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Laplace Transforms
Published in Brian Vick, Applied Engineering Mathematics, 2020
The inverse Laplace transform is the process of finding the time function f(t) from the corresponding transform F(s). The methods of finding the inverse Laplace transform are to Use the basic definition, Equation 8.2. This is usually difficult and is seldom used.Use tables of functions f(t) corresponding to given Laplace transforms F(s). Some important cases are listed in Table 8.1.Use the partial-fraction expansion method. This method is emphasized in many books.Use Mathematica’s “InverseLaplaceTransform” command.
The Laplace Transform Method
Published in Roberts Charles, Elementary Differential Equations, 2018
As we have discovered, manually calculating the Laplace transform of a function from the definition can involve using various integration techniques such as integration by parts, substitution, and so forth, or it can involve looking up definite integrals in a table of integrals. If we have a table of Laplace transforms available, then manually computing the Laplace transform can require using the linearity property or translation property of Laplace transforms. To manually calculate the inverse Laplace transform often requires using partial fraction expansion and the use of a table of Laplace transforms.
Power System Transients
Published in A. P. Sakis Meliopoulos, Power System Grounding and Transients, 2017
For the purpose of solving a transient problem with the Laplace transform, the Laplace transform is applied to the differential equations and initial conditions that describe a specific problem. Subsequently, the resulting equations are solved yielding the quantity of interest in the Laplace domain (i.e., as a function of s). Application of the inverse Laplace transform will provide the quantity of interest in the time domain. The use of Laplace transforms will be illustrated with an example.
Green's function and finite element formulations for the dynamics of pipeline conveying fluid
Published in Ships and Offshore Structures, 2022
Fong Kah Soon, Tan Phey Hoon, Mohamed Latheef, A. Y. Mohd. Yassin
where are functions specified by Combining the subequations in Equations (21) and (22) gives the Green's function where is the inverse Laplace transform, is the unit step function. and are represented by To obtain the dynamic response of the pipeline conveying fluid, various order derivatives of are necessarily to be obtained as written in Equations (25a)–(25d)
A Novel Method for Predicting Power Transient CHF via the Heterogeneous Spontaneous Nucleation Trigger Mechanism
Published in Nuclear Technology, 2020
Emory Brown, Yikuan Yan, Wade R. Marcum
Now an inverse Laplace transform must be applied to get back to the appropriate domain. To perform the inverse Laplace transform, the residue theorem is applied. This method allows for the inverse to be found without needing to resolve the partial fractions.14 Since the problem only has one pole, or singularity, at the residue theorem is applied like so:
On homogenization of the first initial-boundary value problem for periodic hyperbolic systems
Published in Applicable Analysis, 2020
To prove estimate (91), we use the inverse Laplace transform and Theorem 2.9. To guarantee the convergence of the corresponding integrals, we consider the function instead of the cosine. The reason is that the inverse Laplace transform of this function decreases faster than the inverse Laplace transform of the cosine (see, e.g. [48, Section 17.13]).