# Applications of the Laplace Transform

I am curious to know what kind of applications the Laplace transform has. Yes, I know people will reference Wikipedia, and other online sites which discuss the Laplace transform at length. However, all the applications are very one-dimensional. For example, even looking at Wikipedia most the "applications" are towards solving differential equations.

Furthermore, I have been searching for many books, engineering books, physics books, math books, ect., which contain much material on Laplace transforms. All of those books use the Laplace transform only as a means to solve differential equations. I never see any other applications.

To add further to my question, I heard it said, each time the Laplace transform is introduced, of how valuable it is to electrical engineering. In fact, I said so myself, but looking at books, I again only find the applications of the transform to solving differential equations. Nothing really beyond that.

This is what I mean by "one-dimensional applications". Yes, the Laplace transform has "applications", but it really seems that the only application is solving differential equations and nothing beyond that.

Though, that is not entirely true, there is one more application of the Laplace transform which is not usually mentioned. And that is the moment generating function from probability theory. After all that is the original motivation of Laplace to create that transform in the first place. Unfortuantely, moment generating functions are not of superior importance to probability theory (to the best of my knowledge), and so the the only "big" applications of this transform appears to be only to the solution of differential equations (both ordinary and partial).

Contrast this with the Fourier transform. The Fourier transform can be used to also solve differential equations, in fact, more so. The Fourier transform can be used for sampling, imaging, processing, ect. And even in probability theory the Fourier transform is the characteristic function which is far more fundamental than the moment generating function.

The Fourier transform is certaintly a huge powerful tool with vast applications all across mathematics, physics, and engineering. There are books, across all fields, all devoted to the different applications of this transform.

But does the Laplace transform have any other "applications" to it other than solving differential equations? If you say that it does, then please provide a book reference which has an entire chapter, or large part of the book, discussing a non-differential equation application to which the Laplace transform is of fundamental importance?

• Hmm. But why do we care about the moment generating function? In the end, its because we can differentiate it to get moments, which links it quite closely to DEs. You could try to google Borel-Laplace summation, which uses the Laplace transform to take in possibly divergent series, and spit out an analytic function that has the original series as an asymptotic expansion (I should probably not tell you that this was motivated by studying ODEs like Bessel's equation) and also there's Bernstein's thm : every totally monotone function is a Laplace transform of a positive borel measure Nov 9, 2019 at 3:41

Yes, Laplace transform is a very powerful mathematical tool applied in various areas of science and engineering. It has many application in different areas of physics and electrical power engineering. With the increasing complexity of engineering problems, Laplace transforms help in solving complex problems with a very simple approach just like the applications of transfer functions to solve ordinary differential equations. Besides these, Laplace transform is a very effective mathematical tool to simplify very complex problems in the area of stability and control. With the ease of application of Laplace transforms in myriad of scientific applications, many research software have made it possible to simulate the Laplace transformable equations directly which has made a good advancement in the research field.

For more details you may follow the references (and the references there in) given below:

$$\bf{(1)}~~$$"Laplace Transforms and Their Applications" by Alexander Apelblat (Nova Science Publishers, Inc.)

$$\bf{(2)}~~$$"LAPLACE TRANSFORMS AND ITS APPLICATIONS" by Sarina Adhikari

$$\bf{(3)}~~$$"LAPLACE TRANSFORMS AND ITS APPLICATIONS" by Ms. Sandhya Upreti, Ms. Piyali Sarkar

$$\bf{(4)}~~$$"Laplace transforms and it‟s Applications in Engineering Field" by Dr.J.Kaliga Rani, S.Devi

$$\bf{(5)}~~$$"Laplace Transforms and Their Applications to Differential Equations" by N.W. McLachlan (Dover Books on Mathematics)

$$\bf{(6)}~~$$"Theory of Laplace and Fourier Transform With Its Applications" by J. R. Sontakke

$$\bf{(7)}~~$$ "APPLICATIONS OF LAPLACE TRANSFORM IN ENGINEERING FIELDS" by Prof. L.S. Sawant

• I had downvoted this. I specifically asked for applications of the Laplace transform that are outside solving differential equations. I did not ask for "applications of the Laplace transform", I specifically asked for non-DE applications. And your answer just provided me with more DE applications (based on all the links I was able to access). May 3, 2020 at 5:26

I think the Primary application of the Laplace transform is in Solving Linear Ordinary differential equations and Linear Partial differetial equations but you alreadt know that.Moreover you are familiar with its applications to Ptobability theory in finding moment generating functions.But what I want to bring to your attention is that applied Mathematics is very different from Pure Mathematics.You must realize that in modeling Physical problems you first need to understand the problem well enough to inorder to find the most natural and Simple mathematical tool that makes the solution transparent so that important details are easily understood.Linear Control System Problems can be modeled with ordinary differential equations but the problems and the details envolved can be understood most naturally and easily in the context of Laplace transforms.Why dont you try to solve the Problem of Controling the motion of a Servo moter by modeling it with a differential equation.Then try finding an answer from a book on Control systems that uses the Method of Laplace transforms!!That will surley help you in appreciating the advantage.Enrico Fermi and Richard Feynman used to say that Pure Mathrmaticians make little Progress in Physics.This is b/c they dont understand Physical Ptoblems well enough to appreciate a proper method of solution!!I just hope you attempt to solve some Real World applications to engineering and Physics besides looking fora mathematical application.B/c the importance of the Laplace transform is mostly appreciated in the context of Applied mathematics where Pure mathematicans lack the understanding of the nature of the problems involved. That being said the laplace transform can be used in Solving Integral equations where the integrand involved is a convolution. However,since the Laplace transform is just the fourier transform of a class of functions for which the fourier transform will exist if dumped with an exponential function what new Mathematical applications can it really have.The fourier transform covers a broad class of functions--the class of absolutely integrable functions and a great deal of Applications to mathematics are found here.But since the theorems for the Laplace transfom and fourier are very similar the chances of it being mathematically useful are very low--it is just an extension of the fourier transform to an additional class of functions nothing more,The important question to ask is how broad this Class is?And Does this Class include uniques features worth studing?

• I downvoted your answer as well. It does not give me any specific answer to my question. You said that, "Why dont you try to solve the Problem of Controling the motion of a Servo moter by modeling it with a differential equation. Then try finding an answer from a book on Control systems that uses the Method of Laplace transforms." You need to be more specific. What is the problem/solution. And, more importantly, can you solve this problem without the Laplace transform? And furthermore, it again, sounds like another DE application. I specifically asked for non-DE applications. Dec 17, 2021 at 21:08