provided the math chapter it came from and the answer from the back of the book: Thank you so much and Best wishes! 5.7 Matrix Exponentials and Linear...

What information should you include as a table or figure? Explain the decision process for choosing which results should have a visual display.

Upon spotting an insect on a twig overhanging water, an archer fish squirts water drops at the insect to knock it into the water (Figure). Although the fish sees the insect along a straight-line path...

1 Discuss what is meant by the term stakeholder salience and why it might be important to incorporate this theory into a businesss communication strategy.

Use the following graph for Yolandas Frozen Yoghurt Stand to answer the questions that follow. a Use the midpoint formula to calculate the price elasticity of demand for D1 between point A and point...

Alter the program MonteCarlo to estimate the area under the graph of y = 1/(x + 1) in the unit square in the same way as in Exercise 4. Calculate the true value of this area and use your simulation...

In the preceding exercise, it is natural to ask How do we get the information that the given hand has an ace? Gridgeman considers two different ways that we might get this information. (Again, assume...

Using the notation introduced in Example 4.29, let Show that there is exactly one value of x such that if your envelope contains x, then you should switch. if r = 2*. Pa =(4)*. 0, otherwise.

Consider the following cash flows: Year Cash Flow 0 $-29,800 1 13,900 2 15,000 3 11,400 Requirement 1: What is the profitability index for the above set of cash flows if the relevant discount rate is...

Question 6 - The firm has two divisions, divisions A and B, each with the same level of risk. The firms after-tax cost of capital is 16%. The firms EVA was $900,000 last year. The B division had an...

1 1 . Short - Term Financing Requirements. Suppose that Paymore's cash balance ( from Problem 7 ) at the start of the first quarter is $ 4 0 and its minimum is $ 3 0 . Work out the short - term...