Discrete fractional order differencing method in anomalous diffusion and based on step-by-step calculation

An anomalous diffusion and difference method technology, applied in the field of numerical simulation, can solve problems such as low calculation efficiency and limited number of distribution points, and achieve the effect of expanding the number of points, actively engineering practice, and improving efficiency

Inactive Publication Date: 2017-05-31
HOHAI UNIV
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Problems solved by technology

However, the existing discrete fractional order difference method still has

Method used

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  • Discrete fractional order differencing method in anomalous diffusion and based on step-by-step calculation
  • Discrete fractional order differencing method in anomalous diffusion and based on step-by-step calculation
  • Discrete fractional order differencing method in anomalous diffusion and based on step-by-step calculation

Examples

Experimental program
Comparison scheme
Effect test

Embodiment

[0101] Consider the following discrete fractional order sediment transport problem, the governing equation is:

[0102]

[0103] Wherein, the diffusion coefficient D=0.04, the convection coefficient A=0.05, the order α of the time fractional derivative=0.8, and the order β=1.6 of the space fractional derivative.

[0104] In order to verify the effectiveness of the method, we choose different time steps and space steps to verify whether the method reduces the limitation of the gamma function and improves the calculation efficiency. The convection item is discretized by the implicit format finite difference algorithm.

[0105] The exact numerical results are as follows:

[0106] For the case of j=1, the numerical result of the diffusion concentration u is,

[0107]

[0108] For the case of j=2, the numerical result of the diffusion concentration u is,

[0109]

[0110] For the case of j>2, the numerical result of the diffusion concentration u is,

[0111]

[0112]...

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Abstract

The invention discloses a discrete fractional order differencing method in anomalous diffusion and based on step-by-step calculation. The method includes following steps: acquiring related parameters and initial-boundary conditions; adopting the discrete fractional order differencing method to discretize a control equation, and simulating distribution size of substance according to needed numerical value to determine grid point number; determining a time-space fractional order anomalous diffusion equation of anomalous diffusion, and acquiring a discrete format of the time-space anomalous diffusion equation; taking a specific value of two gamma functions in the time-space anomalous diffusion equation as a parameter, utilizing recursive relation of the gamma functions, combining concept of distributed calculation and idea of overall consideration in numerical value simulation, converting a problem of the specific value of the gamma functions into a form of product of multiple decimals, and substituting into a discrete control equation to acquire a numerical value result of diffusion concentration. The discrete fractional order differencing method has the advantages that limitation on calculation of the gamma functions is reduced, point number of simulation is expanded, and simulation efficiency is improved.

Description

technical field [0001] The invention relates to a discrete fractional order difference method, in particular to a discrete fractional order difference method based on step-by-step calculation in anomalous diffusion, and belongs to the technical field of numerical simulation. Background technique [0002] In many engineering fields such as environment, water conservancy, chemical industry, and energy, the migration and diffusion of basic parameters such as momentum, heat, and mass of fluids in the flow process have attracted much attention. The fundamental solution to all kinds of fluid flow and heat and mass transfer problems must be based on the scientific grasp of the laws of migration and diffusion in the fluid environment, so as to understand the changes of the fluid or the substances contained in it with time and space. With the improvement of the level of human cognition of the world, the phenomenon of abnormal diffusion is becoming more and more common and attracting ...

Claims

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Application Information

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IPC IPC(8): G06F17/15
CPCG06F17/15
Inventor 孙洪广刘肖廷达玉乐
Owner HOHAI UNIV
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