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
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 .
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
i.e., if given a positive integer, there exists a real number
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,
(by L. Hospital’s rule)
Hence we can say that is an exponential order as.
Another example. is not of exponential order as
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
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.
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
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
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
Also if is positive integer.
Result 5. Find
Suppose and are Laplace forms of and respectively. Then
Where and are any constants.
First Shifting Theorem (First Translation)
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
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).
Example . We are going to find Laplace transform of where
By second shifting theorem,
Theorem 5. Change of Scale Property.
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.
We prove that
From (1) it is clear that
Periodic Functions :
Problem . Using Laplace transform,
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.
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