# Why laplace transform. Differential Equations 2019-02-08

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## The Laplace Transform

That's what happens when I put A in here. In engineering applications, a function corresponding to a is stable if every bounded input produces a bounded output. The trig s,t is really exp -st , which is a sine wave at a specific frequency. You can latter plot frequency vs number of meteors. Use MathJax to format equations. Many of them have very well and practical introduction to such methods.

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## Why laplace transform

The Fourier transform is a function that takes in a frequency and the function value on that frequency tells you how much of that frequency is present in the function. Fourier transforms are often used to solve boundary value problems, Laplace transforms are often used to solve initial condition problems. Because it's doesn't make any sense that the imaginary axis of the Laplace plane has both imaginary and real axis. Re: Use of Laplace Transforms!! So why limit your intermediate step to a transform that 'means' something? Scope of this Chapter In this chapter, we deal only with the Laplace transform f t to F s and the reverse process. So when t equals 0, it becomes e to the minus s times 0. This is good because equations involving derivatives can be very, very hard, but this just sees these equations as algebraic equations, which can be much, much easier to work with.

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## 2. Definition of the Laplace Transform

With a Fourier transform, you're taking a set of orthogonal functions, sin nx and cos nx , and finding the projection of a function on this set using the inner product of integrating over a period. We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We are going to take the limit as A approaches infinity. The Laplace transform can be used to solve this equation. I'll show you in a few videos, there are whole tables of Laplace Transforms, and eventually we'll prove all of them. Unlike the Fourier transform, the Laplace transform of a is generally a function.

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## The Laplace Transform

That might be before many of your times, but I grew up on that. Doing an explicit calculation of the left hand side is going to be quite a mess. In terms of electronics engineering, the Laplace transform is used to get your model into the s-domain, so that s-domain analysis may be performed finding zeroes and poles of your characteristic equation. Please leave a comment or question below and I will do my best to address it. This means that, on the range of the transform, there is an inverse transform.

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## Relation between Laplace and Fourier transforms

If you just needed to numerically solve a specific equation, then Euler or Runge-Kutta or something like that would be great. Keeping track of the zeros and poles then is a way to easily keep track of information about the original function. So anyway, let's take the anti-derivative and evaluate this improper definite integral, or this improper integral. So, you let that interval go to infinity in length and you get the Fourier transform, rather than Fourier series. The examples in this section are restricted to differential equations that could be solved without using Laplace transform.

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## Why laplace transform

Maybe we can transform it in a way so that we can recover the easy case. In some cases the problem might get such easy under the new point of view, that you are able to solve the problem there and then you take the obtained solution and transform back to your original point of view. Ideally you want a linear system that does not depend on time like a spring or a circuit or something like that , and then you apply some kind of forcing function. When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement. Likewise, Laplace and Z transforms turn nasty differential equations into algebraic equations that you have a chance of solving. Every linear physical system has its spectrum of normal modes, and if coupled with some external random source of energy white noise , its evolution runs through such states from the lowest possible energy to the greatest.

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## What exactly is Laplace transform?

This is not the case. I can happily transform and inverse away, but what does f s represent? As you're encountering the Laplace transform, you've probably encountered a dot product. It's all just fancy cryptography with functions. Once solved, use of the inverse Laplace transform reverts to the time domain. About your question Fourier series has many application in electrical engineering a lot of which fall under signal processing, see the great thing about Fourier series is that it can represent signals as a summation of cosine and sine functions, so if you need to analyze a given system all you have to do is apply a sine or cosine input and a couple of other unique inputs to the system and because most signals can be broken down to these two signals then what applies to them applies to more complicated signals, this allows us to analyze the response of complex systems to a great number of inputs without actually applying these inputs. The most significant advantage is that and become multiplication and division, respectively, by s similarly to changing multiplication of numbers to addition of their logarithms. Application to vibrating strings, organ pipes, and time series analysis came later.

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## Relation between Laplace and Fourier transforms

This is also true for the bilateral two-sided Laplace transform, so the function need not be one-sided. Better explanations deals that Laplace is used for stability studies and Fourier is used for sinusoidal responses of systems. But I won't get into all of that yet. Use MathJax to format equations. Back at the end of the sixteenth century astronomers were starting to do nasty calculations. The transform turns and to , which are much easier to solve. A common overlooked by mathematicians example is in music.

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## Relation between Laplace and Fourier transforms

This is the result of the time shifting property and the. The Laplace transform has applications throughout probability theory, including of such as , and. The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform. Let's assume that s is greater than 0. So if we assume that s is greater than 0, then as A approaches infinity, what's going to happen? Such systems are called causal systems: the response depends only on what happened so far. But we'll keep doing it, and we'll solve more and more difficult problems. Not the answer you're looking for? The Fourier transform is simply the frequency spectrum of a signal.

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