Basic Properties of Laplace Transformation in Mathematics

Laplace Transformation

The Laplace transform is a powerful mathematical tool with diverse applications in fields as diverse as engineering, physics, economics, and control theory. It is a mathematical technique that converts a time function into an s function and enables analysis of the function on a linear level. The Laplace transformation provides a method for solving differential equations and analyzing the behavior of linear proportional- invariant systems [5].

The Laplace transform was introduced to expand the category of frequency signals that can be interpreted by numerical methods. The problem of solving differential equations also manifests itself in the analysis and integration of the system as a whole. The analysis of the behavior of linear dynamics with time invariants is reduced to the problem of solving a system of differential equations with constant coefficients. The solution of these equations is simplified using the Laplace transformation [3].

The Laplace transform of the function f(t) is defined as follows:

∞ − = = ∫ (2)

( ) { } ( ) ( )

st L f t F s x t e dt

0 where the complex variable s=δ+jω is called the complex frequency (frequency), and the real part of the complex variable s, Re{s}=δ, is chosen to ensure the convergence of the integral. The Laplace transform defined by the above expression is called the bilateral or two-sided Laplace transform. The two-sided Laplace transform is used for the analysis of power systems, as well as in certain applications in telecommunications and signal processing [6].

For the analysis of causal signals (x(t) = 0 for t<0), which are transmitted and processed by linear time-invariant causal (physically realizable), real systems, a fundamental role is played by the unilateral or one-sided Laplace transformation, which is defined by with the expression:

∞ − = = ∫ (3)

( ) { } ( ) ( )

st L f t x t e dt F s −

0

Sometimes in the literature, the unilateral Laplace transform is defined as:

∞ − = = ∫ (4)

( ) { } ( ) ( )

st L f t F s x t e dt

0 The difference between these two definitions is only in the lower limit of integration, which in the first definition is 0–, and if the signal x(t) does not contain the impulse delta signal δ(t) in the coordinate origin (t = 0), these two are basically identical. However, if the signal x(t) contains an impulse delta function at the point t=0, it is necessary that the defining integral includes the entire delta impulse, so that the lower limit of integration must be t=0–. Let’s also note that the Laplace transformation was introduced in the theory of linear time-invariant systems primarily in order to determine the response of such a system to arbitrary excitation, and therefore also to impulse delta excitation. On the other hand, as shown earlier, the response of a static system consists of two parts (also known as the principle of superposition): the stimulus response (input or excitation signals) and the initial response, which represents the energy accumulated in the system before it occurs the excitement itself. If we assume that the time at which we set the stimulus for entering the system is t = 0, then the initial conditions must be defined immediately before applying the stimulus, that is, at t = 0–. Therefore, the first definition of the Laplace transform will enable the determination of the response of the system both to the input signal and to the initial conditions.

The Laplace transform is a mathematical function that transforms the process of capturing the Laplace transform. Transform the Laplace curve function using complex numbers back to its original position.

This paper has a wide application in the analysis and solution of time-invariant and differential equations in engineering, science and mathematics [7].

By reproducing the behavior of work in time, the Laplace transform provides a deep and practical approach to real- time work. It serves as a fundamental tool in understanding and solving different systems and phenomena in different scientific disciplines.

( ) { ( ) 1 L F s f t − = (5)

It is used to find the first time domain function from the Laplace transform. It is also used for solving differential equations and finding finite time solutions of systems defined in the Laplace domain [8].

The Laplace transform is used when different equations need to be converted into an algebraic form for easier calculation, study and analysis [9]. The Laplace transform is a fundamental transformation that transforms a function of a complex variable into a function of a real variable, usually time. The Laplace transform is a fundamental transformation commonly used to convert real-time differential equations to of polynomial equations in complex systems. It also helps solve linear differential equations because it transforms differential equations into simple algebraic operations and is an important part of applied mathematics, engineering, electrical engineering, and systems engineering.

The transformed functions and their solutions can be transformed back into the function in the original domain with the help of inverse integral transformations using inverse kernel functions, K-1(i,k). Such a transformation is called the inverse Laplace transform. To understand what the inverse Laplace transform is, it is necessary to understand the Laplace transform. If the function f(t) is defined for all +ve values of t. The Laplace transform is denoted by the formula;

The inverse Laplace transform is a transformation opposite to the Laplace transform, which maps the given function F(s) from the complex domain to the function f(t) in the time domain. It is symbolically denoted as ( ) ( ) { } 1 f t L F s − = (6) It is mainly used for the character structure problem given by the equation i with the real coefficients of two polynomials in the variable s as a real function of the variables to the:

( ) ( ) ( )

P s b s b s b s b F s Q s s a s a s a

1 1 1 0 1 1 1 0

m m m m n n n + + + + = + + + + Λ Λ (7)

− − − − where the degree of the polynomial in the numerator is less than or equal to the degree of the polynomial in the denominator (m ≤ n).

