Help with heat equation dirac delta function?

In summary, the conversation discusses a lengthy question that is too long to type out, so a snapshot of the question is provided. The equations for the question are included in the snapshot. The person has attempted part A of the question and simplified it to 1/(\sqrt(2\pi)\delta)^2\iint\exp(-(r^2)/2\delta^2) rdrd\theta (0<\theta<2\pi) (0<r<inifity), but then got a wrong answer of -1 instead of 1. They are unsure of where they went wrong and are asking for help and hints on how to start the next few questions.
  • #1
fredfuchs
2
0

Homework Statement


The question was way too long so i took a snap shot of it
http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg

Homework Equations


The equations are all included in the snapshot

The Attempt at a Solution


So for question A I've done what the question said:

so g(x)^2 :

[tex]= 1/(\sqrt(2\pi)\delta)^2\iint\exp(-(x^2+y^2)/2\delta^2) dxdy[/tex]
so i simplified to:
[tex]1/(\sqrt(2\pi)\delta)^2\iint\exp(-(r^2)/2\delta^2) rdrd\theta (0<\theta<2\pi) (0<r<inifity)[/tex]
so i let z= r/2\delta^2

after all that i get to

[tex]1[\exp(\infty)-\exp(0)[/tex]

its suppose to = 1, but i got -1 instead
i know it's wrong, but i don't know here, and the next 2 questions i don't have a clue how to start, can you help me?
 
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  • #2
Show us what you tried. Surely, the first part should be simple.
 
  • #3
I just added into what i did
sorry i did't know how to post mathematical equations until i looked it up before...
yeah, after that I pretty much have no clue...
Can you give me a few hints how to start the next few questions?
 

1. What is the heat equation?

The heat equation is a partial differential equation that describes the flow of heat in a given region over time. It is commonly used in physics and engineering to model heat transfer in various systems.

2. What is the Dirac delta function?

The Dirac delta function, also known as the unit impulse function, is a mathematical function that is defined as zero everywhere except at a single point, where it is infinite. It is commonly used in physics and engineering to represent point sources or idealized inputs in mathematical models.

3. How is the Dirac delta function used in the heat equation?

The Dirac delta function is often used as an initial condition or boundary condition in the heat equation, representing a sudden, localized input of heat at a specific point in time and space. It allows for more precise and accurate modeling of heat transfer in certain situations.

4. Can the heat equation with Dirac delta function be solved analytically?

It depends on the specific problem being modeled. In some cases, the heat equation with Dirac delta function can be solved analytically using techniques such as separation of variables. However, in more complex situations, numerical methods may be necessary.

5. What are some real-world applications of the heat equation with Dirac delta function?

The heat equation with Dirac delta function has many practical applications, including modeling heat transfer in materials, predicting temperature changes in electronic devices, and studying thermal diffusion in the human body. It is also used in fields such as meteorology, geology, and astrophysics to understand heat transfer processes in natural systems.

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