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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 phối of rules or equations. The best way to convert differential equations into algebraic equations is the use of Laplace transformation.

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Laplace transformation plays a major role in control system engineering. Lớn analyze the control system, Laplace transforms of different functions have to be carried out. Both the properties of the Laplace transform and the inverse Laplace transformation are used in analyzing the dynamic control system. In this article, we will discuss in detail the definition of Laplace transform, its formula, properties, Laplace transform table & its applications in a detailed way.

Table of Contents:

What is the Laplace Transform?

A function is said to be a piecewise continuous function if it has a finite number of breaks and it does not blow up khổng lồ infinity anywhere. Let us assume that the function f(t) is a piecewise continuous function, then f(t) is defined using the Laplace transform. The Laplace transform of a function is represented by Lf(t) or F(s). Laplace transform helps lớn solve the differential equations, where it reduces the differential equation into an algebraic problem.

Laplace Transform Formula

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

The Laplace transform of f(t), that is denoted by Lf(t) or F(s) is defined by the Laplace transform formula:

whenever the improper integral converges.

Standard notation: Where the notation is clear, we will use an uppercase letter khổng lồ 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) và 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)
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)

Laplace Transform Table

The following Laplace transform table helps khổng lồ solve the differential equations for different functions:

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 lớn transform the problem into another problem that is easier khổng lồ solve. On the other side, the inverse transform is helpful to calculate the solution khổng lồ the given problem.

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

Consider y’- 2y = e3x & 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

Step Functions

The step function is often called the Heaviside function, and it is defined as follows:

(eginarraylu_c(t)= left{eginmatrix 0 và if t

The step function can take the values of 0 or 1. It is like an on và off switch. The notations that represent the Heaviside functions are uc(t) or u(t-c) or H(t-c)

Bilateral Laplace Transform

The Laplace transform can also be defined as bilateral Laplace transform. This is also known as two-sided Laplace transform, which can be performed by extending the limits of integration to lớn be the entire real axis. Hence, the common unilateral Laplace transform becomes a special case of Bilateral Laplace transform, where the function definition is transformed is multiplied by the Heaviside step function.

The bilateral Laplace transform is defined as:

(eginarraylF(s)=int_-infty ^+infty e^-stf(t)dtendarray )

The other way to represent the bilateral Laplace transform is BF, instead of F.

Inverse Laplace Transform

In the inverse Laplace transform, we are provided with the transform F(s) and asked lớn find what function we have initially. The inverse transform of the function F(s) is given by:

f(t) = L-1F(s)

For example, for the two Laplace transform, say F(s) & G(s), the inverse Laplace transform is defined by:

L-1aF(s)+bG(s)= a L-1F(s)+bL-1 G(s)

Where a và b are constants.

In this case, we can take the inverse transform for the individual transforms, and địa chỉ cửa hàng their constant values in their respective places, & perform the operation khổng lồ get the result.

Also, check:

Convolution Integrals

If the functions f(t) và g(t) are the piecewise continuous functions on the interval <0, ∞), then the convolution integral of f(t) and g(t) is given as:

(f * g) (t) =0∫t f(t-T) g(T)dT

As, the convolution integral obey the property, (f*g)(t) = (g*) (t)

We can write, 0∫t f(t-T) g(T)dT = 0∫t f(T) g(t-T)dt

Thus, the above fact will help us to lớn take the inverse transform of the hàng hóa of transforms.

(i.e.) L(f*g) = F(s) G(s)

L-1 F(s)G(s) = (f*g)(t).

Laplace Transform in Probability Theory

In pure và applied probability theory, the Laplace transform is defined as the expected value. If X is the random variable with probability density function, say f, then the Laplace transform of f is given as the expectation of:

Lf(S) = E, which is referred lớn as the Laplace transform of random variable X itself.

Applications of Laplace Transform

It is used lớn convert complex differential equations to lớn a simpler form having polynomials.It is used khổng lồ convert derivatives into multiple domain name variables & then convert the polynomials back to lớn the differential equation using Inverse Laplace transform.It is used in the telecommunication field to lớn send signals lớn 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 và then superimposed 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.

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Laplace Equation

Laplace’s equation, a second-order partial differential equation, is widely helpful in physics và maths. The Laplace equation states that the sum of the second-order partial derivatives of f, the unknown function, equals zero for the Cartesian coordinates. The two-dimensional Laplace equation for the function f can be written as: