# Laplace Transformation

by | Jun 15, 2022 | Math Learning

# HISTORY

Integral transformation data back to the Work of Leonard Euler (1763 & 1769), who considered them essentially in the form of the inverse Laplace transform in solving second-order, linear ordinary differential equations. Even Laplace, in his great work, “Theorie Analytique Des Probabilities”(1812), credits Euler with introducing integral transforms. It is Spitzer (1878) who attached the name of Laplace to the expression

Employed by Euler. In this form, it is substituted into the differential equation where is the unknown function of the variable .

In the late 19th century, the Laplace transform was extended to his complex form by Poincare and Pincherle, Redis covered by Petzval, and extended to two variables by Picard, with the further investigation conducted by Abel and many others.

The first application of the modern Laplace transformations equations arose of Batemer (1910), who was working on radioactive decay

By setting

and obtaining the transformed equation. Bernstein (1920) used the expression

called it the Laplace transformation, in his work on theta functions. The modern approach was given particular impetus by Doetsch in the 1920s and 30s; he applied the Laplace transform to differential, integral, and integro-differential equations. This body of work culminated in his foundational 1937 text, “Theorie and Anwendungender Laplace Transformation.”

No account of the Laplace transformation would be complete without mention of the work of Oliver Heaviside, who produced a vast body of what is termed the “Operational Calculus”. The material is scattered throughout his three volumes, Electromagnetic Theory (1894, 1899, 1912), and bears many similarities to the Laplace transform method. Although Heaviside’s calculus was not the theory of Laplace Transformation, he did find favour with electrical engineers as a useful technique for solving their problems. Considerable research went into trying to make the Heaviside calculus rigorous and connecting it with the Laplace transform. One such effort was that of Bromwich, who, among others, discovered the Inverse Laplace Transformation!

Where lying to the right of all singularities of the function .

# INTRODUCTION

In this article, we shall consider theoretic aspects of the Laplace transform, reserving for the application of the subject to the solution of linear differential equations. This transformation is defined by the equation;

## PIECE-WISE OR SECTIONAL CONTINUITY

A function  is called sectionally continuous or piece-wise continuous in any interval if it is continuous and has finite left and right-hand limits in every subinterval  as shown in the graph of the function

## FUNCTION OF EXPONENTIAL ORDER

A function  is said to be of exponential order as

if

i.e., if given a positive integer, there exists a real number

Or

Briefly, we say that is of exponential order.

Sometimes we write,

For example. is of exponential order as being any positive integer.

So we are going to do L’Hospital Rule where we differentiate numerator and denominator individually,
hence,

(by L. Hospital’s rule)

=

Since.
Hence we can say that is an exponential order as.

Another example.   is not of exponential order as

Taking,

Therefore,

is not of exponential order

# DEFINITION OF THE  TRANSFORM  CONCEPT

An improper integral of the form is called Integral Transform of if it is convergent. Sometimes it is denoted by  or  Thus

The function appearing in the integrand is called Kernel of the transform. Here’s a parameter and is independent of may be real or complex number.

If we take

Then becomes

This transform is known as Laplace transform.

Some examples of well-known transformations

Hankel Transform of is

Mellin Transform of is

Fourier transform of is

# DEFINITION. LAPLACE TRANSFORM

Suppose  is a real valued function defined over the interval

The Laplace Transform of, denoted by, is defined as

We also write

Here L is called Laplace transformation operator. The parameter s is a real or complex number. In general, the parameter is taken to be a real positive number. Sometimes we use symbol for parameter .

The Laplace transform is said to exist if the integral (1) is convergent for some value of .

The operation of multiplying  by and integrating from is called Laplace Transformation.

Now we are going to see the existence of Laplace transformation on a function

Existence Theorem of Laplace Transform.

Statement: – If  is a function of class , then Laplace transform of exists.

Or,

Suppose is piece-wise continuous in every finite interval and is of exponential order an as. Then exists , i.e., Laplace transform exists.

Proof. Let be piece-wise continuous in every finite interval and of exponential order as

To show that

Let Then

Continuity of in the finite interval implies that

exists. It remains to show that

exists  is of exponential order a implies

is finite, i.e., given a number , there exists a real number

i.e.,

can be made as small as we please choosing sufficiently large. Hence  exists

Fact: The condition given in this part is sufficient for the existence of, but are not the necessary conditions.

# LAPLACE TRANSFORMS OF STANDARD FUNCTION

Result 1.

Proof.

Result 2.

Proof.

Also  if is positive integer.

Result 3.

For

Result 4.

Proof.

Result 5.  Find

Proof.

Linear Property.

Suppose and are Laplace forms of and respectively. Then

Where and are any constants.

First Shifting Theorem (First Translation)

Proof.  Let

Then,

Fact: This theorem can also be restated as:

If is the Laplace transform of and a is any real, or complex number, then is the Laplace transform of). That is to say,

Some examples related with the previous properties and we are going to know how to use these properties to solve a problem

Example.

Solution.

Solution. We know that,

Example.  Let’s we are going to prove,

Solution. We know that

Using the result,  , we get

Theorem 4. Second shifting Theorem (Second Translatation).

Statement:- If

then

Or

And

Example . We are going to find Laplace transform of where

Solution. then

Also

By second shifting theorem,

Theorem 5. Change of Scale Property.

If

Laplace Transform of Derivatives:

An Alternate form of Theorem 6.

To express the Laplace transform of the function  in terms of the Laplace transform of the function

Generalising the results (1) and (2), we get the required result.

Theorem 7. Laplace Transform of Integral.

and

… (1)

We prove that

From (1) it is clear that

Periodic Functions :

Problem . Using Laplace transform,

solve where

Solution. Taking Laplace transform of given question,

A few worked examples should convince the reader that the Laplace transform furnishes a useful technique for solving liner differential equations. Lastly thanks to all of you and hope you all enjoy this article. We will meet next time with a new article.

Thank You.

References/Bibliography ⟶

This is merely a brief selection. An extensive list is given in reference –

Laplace Transforms and their applications to Differential equations. – N. W. MCLACHLAN
The Laplace Transform: Theory and Applications – Joel L. Schiff

(iii)       Ordinary & Partial Differential

(iv)       Advanced Calculus – Davis V. Widder

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