Diffusion-reaction approach to electronic relaxation in solution. Exact solution for delta function sink models

1994 ◽  
Vol 106 (2) ◽  
pp. 493-505
Author(s):  
K. L. Sebastian
Keyword(s):  
Exact Solution ◽  
Delta Function ◽  
Diffusion Reaction ◽  
Electronic Relaxation

3D Unsteady Convection-Diffusion-Reaction via GFEM Solver

2015 ◽  
Vol 751 ◽  
pp. 313-318
Author(s):  
Estaner Claro Romão ◽  
Luiz Felipe Mendes de Moura
Keyword(s):  
Numerical Simulation ◽  
Exact Solution ◽  
Galerkin Method ◽  
Diffusion Reaction ◽  
Convection Diffusion ◽  
Temporal Discretization ◽  
Physical Problems ◽  
Important Study ◽  
Initial Idea ◽  
The Galerkin Method

In this paper, an important study on the application of the α family of temporal discretization is presented. For spatial discretization the Galerkin Method (GFEM) was used. With the variation of the α coefficient in temporal discretization and through one numerical applications with exact solution, it will be possible to have an initial idea on how each one of the two suggested methods behaves. It is expected that this study can be able to advance several other studies within the domain of numerical simulation of physical problems. It is important to note that for all applications we will use a mesh that is considered gross, with the purpose of presenting a method that is robust, precise and mainly computationally economic.

Theory of barrierless electronic relaxation in solution. Delta function sink models

1993 ◽  
Vol 204 (5-6) ◽  
pp. 496-504 ◽  
Author(s):  
Nalini Chakravarti ◽  
K.L. Sebastian
Keyword(s):  
Delta Function ◽  
Electronic Relaxation

The Exact Solution of the Translational Acceleration of a Low Modulus Elastic Medium in Rigid Spherical Shells—Implications for Head Injury Models

1975 ◽  
Vol 42 (4) ◽  
pp. 759-762 ◽  
Author(s):  
K. B. Chandran ◽  
Y. King Liu ◽  
D. U. von Rosenberg
Keyword(s):  
Exact Solution ◽  
Head Injury ◽  
Elastic Medium ◽  
Closed Head Injury ◽  
Delta Function ◽  
Step Function ◽  
Spherical Shells ◽  
Dirac Delta ◽  
Low Modulus ◽  
Translational Acceleration

The exact solution in the form of a finite series has been obtained for the problem of low modulus elastic medium contained in rigid spherical shells subjected to translational acceleration about its diametrical axis. Laplace transformation technique and the shifting theorem were used to obtain the Green’s functions for the potentials when the external acceleration is a Dirac delta function. The solutions are formally extended to external accelerations which are general functions of time by the convolution integral. The shear stress distribution for a unit step function acceleration is illustrated. The results obtained are used to judge the adequacy of this and other similar models for the study of closed head injury mechanism.

scholarly journals Finite element based hybrid techniques for advection-diffusion-reaction processes

2018 ◽  
Vol 8 (2) ◽  
pp. 127-136
Author(s):  
Murat Sari ◽  
Huseyin Tunc
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Numerical Solutions ◽  
Diffusion Reaction ◽  
Optimization Approach ◽  
B Spline ◽  
Hybrid Techniques ◽  
Advection Diffusion ◽  
Pure Diffusion ◽  
Spline Basis

In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

Approximation Techniques for Stationary and Dynamic Problems

2021 ◽  
pp. 181-208
Author(s):  
Duncan G. Steel
Keyword(s):  
Magnetic Field ◽  
Exact Solution ◽  
Perturbation Theory ◽  
Golden Rule ◽  
Delta Function ◽  
Time Dependent ◽  
Dynamic Problems ◽  
Approximation Techniques ◽  
Dirac Delta ◽  
Physical Features

For many aspects of device design, an exact solution to Schrödinger’s equation is not needed. However, it may simultaneously be required that all of the physical features are clearly understood. The most important technique for approaching these problems is perturbation theory, since it is difficult to develop physical intuition by just numerical means. For the case of solutions to the time independent Schrödinger equation, such as where an electric or magnetic field is applied, time independent perturbation theory is very useful, and is typically adequate for many problems. In some cases, problems may need an exact solution, but it may not be necessary to consider all the levels, leading to the approximation of using just a few levels. If the Hamiltonian is time dependent, we use time dependent perturbation theory which leads to Fermi’s golden rule. The result leads to a Dirac delta-function which can be eliminated by using the density of states.

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