Trouble with dirac delta in R^2

In summary, a distribution F in R^2 that satisfies (Dx) F(x,t) = t*Delta(x) is not t*H(x) as in R. The ODE u' + u = t*Delta(x) + Delta''(x) can be solved by multiplying both sides by exp(x) and integrating.
  • #1
obomov2
3
0
Find a distribution F in R^2 that satisfies (Dx) F(x,t) = t*Delta(x)
It is apperantly not t*H(x) as in R.

* is multiplication, D is dirac delta, H is Heavyside , (Dx) is derivation with respect to x (in the sense of distributions)

Sorry for not using Latex.

Indeed I am trying to solve following ODE for u(t,x) :
u' + u = t*Delta(x) + Delta'' (x)
 
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  • #2
obomov2 said:
Find a distribution F in R^2 that satisfies (Dx) F(x,t) = t*Delta(x)
It is apperantly not t*H(x) as in R.

why not?

obomov2 said:
Indeed I am trying to solve following ODE for u(t,x) :
u' + u = t*Delta(x) + Delta'' (x)

multiply both sides by exp(x) to get d/dx(exp(x)u) on the left hand side and integrate.
 

Related to Trouble with dirac delta in R^2

1. What is the Dirac delta function in two dimensions?

The Dirac delta function, denoted as δ(x,y), is a mathematical function used in two-dimensional space to represent a point mass or impulse. It is defined as zero everywhere except at the origin (x=0, y=0) where it has an infinite value, but with an area of 1 under its curve.

2. How is the Dirac delta function used in two-dimensional space?

The Dirac delta function is often used as a mathematical tool to simplify calculations involving point masses, such as in electromagnetism or fluid dynamics. It can also be used to represent a point source of energy or a point charge in two-dimensional space.

3. What are the properties of the Dirac delta function in two dimensions?

The Dirac delta function in two dimensions has similar properties to the one-dimensional version, including symmetry, scaling, and shifting. Additionally, it satisfies the integral property where its integral over any area containing the origin is equal to 1.

4. What are some applications of the Dirac delta function in two dimensions?

The Dirac delta function is commonly used in physics and engineering, such as in the study of electromagnetic fields, signal processing, and fluid dynamics. It is also used in image processing and computer graphics to represent point objects or singularities.

5. What are some challenges when working with the Dirac delta function in two dimensions?

One challenge is the interpretation of the infinite value at the origin, which may lead to mathematical inconsistencies if not properly handled. Another challenge is the need for careful handling when integrating or manipulating equations involving the Dirac delta function, as it can lead to incorrect results if not done correctly.

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