The zeros of the polynomials P(s) and Q(s) are called the poles and zeros of the real rational function F(s), respectively. Since these polynomials are with real coefficients, their zeros, that is, the poles and zeros of the complex character F(s), can appear either as real or in conjugate-complex pairs, and can be simple and/or multiple.

( )

1 1 1 0 0 n n n Q s s a s a s a − − = + + + + = Λ (8)

For finding the ILT, the poles of the function F(s), i.e., are of particular interest the zeros of the polynomial Q(s), i.e. the roots of the equation. This equation is called the characteristic equation, the characteristic polynomial polynomial.

Based on a known complex character, it is possible to determine the function (original) in the time domain by applying the inverse Laplace transformation. The original function F(s) in the time domain f(t) is determined using the following formula:

j st σ ω + −

( ) { } ( ) ( ) 1 1 ; 0 2

+ = = > ∫ (9)

j L F s f t F s e ds t j σ ω π the previous relations are usually by the inverse Laplace transform, and together with the relation:

∞ − −∞ = ∫ (10)

( ) ( ) st F s f t e dt It defines the so-called Laplace transform pairs. The fact that the functions f(t) and F(s) satisfy these two relations is often indicated in the literature by one of the following notations:

( ) ( ) { } ( ) ( ) { } ( ) ( ) 1 , , F s L f t x t L F s f s F s − = = ↔ (11)

the formula of the inverse Laplace transform tells us that the signal f(t) can be reconstructed on the basis of its Laplace transform pair, however, calculating this integral is very often a serious task and implies the so-called contour integration, which is studied in the theory of functions of complex variables. What we will mention here is that if the function F(s) exists, then the inverse Laplace transform must be calculated according to the line σ=const. That line must belong to the area of convergence of the function F(s), which means that the area of convergence must be such that there is a band of finite width and infinite length in it: σ1< Re{s} < σ2.

In general, the search for the Laplace transform is limited to the use of the definition formula, which requires the solution of a set of parameters that represent a complex processing technique. Solving this problem requires complex analytical skills, and actually solving a closed variable example (curvilinear solution) is something new. However, although we follow the theory of linear signals and systems in electrical engineering, i.e. to the in various fields such as automation, electronics, energy, communications, electrical circuits, signal processing, Laplace transform. All common signals of interest can be determined using methods to develop complex signals in some areas using the Laplace transform table and related properties. In general, finding the Laplace variable reduces to by means of a definitional formula that requires the solution of various elements that represent the work of a complex technique. Solving this problem requires complex analytical skills, and actually solving a closed variable example (curvilinear solution) is something new. But even though we follow the theory of linear signals and systems in electrical engineering. In various fields such as automation, electronics, energy, communications, electrical circuit theory, signal processing, the Laplace transform of all signals of interest and can be determined by methods used to develop complex structures of individual parts. The signal is obtained using the Laplace transform of a table of standard form and the corresponding properties of the Laplace transform [10].

The method of development of the complex character of the signal (Laplace transformation) to partial fractions can be applied only to the signal f(t) whose complex character is a strictly rational function, i.e. represents the quotient of two polynomials with real coefficients:

( ) ( ) { } ( ) ( )

B s b s b s b s b F s L f t m n A s s a s a s a

1 1 1 0 1 1 1 0 ;

m m m m m n n n n + + + + = = < + + + + Λ Λ

− − − − (12) and such that the order of the polynomial in the numerator m is strictly less than the order n of the polynomial in the denominator A(s), i.e. m<n. For many real signals and systems, their Laplace transform takes the form of a strictly rational function. However, there is also a more common form of the equation in real life that represents a real function, where m = n, and the standard design is defined as a purely rational function. At the same time, in order to apply the technique of developing into fractions, it is first necessary to divide the polynomial into both a number and a difference, and the result of that division will be a constant factor, while the remaining part will enable rational calculation. . Some of these types of operations have been discontinued.

Ownership of Laplace Transformation

The Laplace transform is a fundamental transformation that connects the function F(p) of a random variable with the function f(t) of a real variable. With its help, the structure of the dynamic system is studied and its various and integral differences are resolved [11]. One of the properties of the Laplace transform, which determines its distribution in the mathematics of science and technology, is that it corresponds to simple relationships in the images of many relationships and functions in nature. Checking two functions in image space thus makes multiplication and linear differential equation algebraic.

Properties of linearity, similarity, differentiation, integration, delay, shift, convolution. Properties of the Laplace transform. We will mark with the f(t), g(t), ... originals, and through F(p), G(p), ... - their pictures.

Directly from the Integral Properties

∞ ∞ − − = = ∫ ∫ (13)

( ) ( ) ( ) ( )

0 0 , ,... pt pt F p f t e dt G p g t e dt The properties of the Laplace transform greatly facilitate the task of finding images for a large number of different functions, as well as the task of finding originals from their images.

