scholarly journals Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1195 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Hijaz Ahmad ◽  
Phatiphat Thounthong ◽  
Yu-Ming Chu ◽  
Clemente Cesarano
Keyword(s):  
Meshless Method ◽  
Three Dimensional ◽  
Fractional Part ◽  
Diffusion Equations ◽  
Numerical Treatment ◽  
Physical Processes ◽  
Dimensional Diffusion ◽  
Symmetry Properties ◽  
Fractional Pde ◽  
Fractional Differential

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

scholarly journals Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 95-105 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Hijaz Ahmad ◽  
Mustafa Inc ◽  
Shao-Wen Yao ◽  
Bandar Almohsen
Keyword(s):  
Mass Transfer ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Meshless Method ◽  
Fractional Part ◽  
Higher Dimensions ◽  
Numerical Treatment ◽  
Dimensional Diffusion ◽  
Derivatives Of

In this article, we presented an efficient local meshless method for the numerical treatment of two term time fractional-order multi-dimensional diffusion PDE. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solu?tion on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. The numerical re?sults are obtained for 1-, 2- and 3-D cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

scholarly journals Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method

2020 ◽  
pp. 210-210 ◽  
Author(s):  
Mohan Srivastava ◽  
Hijaz Ahmad ◽  
Imtiaz Ahmad ◽  
Phatiphat Thounthong ◽  
Nawaz Khan
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Three Dimensional ◽  
Fractional Part ◽  
Higher Dimensions ◽  
Test Problems ◽  
Numerical Treatment ◽  
Convection Diffusion ◽  
Derivatives Of

In this article, we present an efficient local meshless method for the numerical treatment of three-dimensional convection-diffusion PDEs. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solution on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. Some test problems on regular and irregular computational domains are presented to verify the validity, efficiency and accuracy of the method.

scholarly journals Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 95-105 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Hijaz Ahmad ◽  
Mustafa Inc ◽  
Shao-Wen Yao ◽  
Bandar Almohsen
Keyword(s):  
Mass Transfer ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Meshless Method ◽  
Fractional Part ◽  
Higher Dimensions ◽  
Numerical Treatment ◽  
Dimensional Diffusion ◽  
Derivatives Of

In this article, we presented an efficient local meshless method for the numerical treatment of two term time fractional-order multi-dimensional diffusion PDE. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solu?tion on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. The numerical re?sults are obtained for 1-, 2- and 3-D cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

Accurate numerical schemes based on the lattice Boltzmann methods for multi-dimensional diffusion equations

2014 ◽  
Vol 25 (04) ◽  
pp. 1350104
Author(s):  
Shinsuke Suga
Keyword(s):  
Fourth Order ◽  
Lattice Boltzmann ◽  
Three Dimensional ◽  
Relaxation Parameter ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Velocity Models ◽  
Dimensional Diffusion ◽  
Error Terms

We propose accurate explicit numerical schemes based on the lattice Boltzmann (LB) method for multi-dimensional diffusion equations. In LB schemes, the velocity models D2Q9 and D2Q13 are used for two-dimensional equations and D3Q19 and D3Q25 for three-dimensional equations. We introduce free parameters that characterize the weight of the equilibrium distribution functions to reduce numerical errors. Consistency analysis through the fourth-order Chapman–Ensgok expansion of the distribution functions gives an approximate diffusion equation with error terms up to fourth-order. The relaxation parameter and weight parameters are determined so that second-order error terms are eliminated in the approximate equation. Stability analysis shows that we can find a relaxation parameter so that each of the presented schemes is stable for given diffusion coefficients and discretizing parameters. Numerical experiments for the isotropic and anisotropic benchmark problems show that the presented schemes derived from the velocity models D2Q13 and D3Q25 are useful for numerical simulations of practical problems governed by two- and three-dimensional diffusion equations, respectively. In particular, schemes in which the value of the relaxation parameter is set to be 1 demonstrate a fourth-order accuracy under the stability condition.

scholarly journals RNA Polymerase and Transcription Mechanisms: The Forefront of Physicochemical Studies of Chemical Reactions

Biomolecules ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 32
Author(s):  
Nobuo Shimamoto ◽  
Masahiko Imashimizu
Keyword(s):  
Three Dimensional ◽  
Rate Equations ◽  
Physical Processes ◽  
Antenna Effect ◽  
Binding Reaction ◽  
Interdisciplinary Field ◽  
Dimensional Diffusion ◽  
Physicochemical Studies ◽  
Reaction Theory ◽  
Single Set

The study of transcription and its regulation is an interdisciplinary field that is closely connected with genetics, structural biology, and reaction theory. Among these, although less attention has been paid to reaction theory, it is becoming increasingly useful for research on transcription. Rate equations are commonly used to describe reactions involved in transcription, but they tend to be used unaware of the timescales of relevant physical processes. In this review, we discuss the limitation of rate equation for describing three-dimensional diffusion and one-dimensional diffusion along DNA. We then introduce the chemical ratchet mechanism recently proposed for explaining the antenna effect, an enhancement of the binding affinity to a specific site on longer DNA, which deviates from a thermodynamic rule. We show that chemical ratchet cannot be described with a single set of rate equations but alternative sets of rate equations that temporally switch no faster than the binding reaction.

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