Analysis of transformations of the ultrafast electron transfer photoreaction mechanism in liquid solutions by the rate distribution approach

2014 ◽  
Vol 13 (5) ◽  
pp. 770-780 ◽  
Author(s):  
Michael G. Kuzmin ◽  
Irina V. Soboleva
Keyword(s):  
Electron Transfer ◽  
Laplace Transform ◽  
Rate Control ◽  
Distribution Functions ◽  
Inverse Laplace Transform ◽  
Rate Distribution ◽  
Control Factors ◽  
Liquid Solutions ◽  
P Rate ◽  
Experimental Kinetics

Rate distribution functions P(k), obtained directly from the experimental kinetics N(t) by an inverse Laplace transform, demonstrate transformations of the rate control factors in the course of ultrafast ET reactions.

The Absence of a Marcus Inversion Region for Electron Transfer at the Metal-Redox Electrolyte Interface

1994 ◽  
Vol 47 (12) ◽  
pp. 2171 ◽  
Author(s):  
D Matthews
Keyword(s):  
Electron Transfer ◽  
Potential Energy ◽  
Electron Distribution ◽  
Distribution Functions ◽  
Rate Distribution ◽  
Inversion Region ◽  
Redox Electrolyte ◽  
Electrolyte Interface ◽  
Metal Redox ◽  
Nuclear Configuration

The theory of electron transfer at the metal- redox electrolyte interface is described by starting with the work of Gurney and incorporating that of Gerischer and Marcus. This GGM model brings together diverse approaches to the description of electron transfer at electrodes. The electron transfer is described in terms of nuclear configuration potential energy diagrams, electronic configuration potential energy diagrams, electron distribution functions and rate distribution functions. The distinction between microscopic energies and macroscopic (thermodynamic) energies is made and the concept of the Fermi level of the redox electrolyte is clarified. The model of identical parabolas is used for the nuclear configuration diagrams and this is shown to lead to Gaussian electron distribution functions for the redox electrolyte. The rate distribution is obtained from the overlap between occupied and unoccupied states of the metal and redox electrolyte. Integration of the rate distribution gives the rate which is calculated as a function of the electrode potential for various values of the reorganization energy λ. It is shown that the variation of symmetry factor β is small for high λ and that the Tafel plots do not show significant decrease in rate at high overpotentials in the anomalous or inversion region. The Tafel plots for charge transfer (mass transfer is assumed to be fast at all potentials) tend to a limiting value with only a small decrease at high overpotential. This contrasts with the prediction based on nuclear configuration potential energy curves and is attributed to the fact that the overlap is between a Gaussian and a Fermi function rather than between two Gaussians, the latter being the case for homogeneous reactions.

Evaluation of the Fast Inverse Laplace Transform for Three-Dimensional NMR Distribution Functions

2013 ◽  
Vol 44 (11) ◽  
pp. 1335-1343 ◽  
Author(s):  
Zongfu Zhang ◽  
Lizhi Xiao ◽  
Guangzhi Liao ◽  
Huabin Liu ◽  
Wei Xu ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Three Dimensional ◽  
Distribution Functions ◽  
Inverse Laplace Transform

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
Sign in / Sign up

Export Citation Format

Share Document