Example 1: Find function images: а) sin ωt, cosωt ; b) sin 2t cos 5t ; c) e-3t sin πt . The solution: a) Using the property of linearity and the formula obtained in example 2b), we find:

2 2 1 1 1 sin 2 2

−   − = = − =   − + +  

i t i t e e t i i p i p i p ω ω ω ω ω ω ω (14)

those 2 2 sin t p ω ω ω = + .Similarly we get the formula

2 2 sin t p ω ω ω = + .

b) First, let’s present the work sin 2t cos 5t as the difference of sine: ( ) 1 sin 2 cos5 sin 7 sin3 2 t t t t = − , and then use the linearity and similarity properties:

2 2 1 1 1 3 sin 2 cos5 2 49 2 9 t t p p = − + + .

c) Since 2 2 sin t p π π π = + , then by the displacement property π π π

3 2 2 sin 3 t e t p − = + +

.

( ) Example 2: Find the original by its image

2 5 6 11 p F p p p − = − + ; b) ( ) ( )

2 2 2 1 F p p ω = + ; c) а) ( ) 2

2 2 2 p F p p = + .

2 ( ) ( )

1 The solution: a) Transform this fraction so that you can use the shift property ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 3 1 2 5 3 1 2 2 6 11 2 3 2 3 2 3 2

p p p F p p p p p p − + − − = = = ⋅ + ⋅ − + − + − + − +

(15) b) Since ( ) 2 2 2 2 2 2 1 1 1 1 sin F p and t p p p ω ω ω ω ω = ⋅ = + + + that t t

1 1 1 sin sin cos 2 cos 2

( ) ( ) ( ) ( )

∫ ∫ F p t d t t d ωτ ω τ τ ω τ ω τ ω ω ω = ⋅ − = − − =

2 0 0

1 1 sin 2 cos 2 2

  ⋅ − − ⋅ =    

( ) t t t t ω τ ω τ ω ω

0 0 2

1 1 1 sin cos sin cos 2 2

  − = −    

( ) t t t t t t ω ω ω ω ω ω ω ω

2 3 (16) ( ) 2 2 2 2 1 1 sin cos 2 t t t p ω ω ω ω ω = = − +

those .

( ) c) Since

2 2 2 2 2 2 1 1 2 sin , cos 1 1 1 1 1 p p p p and t t p p p p p

2 = ⋅ ⋅ = = + + + + + ,

( ) based on the formula we have t p p t d t t t p p τ τ τ ⋅ ⋅ = − + = + + + ∫ .

1 2 2 cos cos 0 cos sin 1 1

( ) 2 2

0 (17)

Application of Laplace Functions for Joint Differential Functions

The Laplace transform is used to directly solve two ordinary differential equations with constant coefficients and the same type of linear differential equation. It should be emphasized that one of the most important features of the Laplace transform in general is that the operations performed on images can be simplified more easily than the original operations. These conditions contribute to the efficient use and widespread use of analytical functions based on Laplace transformation in solving various problems in physics and technology. For example, differentiation and integration correspond to the simple operations of multiplication and division. Similarly, when different parameters are combined using the Laplace transform, the gains and losses are obtained according to known methods.

The Laplace transform can be used to solve various normals. It is used for differential, discrete and full-line systems. Now let’s look at some examples of functions [12].

10 Solve an inhomogeneous linear differential equation with constant coefficients ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , 0 1 1 ... n n n n a y t a y t a y t a y t f t − − + + + + = , where it is understood that f(t) is the original.

Solution: Let it be ( ) ( ) ( ) ( ) ( ) ( ) , Y Y s L y t F s L f t = = = . Assume that the initial conditions are given ( ) ( ) ( ) ( ) 1 ' 0 , 0 ,... 0 n y y y − . If we apply the Laplace transformation to the equation, we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 2 0 1 1 0 ... 0 0 ... 0 ... 0 n n n n n n n n a s Y s y a s Y s y y a sY y a Y F s − − − − − − − − − + − − − + + − + = (18)

Introducing polynomials

( )

1 0 1 1 ... a n n n n n P s a s a s s a − − = + + + + (19)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 0 1 1 0 ... 0 0 ... 0 ... 0 n n n n n n Q s a s y y a s y y a y − − − − − − = + + + + + + (20) from the previous equation we get ( ) ( ) ( ) ( ) ( ) Q s F s Y s P s P s

1 n − + (21) n n By applying the inverse Laplace transform, we get the solution ( ) ( ) ( ) ( ) 1 1 1 n h p n n ( ) ( ) ( ) Q s F s y t L L u t y t P s P s − − − = + = + (22) where yh(t) is the solution of the homogeneous equation, and yp(t) is the particular solution of the homogeneous equation.