# Partial differential equation-delta Dirac& Heaviside function

• dragonxhell
In summary, The conversation is about a student asking for help with three math questions involving Laplace transforms and partial fraction expansion. The first question is about rearranging functions for inverse Lagrange and the student is unsure if their solution is correct. The second question involves using Laplace transforms to solve for a specific graph and the student is asking for someone to check their solution. The third question is about using the shifting theorem and partial fraction expansion to solve an equation with an e-sT term. The conversation also briefly touches on the student's confusion about the name of the thread and their teacher's method for solving partial fraction expansion.

#### dragonxhell

I got 2 questions to ask! I have finished one but not sure if it's correct so I need to double check with someone :)

http://imageshack.us/a/img708/1324/83u8.png [Broken]

Here is my worked solution, I took this picture with my S4 and I wrote is very neatly as I could! The reason I didn't type it all out since the maths is too much, so I'm sorry ): Question 2 answer: http://img23.imageshack.us/img23/134/8knw.jpg [Broken]

and http://img202.imageshack.us/img202/1538/klg5.jpg [Broken] If you are wondering what graph I'm talking about, here it is: http://img16.imageshack.us/img16/4435/s2yd.png [Broken]

For the 3 question, I gave it a try but I'm sure it's wrong so if you could solve it for me. It is very nice of you! I know it's in this form t^2 [H(t-3)-H(t-4)] Thank you for taking your time to help me! Cheers

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Did you cover the Laplace transform?

rude man said:
Did you cover the Laplace transform?

Yes, I did. Sometime its hard to rearrange function for inverse lagrange.
I managed to work out the solution for question 3. So could you or someone check my question 2 please!

EDIT: I need to correct a typo:

Don't go e-sT = 1/esT. T a constant, in your case 10 and 30.
Any F(s)e-sT inverses to f(t-T)U(t-T) where F(s) inverses to f(t) (shifting theorem). Use that fact whenever you get an e-sT coefficient in the s domain.

Your original transformed equation is correct.

I wouldn't try partial fraction expansion with an esT term in the original F(s). Do your p/f expansion with the polynomial expression only, then use the shifting theorem.

OK, I looked at your solution. I got 2/5 instead of 4/5 for the sin(t-10) term. I might as well confess I used Wolfram Alpha.

But the main problem was whenver you had an exp term you wrote e-at instead of e-a(t-T).

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rude man said:
Don't go e-sT = 1/esT. T a constant, in your case 10 and 30.
Any F(s)e-sT inverses to f(t)U(t-T) where F(s) inverses to f(t) (shifting theorem). Use that fact whenever you get an e-sT coefficient in the s domain.

Your original transformed equation is correct.

I wouldn't try partial fraction expansion with an esT term in the original F(s). Do your p/f expansion with the polynomial expression only, then use the shifting theorem.

Well I know the Laplace you are talking about, I was using it for question 1.
For an example 3/(s+3)^3
Y(t)= 3/2 t^2 e^-3t
I usually do the algebra way but this question is too hairy so that's why I'm not sure if my working out is correct so if you could work your way and compare to my answer see you would get the same?

My teacher do this method but I'm not sure if its right though
1/s e^10s (s^2+4s+5)
=1/5 [(1/s -s+2/(s+2)^2+1 -2/(s+2)^2+1 ]e^-10s

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PS why did you call this thread "partial" differential equation?

PPS look again at my post #4.

rude man said:
PS why did you call this thread "partial" differential equation?

PPS look again at my post #4.

Well... that's the name of the math course I'm doing so yeah
I think my p/f is wrong and my teacher way is right
1/s e^10s (s^2+4s+5)
transform to and then you just look up the table
=1/5 [(1/s -s+2/(s+2)^2+1 -2/(s+2)^2+1 ]e^-10s

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## 1. What is a partial differential equation?

A partial differential equation (PDE) is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to model and study physical systems that involve changing quantities, such as heat, fluid flow, and electromagnetic fields.

## 2. What is the delta Dirac function in PDEs?

The delta Dirac function, also known as the delta function, is a mathematical function that is used to represent a point mass or impulse in PDEs. It is often used to model point sources or point charges in physical systems.

## 3. How is the Heaviside function used in PDEs?

The Heaviside function, also known as the unit step function, is a mathematical function that is used to represent a sudden change or discontinuity in a quantity in PDEs. It is often used to model the behavior of physical systems at a specific point or boundary.

## 4. What is the relationship between the delta Dirac and Heaviside functions?

The delta Dirac and Heaviside functions are closely related and are often used together in PDEs. The delta Dirac function can be defined as the derivative of the Heaviside function, and the Heaviside function can be defined as the integral of the delta Dirac function.

## 5. Why are the delta Dirac and Heaviside functions important in PDEs?

The delta Dirac and Heaviside functions are important in PDEs because they allow us to model and solve physical systems with point sources, discontinuities, and other complex behavior. They are also used in many other areas of mathematics and engineering, such as signal processing, control theory, and quantum mechanics.