A closed-form solution for the two-dimensional Fokker–Planck equation for electron transport in the range of Compton Effect

Rodriguez, Bárbara Denicol do Amaral; Vilhena, Marco T.; Borges, Volnei; Hoff, Gabriela

doi:10.1016/j.anucene.2007.09.002

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…(1.1), B(x, t, u) > 0 is called the diffusion coefficient and A(x, t, u) is the drift coefficient. This kind of equation often occurs in a wide range of practical problems such as, electron transport [9] , stochastic nonlinear dynamical systems [1,12] , inhomogeneous environments [11] , maximally reduced air-sea coupling climate model [2] and marketing [3] . He [4] considered the facts that nearly all iterative methods are sensitive to initial solutions, so it is very difficult to obtain convergent results in cases of strong nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

On exact solutions to partial differential equations by the modified homotopy perturbation method

Yang

Chen

Yao

2011

Acta Math. Appl. Sin. Engl. Ser.

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Section: Introductionmentioning

confidence: 99%

On exact solutions to partial differential equations by the modified homotopy perturbation method

Yang

Chen

Yao

2011

Acta Math. Appl. Sin. Engl. Ser.

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“…The electron contribution to energy deposition induced by incident photons is quantified solving the two-dimensional Fokker-Planck equation for electron transport [3,4] by the P N approximation in the angular variable followed by applying the Laplace Transform to one of the spatial variables (here x). This procedure leads to a closed-form formulation for the build-up factor and absorbed energy, in one and two dimensional Cartesian geometry for photons and electrons, in the energy range where Compton scattering is dominant [5].…”

Section: Introductionmentioning

confidence: 99%

A Closed-Form Formulation for the Build-Up Factor and Absorbed Energy for Photons and Electrons in the Compton Energy Range in Cartesian Geometry

Borges¹,

Fernandes²,

Bodmann³

et al. 2012

WJNST

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In this work, we report on a closed-form formulation for the build-up factor and absorbed energy, in one and two di- mensional Cartesian geometry for photons and electrons, in the Compton energy range. For the one-dimensional case we use the LTSN method, assuming the Klein-Nishina scattering kernel for the determination of the angular radiation intensity for photons. We apply the two-dimensional LTSN nodal solution for the averaged angular radiation evaluation for the two-dimensional case, using the Klein-Nishina kernel for photons and the Compton kernel for electrons. From the angular radiation intensity we construct a closed-form solution for the build-up factor and evaluate the absorbed energy. We present numerical simulations and comparisons against results from the literature

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“…On the other hand, in a recent work [9] we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The Fokker-Planck (FP) approximation represents the impact of soft reactions as continuously slowing down the electrons, while also continuously changing their direction; e.g., a monodirectional beam will be dispersed into a finite beam width.…”

Section: Introductionmentioning

confidence: 99%

An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

Rodriguez

Vilhena

Bodmann

2011

Progress in Nuclear Energy

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Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P N approximation, the S N method (discrete ordinates method) and their variants. In the discrete ordinates (S N) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to S N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The main idea relies on applying the P N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x-variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.

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A closed-form solution for the two-dimensional Fokker–Planck equation for electron transport in the range of Compton Effect

Cited by 3 publications

References 11 publications

On exact solutions to partial differential equations by the modified homotopy perturbation method

On exact solutions to partial differential equations by the modified homotopy perturbation method

A Closed-Form Formulation for the Build-Up Factor and Absorbed Energy for Photons and Electrons in the Compton Energy Range in Cartesian Geometry

An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

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