Laplace transformation (definition, formula, properties and examples)Laplace Transform Differential comparisons (2024)

Laplace transformation of differential comparison

The Laplace transformation is a well -set mathematical technique for solving a differential comparison.is useful to calculate the solution for the problem given.

For a better understanding, let's solve the first -order differential comparison using Laplace transformation,

Consider 2Y = E^ZKSand y (0) = -5.Find the value of L (Y).

First step in the comparison can be resolved using the linearity comparison:

L (I - = (e^ZKS))

L (j) - l (u) = 1/(s)

(omdat l (e^axe) = 1/(s-a))

L (y ') -2s (y) = 1/(s -3)

SL (y) -Y (0) -2l (y) = 1/(s -3)

(With the help of linearity properties of Laplace transformation)

L (Y) (S-2) + 5 = 1/(S-3) (Use value of y (0) IE -5 (given))

L (Y) (S-2) = 1/(S-3) -5

L (y) = (-5s+16)/(S-2) (S-3)… .. (1)

Here (-5s + 16)/(S-2) (S-3) can be written as -6/s-2 + 1/(S-3) using partial fraction method

(1) includes l (y) = -6/(s-2) + 1/(S-3)

L (y) = -6e^X+ e^ZKS

Trinal functions

The trim function is often called the gravity function and is defined as follows:

\) Right. \ End {array} \)

The step function can be the values ​​of 0 or 1. It is like a switch in and out.The notations that represent the heavy functions are you_C(T) Eller U (T-C) Eller H (T-C)

Bilateral Laplace transformation

The Laplace transformation can also be defined as a bilateral Laplace transformation.Transformed multiplied by the step function of the heavy ride.

The bilateral Laplace transformation is defined as:

\)

The other way to display the bilateral Laplace transformation is b {f} instead of F.

Inverse Laplace transformation

In the reverse Laplace transformation we get the transformation F (s) and asked to find which function we originally have.

f (t) = l^-1{F (s)}

For example, the inverse Laplace transformation for the two Laplace transformation, e.g. defined F (s) and G (s).

L^-1{of (s)+bg (s)} = a l^-1{F (s)}+bl^-1{G (s)}

Where A and B are constants.

In this case we can take the reverse transformation of the individual transformations and add their constant values ​​to their respective places and perform the operation to get the result.

Also check:

Convolution Integrals

If the functions f (t) and g (t) The continuous functions of the interval are [0, ∞), it gets the convolution integral of f (t) and g (t) as:

(f * g) (t) =₀∫^Tf (t-t) g (t) dt

As the convolution fully perceives the property, (f*g) (t) = (g*) (t)

We can write,₀∫^Tf (t-t) g (t) dt =₀∫^Tf (t) g (t-t) dt

The above fact will help us take the reverse transformation of the product of transformations.

(dvs.) l (f*g) = f (s) g (s)

L^-1{F (s) g (s)} = (f*g) (t).

Laplace transformation in probability theory

In the pure and applied probability theory, the Laplace transformation is defined as the expected value.

L {f} (s) = e [e^-SX], referred to as the Laplace transformation of random variable X itself.

Applications of Laplace transformation

• It is used to convert complex differential comparisons into a simpler form of polynomes.
• Ofisused toConvert derivatives toMore domain variablesAnd then convert the polynomes back to the differential comparison with the help of inverted Laplace transformation.
• It is used in the Telecommunication field fieldUnpleasantSend signals to bothThe pages of the media.For example when the signals are sentover the phoneAndThey are first transformed into a time -oriented wave and the media exceived.
• It is also used for many technical tasks, such as analysis of electrical circulation, digital signal treatment, system modeling, etc.

Laplace -Transformation Examples

Below are examples based on some important basic characteristics of Laplace transformation.

Laplace -comparison

Laplaces -comparison, a different order partial differential comparison, is much useful in physics and mathematics.For the function F can be written as:

The Laplace comparison for three-dimensional coordinates can be displayed as:

Download Byju's the-Learning app and receive personalized videos to understand the mathematical concepts.