Laplace transform convolution property

The convolution and general properties of the Fourier transform, as presented in Section 11.1, are equally applicable to the Laplace transform. Thus,... 

There are good reasons for choosing to carry out certain operations in the Laplace domain rather than performing equivalent operations in the time domain. In particular, integration of a function with respect to time is equivalent in the Laplace domain to division by the Laplace variable s. Conversely, differentiation corresponds to multiplication by s. This latter property enables differential equations to be Laplace transformed and then solved by algebraic means. These Laplace domain operations are all more simple than their time domain counterparts. In addition convolution in the time domain is equivcdent to multiplication in the Laplace domain. Formally, this may be represented by eqn. (24), the left-hand side of which is termed the convolution integral. 

Besides the above properties, an inverse Laplace transform is often applied by using convolution to solve the diffusion equations, thus,... 

Laplace transforms and the properties of the integral of convolution permit us to establish simple relationships between distinct functions defined in the present context such as those outlined in Chapter 5, Thus,... 

Integral transforms can be used to solve ordinary differential equations by converting them to algebraic equations. In what follows, the convolution properties of the different transforms have been listed, followed by the methods of integral transform used to solve (a) one-dimensional diffusion equations in the infinite and semi-infinite domains and (b) Laplace equations in the cylindrical geometries. 

It is readily shown that several simple mathematical operations such as differentiation, integration, and linear transformations (scaling and translation), as well as more complex operations such as convolution, deconvolution, and Laplace transformations (and inverse Laplace transformation) have the above linear operator property. 

The Laplace transform, which is a linear operator, is frequently used as a mathematical tool when dealing with linear systems. The Laplace transform is often useful in dealing with more complex convolution relationships. For example, consider the following property of the Laplace transform operation L ... 

An interesting property of the convolution product is that it can be transformed into an ordinary product of transformed functions. The transformation adapted to the time range of the convolution is not the Fourier transform, which works on a full range to +°o (two-sided transform), but the Laplace transform, which is analogous but working on a half range from 0 to infinite (one-sided transform). 

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