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Table of contents
1.
Introduction
2.
Laplace Transform
2.1.
Transformation Table
2.2.
Properties
3.
Applications of Laplace Transform
4.
Problem
5.
FAQs
6.
Key Takeaways
Last Updated: Mar 27, 2024

Laplace Transform

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Introduction

Laplace Transform is one of the best ways to convert a differential equation into an algebraic equation. It is widely used in control system engineering, electrical circuit analysis, and many other engineering domains. Let's learn about the Laplace Transform in detail.

Laplace Transform

Laplace Transformation is an integral transformation of a function in real variable 't' to a function in complex variable 's.' It is a tool for solving differential equations. It transforms a linear differential equation into an algebraic equation.

The Laplace transform of a function f(t) is denoted by L{f(t)} or F(s) and can be calculated by the formula given below.

This integral formula is sometimes called a one-sided Laplace Transform because the integration is from 0 to +∞. When we take the integration limits from -∞ to +∞ , it is called a two-sided Laplace Transform.

Let’s calculate the Laplace Transform for a function f(t) = et

The Laplace transform of f(t) = et = F(s)

Now,

Hence the Laplace Transform of the given function is 1/(s-1)

Now I know what you must be thinking. You may wonder if you have to perform the tedious integration whenever you need to perform a Laplace Transform. The answer is a big NO. We have provided you with a table of transformations. In that transformation table, we put the function f(t) on the left side and the result F(s), found using the integration formula discussed above, on the right side. The transformation table is given below.

Transformation Table

Source

This table contains the most commonly used transformations, but there are many other transformations as well.

Properties

We know that all the Laplace Transformations we will have to perform are not listed in this table. But there is an exciting thing about it. We have some properties of Laplace Transform using which we can deduce the given function into standard functions (whose Laplace Transform is provided in the table). The properties of Laplace Transform are given below.

  1. Linearity Property
    If f(t) and g(t) are two functions and 'a' and 'b' are two real numbers, then
  2. Time Shifting Property
    For t ≥ a
  3. Time Scaling
  4. Frequency Shifting
  5. Frequency differentiation

    Where F(n)(s) is the nth derivative of F(s).
  6. Frequency Integration
  7. Differentiation
    If f(t) is n times differentiable, then

    For n=1, i.e. first derivative,
  8. Integration

  9. Convolution
  10. Initial Value Theorem
    This theorem enables us to find the initial value at time t = (0+) for a given Laplace transformation without finding f(t), which is a tedious job in those cases.

You can also read about Initial Value Theorem in detail.

  1. Final Value Theorem
    There are some applications where we need to know the value of f(∞) without calculating the whole f(t) function. For this purpose, we can use the final value theorem.
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Applications of Laplace Transform

Laplace Transform has many applications in science and engineering as it is a tool for solving differential equations. Some of the use cases of Laplace Transform are given below.

  • It is used to breakdown complex differential equations into polynomial equations, which are much easier to solve and save a lot of trouble.
  • It is used in the analysis of electrical and electronic circuits.
  • It is also used in Digital Signal Processing and System Modelling.

Problem

We have discussed numerous times in this article that Laplace Transform is used to solve differential equations. So, let's try to solve a differential equation to understand its work better.

Solve the differential equation d2x/dt2 + dx/dt + x = 0, given x(0+)=0 and x`(0+)=1.

The given equation is d2x/dt2 + dx/dt + x = 0

Lets take Laplace Transform on both sides.

L{d2x/dt2 + dx/dt + x} = L{0}

L{d2x/dt2} + L{dx/dt} + L{x} = 0

s2L{x} - 1 + sL{x} + L{x} = 0

L{x} [s2 + s + 1] = 1

L{x} = 1 / (s2 + s + 1)

L{x} = (2/√3) . {(√3/2) / [(s+0.5)2 + (√3/2)2]} [converting in this form for inverse]

Now, taking Inverse Laplace Transform on both sides.

x = (2/√3) e-0.5t Sin(√3/2 . t)

As you can see, solving this differential equation required only the knowledge of some algebra, Laplace Transform, and Inverse Laplace Transform.

FAQs

  1. What is Laplace transform?
    Laplace Transformation is an integral transformation of a function in real variable 't' to a function in complex variable 's.'
  2. What is the linearity property?
    Linearity property states that Laplace transform of a.f(t) + b.g(t) is a times L{f(t)} added to b times L{g(t)}.
  3. What is the use of Laplace Transformation?
    Laplace Transform is used to solve differential equations. It simplifies a linear differential equation into an algebraic function which becomes easy to solve.
  4. What is Initial Value Theorem?
    Initial Value Theorem (IVT) is used to find the initial value at time t = (0+) for a given Laplace transformation without the need to find f(t). 
  5. What is the Final Value Theorem?
    The final Value Theorem is used to find the value of f(∞) without calculating the whole f(t) function.

Key Takeaways

In this article, we have extensively discussed the Laplace Transform topic. We hope that this blog has helped you enhance your engineering mathematics knowledge. If you would like to learn more, check out our other articles.

  1. Inverse Laplace Transform
  2. Laplace transform of a Unit Step function
  3. Laplace Transform Properties
  4. Arithmetic Progression
  5. Mean, Variance, and Standard Deviation

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