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.

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 Partial Differential Equations via Local Meshless Method

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 257 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Muhammad Riaz ◽  
Muhammad Ayaz ◽  
Muhammad Arif ◽  
Saeed Islam ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Meshless Method ◽  
Three Dimensional ◽  
Salient Feature ◽  
Test Problems ◽  
Spatial Discretization ◽  
Partial Differential ◽  
Meshless Collocation

In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs into a system of ODEs. Multiquadric, Gaussian and inverse quadratic RBFs are used for spatial discretization. The obtained system of ODEs has been solved by different time integrators. The salient feature of the local meshless method (LMM) is that it does not require mesh in the problem domain and also far less sensitive to the variation of shape parameter as compared to the global meshless method (GMM). Both rectangular and non rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy and ease implementation of the proposed method are shown via test problems.

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.

Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Imtiaz Ahmad ◽  
Aly R. Seadawy ◽  
Hijaz Ahmad ◽  
Phatiphat Thounthong ◽  
Fuzhang Wang
Keyword(s):  
Meshless Method ◽  
Material Science ◽  
Numerical Study ◽  
Research Work ◽  
Time Integration ◽  
Three Dimensional ◽  
Nuclear Material ◽  
Telegraph Equations ◽  
Model Equations ◽  
Derivatives Of

Abstract This research work is to study the numerical solution of three-dimensional second-order hyperbolic telegraph equations using an efficient local meshless method based on radial basis function (RBF). The model equations are used in nuclear material science and in the modeling of vibrations of structures. The explicit time integration technique is utilized to semi-discretize the model in the time direction whereas the space derivatives of the model are discretized by the proposed local meshless procedure based on multiquadric RBF. Numerical experiments are performed with the proposed numerical scheme for rectangular and non-rectangular computational domains. The proposed method solutions are converging quickly in comparison with the different existing numerical methods in the recent literature.

A New Five Dimensional Hyperchaotic System and its Fractional Order Form

2013 ◽  
Vol 464 ◽  
pp. 375-380 ◽  
Author(s):  
Ling Liu ◽  
Chong Xin Liu ◽  
Yi Fan Liao
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Simulation Analysis ◽  
Three Dimensional ◽  
Hyperchaotic System ◽  
Order Form ◽  
Simulation Results ◽  
Predictor Corrector ◽  
New Hyperchaotic System

In this paper, a new five-dimensional hyperchaotic system by introducing two additional states feedback into a three-dimensional smooth chaotic system. With three nonlinearities, this system has more than one positive Lyapunov exponents. Based on the fractional derivative theory, the fractional-order form of this new hyperchaotic system has been investigated. Through predictor-corrector algorithm, the system is proved by numerical simulation analysis. Simulation results are provided to illustrate the performance of the fractional-order hyperchaotic attractors well.

scholarly journals Solving acoustic scattering problem by the meshless method based on radial basis function

2019 ◽  
Vol 288 ◽  
pp. 01008
Author(s):  
Weifang Liu ◽  
Sanfei Zhao ◽  
Cong Zhang ◽  
Huipo Qiao
Keyword(s):  
Meshless Method ◽  
Solid Mechanics ◽  
Interpolation Method ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Neumann Boundary ◽  
Convection Diffusion ◽  
Point Collocation ◽  
Solution Accuracy

In this paper, radial point collocation method (RPCM) is introduced to solve the acoustic scattering problem. This is a mathematically simple, easy-to-program and truly meshless method, which has been successfully applied to solve the solid mechanics and convection diffusion problems. However, application of this method to investigate acoustic problems, in particle the acoustic scattering problem is relatively new. The main advantage of this method is its mathematically simple, easy to program, and truly meshless. A Hermite-type interpolation method is employed to improve the solution accuracy while the Neumann boundary conditions exist. In addition, acoustic scattering problem is a typical unbounded domain problem, in order to solve it with RPCM, the domain is truncated to a finite region and an artificial boundary condition (ABC) is imposed. Finally, numerical example is presented to validate the accuracy and effectiveness of RPCM. In the future, the extension of RPCM to more complex and practical problems, especially the three-dimensional situations need to be investigated in more detail.

scholarly journals Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 368
Author(s):  
Moein Khalighi ◽  
Leila Eftekhari ◽  
Soleiman Hosseinpour ◽  
Leo Lahti
Keyword(s):  
Fractional Derivative ◽  
Differential Operators ◽  
Three Dimensional ◽  
Volterra Model ◽  
Volterra System ◽  
Existence Of A Solution ◽  
Dynamical Behaviors ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Derivatives Of

In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.

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