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Resource and Interference Management
Published in Wen Sun, Qubeijian Wang, Nan Zhao, Haibin Zhang, Chao Shen, Ultra-Dense Heterogeneous Networks, 2023
Wen Sun, Haibin Zhang, Nan Zhao, Chao Shen, Lawrence Wai-Choong Wong
The Laplace transform is a widely used integral transform in mathematics. It can be used to transform ordinary differential equations into algebraic equations, which makes it easier to solve ordinary differential equations [49]. In engineering, the importance of the Laplace transform is to transform a function of time into a function of complex frequency. It is defined by F(s)=∫0∞e−stf(t)dt
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.
Mathematical Background
Published in P.N. Paraskevopoulos, Modern Control Engineering, 2017
A popular application of the Laplace transform is in solving linear differential equations with constant coefficients. In this case, the motivation for using the Laplace transform is to simplify the solution of the differential equation. Indeed, the Laplace transform greatly simplifies the solution of a constant coefficient differential equation, since it reduces its solution to that of solving a linear algebraic equation. The steps of this impressive simplification are shown in the bottom half of Figure 2.9. These steps are analogous to the steps taken in the case of multiplying numbers using logarithms, as shown in the top half of Figure 2.9. The analogy here is that logarithms reduce the multiplication of two numbers to the sum of their logarithms, while the Laplace transform reduces the solution of a differential equation to an algebraic equation.
An alternative for the determination of thermal diffusivity using 1D Fourier solution: Talbot’s method
Published in Chemical Engineering Communications, 2023
Giovanna O. Moreira, Guilherme L. Dotto, Marcos F. P. Moreira
Among the various methods for solving the Fourier heat conduction equation to obtain the thermal diffusivity, Laplace Transform is one (Carslaw and Jaeger 1959). This method uses the Laplace domain to solve the problem. In the end, the inverse Laplace Transform is applied to return to the time domain. Different methods (Galdino 1995) can solve the inverse Laplace Transform: searching a table of inverse Laplace transforms; decomposing inverse Laplace Transform into simple forms whose inverses are in a table; integrating the definition of the inverse Laplace transform analytically (Rice and Duong 1995; Grigoletto and Oliveira 2018), and solving the definition of the inverse Laplace transform numerically (Piessens and Dang 1976; Cohen 2007). Among the numerical methods for inverse Laplace Transform, it is possible to mention Talbot’s method (Talbot 1979; Cohen2007). Talbot’s method can be solved in computational form, as Murli and Rizzardi (1990) presented for Fortran, a difficult language, or in licensed software such as MatLab. A good alternative to Fortran and MatLab is Scilab (SCILAB 2021), free and open-source software with a simple language similar to MatLab. Scilab supplies a good computation environment to scientific applications with several numerical tools.
Numerical treatment of multi-term time fractional nonlinear KdV equations with weakly singular solutions
Published in International Journal of Modelling and Simulation, 2023
Sudarshan Santra, Jugal Mohapatra
• , . Definition 2.5. (see [39]). The convolution of the two functions and is defined by: Theorem 2.6. (see [39]). The Laplace transform of the convolution of two functionsis the product of individual Laplace transform. Mathematically, it is expressed as: .Definition 2.7. If, then is the inverse Laplace transform of and is generally written as: .
An impulse response formulation for small-sample learning and control of additive manufacturing quality
Published in IISE Transactions, 2023
A Laplace transform maps function to function of a complex variable s. The double Laplace transform of a 2D f is denoted as By the property of the Laplace transform of a convolution, i.e., the prediction model in (8) for 2D shapes and prediction model in (11) for 3D shapes can be transformed as where represents the prediction without considering the error terms.