Laplace Transform

Laplace transform is named in honour of the great French mathematician, Pierre Simon De Laplace (1749-1827). Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations. The best way to convert differential equations into algebraic equations is the use of Laplace transformation. In this section, students get a step-by-step explanation for every concept and will find it extremely easy to understand this topic in a detailed way.

Contents:

Laplace Transform Formula

Laplace transform is the integral transform of the given derivative function with real variable t to convert into complex function with variable s. For t ≥ 0, let f(t) be given and assume the function satisfies certain conditions to be stated later on.

The Laplace transform of f(t), that it is denoted by f(t) or F(s) is defined by the equation

Laplace Transform Formula

whenever the improper integral converges.

Standard notation: Where the notation is clear, we will use an uppercase letter to indicate the Laplace transform, e.g, L(f; s) = F(s).

The Laplace transform we defined is sometimes called the one-sided Laplace transform. There is a two-sided version where the integral goes from −∞ to ∞.

Properties of Laplace Transform

Some of the Laplace transformation properties are:

If f1 (t) ⟷ F1 (s) and [note: ⟷ implies Laplace Transform]

f2 (t) ⟷ F2 (s), then

Linearity Property A f1(t) + B f2(t) ⟷ A F1(s) + B F2(s)
Frequency Shifting Property es0t f(t)) ⟷ F(s – s0)
nth Derivative Property (dn f(t)/ dtn) ⟷ sn F(s) − ni = 1 sn − i fi − 1 (0)
Integration t0 f(λ) dλ ⟷ 1⁄s F(s)
Multiplication by Time T f(t) ⟷ (−d F(s)⁄ds)
Complex Shift Property f(t) e−at ⟷ F(s + a)
Time Reversal Property f (-t) ⟷ F(-s)
Time Scaling Property f (t⁄a) ⟷ a F(as)

Laplace Transform Table

Sl No. f(t) L(f(t)) = F(s) Sl No. f(t) L(f(t)) = F(s)
1 1 1/s 11 e(at) 1/(s − a)
2 tn at t = 1,2,3,… n!/s(n+1) 12 tp, at p>-1 Γ(p+1)/s(p+1)
3 √(t) √π/2s(3/2) 13 t(n-1/2) at n = 1,2,.. (1.3.5…(2n-1)√π)/(2n s(n+1/2)
4 sin(at) a/(s2+a2) 14 cos(at) s/(s2+a2)
5 t sin(at) 2as/(s2+a2)2 15 t cos(at) (s2-a2)/(s2+a2)2
6 sin(at+b) (s sin(b)+ a cos(b)/(s2+a2) 16 cos(at+b) (s cos(b)-a sin(b)/(s2+a2)
7 sinh(at) a/(s2-a2) 17 cosh(at) s/(s2-a2)
8 e(at)sin(bt) b/((s-a)2+b2) 18 e(at)cos(bt) (s-a)/((s-a)2+b2)
9 e(ct)f(t) F(s-c) 19 tnf(t) at n = 1,2,3.. (-1)n Fn s
10 f'(t) sF(s) – f(0) 20 f”(t) s2F(s) − sf(0) − f'(0)

Laplace Transform of Differential Equation

The Laplace transform is a well established mathematical technique for solving a differential equation. Many mathematical problems are solved using transformations. The idea is to transform the problem into another problem that is easier to solve. On the other side, the inverse transform is helpful to calculate the solution to the given problem.

For better understanding, let us solve a first-order differential equation with the help of Laplace transformation,

Consider y’- 2y = e3x and y(0) = -5. Find the value of L(y).

First step of the equation can be solved with the help of the linearity equation:

L(y’ – 2y] = L(e3x)

L(y’) – L(2y) = 1/(s-3)

(because L(eax) = 1/(s-a))

L(y’) – 2s(y) = 1/(s-3)

sL(y) – y(0) – 2L(y) = 1/(s-3)

(Using Linearity property of the Laplace transform)

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) implies L(y) = -6/(s-2) + 1/(s-3)

L(y) = -6e2x + e3x

Inverse Laplace Transform

The inverse of complex function F(s) to produce a real valued function f(t) is an inverse laplace transformation of the function. If a unique function is continuous on o to ∞ limit and have the property of Laplace Transform,

F(s) = L {(t)} (s);

is said to be an Inverse laplace transform of F(s).  It can be written as, L-1 [f(s)] (t). This function is an exponentially restricted real function. To learn more in detail visit the link given for inverse laplace transform.

Also, check:

Applications of Laplace Transform

  • It is used to convert complex differential equations to a simpler form having polynomials.
  • It is used to convert derivatives into multiple domain variables and then convert the polynomials back to the differential equation using Inverse Laplace transform.
  • It is used in the telecommunication field to send signals to both the sides of the medium. For example, when the signals are sent through the phone then they are first converted into a time-varying wave and then super-imposed on the medium.
  • It is also used for many engineering tasks such as Electrical Circuit Analysis, Digital Signal Processing, System Modelling, etc.

Laplace Transform Examples

Below examples are based on some important elementary functions of Laplace transform.

Laplace Transform Examples -1

Laplace Transform Examples-2

Download BYJU’S-The Learning App and get personalised videos to understand the mathematical concepts.

Frequently Asked Questions on Laplace Transform- FAQs

What is the use of Laplace Transform?

The Laplace transform is used to solve differential equations. It is accepted widely in many fields. We know that the Laplace transform simplifies a given LDE (linear differential equation) to an algebraic equation, which can later be solved using the standard algebraic identities.

How do you calculate Laplace transform?

The steps to be followed while calculating the laplace transform are:
Step 1: Multiply the given function, i.e. f(t) by e^{-st}, where s is a complex number such that s = x + iy
Step 2; Integrate this product with respect to the time (t) by taking limits as 0 and ∞.
This process results in Laplace transformation of f(t), and is denoted by F(s).

What is the Laplace method?

The Laplace transform (or Laplace method) is named in honor of the great French mathematician Pierre Simon De Laplace (1749-1827). This method is used to find the approximate value of the integration of the given function. Laplace transform changes one signal into another according to some fixed set of rules or equations.

What are the properties of Laplace Transform?

The important properties of laplace transform include:
Linearity Property: A f_1(t) + B f_2(t) ⟷ A F_1(s) + B F_2(s)
Frequency Shifting Property: es0t f(t)) ⟷ F(s – s0)
nth Derivative Property: (d^n f(t)/ dt^n) ⟷ s^n F(s) − n∑i = 1 s^{n − i} f^{i − 1} (0^−)
Integration: t∫_0 f(λ) dλ ⟷ 1⁄s F(s)
Multiplication by Time: T f(t) ⟷ (−d F(s)⁄ds)
Complex Shift Property: f(t) e^{−at} ⟷ F(s + a)
Time Reversal Property: f (-t) ⟷ F(-s)
Time Scaling Property: f (t⁄a) ⟷ a F(as)

What is the Laplace transform of sin t?

The laplace transform of f(t) = sin t is L{sin t} = 1/(s^2 + 1). As we know that the Laplace transform of sin at = a/(s^2 + a^2).

1 Comment

  1. I would like to meet Divya Mam if I would get a chance

Leave a Comment

Your email address will not be published. Required fields are marked *

BOOK

Free Class