A Residue Calculus of Isolated Singular Zero Point and Its Application in Z Transform

2015 ◽  
Vol 24 (03) ◽  
pp. 1550033
Author(s):  
Zong-Chang Yang
Keyword(s):  
Signal Processing ◽  
Laplace Transform ◽  
Functional Form ◽  
Zero Point ◽  
Inverse Laplace Transform ◽  
Residue Theorem ◽  
New Formula

The residue theorem is a powerful tool in mathematics and signal processing, such as it is usually employed for the inverse Laplace transform and the inverse Z transform. With respect to the most commonly rational functional form in the Z transform, a theorem and a new formula for residue calculus based on the singular zero point is introduced. Then two consequences are proposed. Finally, based on the derived theorems, a new convenient algorithm for computing the inverse Z transform is presented. Applications with satisfying results indicate convenience and workability of the proposed method.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals Quadrature Integration Techniques for Random Hyperbolic PDE Problems

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 160
Author(s):  
Rafael Company ◽  
Vera N. Egorova ◽  
Lucas Jódar
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Quadrature Rule ◽  
Inverse Laplace Transform ◽  
Process Time ◽  
Efficient Computation ◽  
Hyperbolic Pde ◽  
Numerical Convergence ◽  
Laplace Transform Technique ◽  
Integration Techniques

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss–Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.

scholarly journals Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 354
Author(s):  
Alexander Apelblat ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Operational Calculus ◽  
Inverse Laplace Transform ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Error Functions ◽  
Convolution Integrals ◽  
Special Case ◽  
Integral Identities ◽  
Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Modeling electro-osmotic flow and thermal transport of Caputo fractional Burgers fluid through a micro-channel

2021 ◽  
pp. 095440892110259
Author(s):  
Mohammed Abdulhameed ◽  
Garba Tahiru Adamu ◽  
Gulibur Yakubu Dauda
Keyword(s):  
Electric Field ◽  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Micro Channel ◽  
Osmotic Flow ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Burgers Fluid ◽  
Electro Osmotic Flow

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

MACSYMS's inverse Laplace transform

1989 ◽  
Vol 23 (1) ◽  
pp. 33-38
Author(s):  
M. Clarkson
Keyword(s):  
Laplace Transform ◽  
Inverse Laplace Transform

Numerical Inversion of Laplace Transform for Time Resolved Thermal Characterization Experiment

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
J. Toutain ◽  
J.-L. Battaglia ◽  
C. Pradere ◽  
J. Pailhes ◽  
A. Kusiak ◽  
...  
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Periodic Function ◽  
Thermal Characterization ◽  
Inverse Laplace Transform ◽  
Numerical Inversion ◽  
Time Resolved ◽  
Reliable Technique ◽  
Numerical Inverse Laplace Transform ◽  
Thermal Disturbance

The aim of this technical brief is to test numerical inverse Laplace transform methods with application in the framework of the thermal characterization experiment. The objective is to find the most reliable technique in the case of a time resolved experiment based on a thermal disturbance in the form of a periodic function or a distribution. The reliability of methods based on the Fourier series methods is demonstrated.

Using inverse Laplace transform in positronium lifetime imaging

2022 ◽  
Author(s):  
Kengo Shibuya ◽  
Haruo Saito ◽  
Hideaki Tashima ◽  
Taiga Yamaya
Keyword(s):  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Emission Tomography ◽  
Statistical Accuracy ◽  
Lifetime Imaging ◽  
Positron Emission ◽  
Positronium Lifetime ◽  
Physical And Chemical ◽  
Limited Sensitivity

Abstract Positronium (Ps) lifetime imaging is gaining attention to bring out additional biomedical information from positron emission tomography (PET). The lifetime of Ps in vivo can change depending on the physical and chemical environments related to some diseases. Due to the limited sensitivity, Ps lifetime imaging may require merging some voxels for statistical accuracy. This paper presents a method for separating the lifetime components in the voxel to avoid information loss due to averaging. The mathematics for this separation is the inverse Laplace transform (ILT), and the authors examined an iterative numerical ILT algorithm using Tikhonov regularization, namely CONTIN, to discriminate a small lifetime difference due to oxygen saturation. The separability makes it possible to merge voxels without missing critical information on whether they contain abnormally long or short lifetime components. The authors conclude that ILT can compensate for the weaknesses of Ps lifetime imaging and extract the maximum amount of information.

Generalized coupled transient thermoelastic problem of multilayered spheres by the hybrid numerical method

2003 ◽  
Vol 217 (12) ◽  
pp. 1315-1323
Author(s):  
Z Y Lee ◽  
C L Chang
Keyword(s):  
Laplace Transform ◽  
Finite Difference Methods ◽  
Generalized Thermoelasticity ◽  
Inverse Laplace Transform ◽  
Hybrid Numerical Method ◽  
The Matrix ◽  
The Laplace Transform ◽  
Computational Procedures ◽  
Thermoelastic Problems ◽  
Matrix Similarity

This paper deals with axisymmetric quasi-static coupled thermoelastic problems for multilayered spheres. Laplace transforms and finite difference methods are used to analyse the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. Solutions are obtained for the temperature and thermal deformation distributions for the transient and steady state. It is demonstrated that the computational procedures established in this paper are capable of solving the generalized thermoelasticity problem of multilayered spheres.

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