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Computational Multi-term Time-space Fractional Bloch-Torrey Models in Three-dimensions

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Computational Multi-term Time-space Fractional Bloch-Torrey ...

Progress in Electromagnetic Research Symposium (PIERS)

作者： Fawang Liu Jing Li Vo Anh Fuzhou University Fuzhou China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and Technology Changsha China Swinburne University of Technology P. O. Box 218 Hawthorn VIC Australia

ISBN: (数字)9781728134031

ISBN: (纸本)9781728134048

In this paper, we will develop new mathematical tools for mapping tissue microstructural properties via the use of space-time fractional calculus methods. We propose the following two new computational multi-term time-space fractional diffusion models in three dimensions, which can be used to simulate multi-term time-space fractional Bloch-Torrey models in three dimensions: Model I: Finite element method for the multi-term time-space fractional diffusion model with Riesz fractional operator; Model II: Finite difference method for the multi-term time-space fractional diffusion equation with fractional Laplacian operator. Firstly, the three-dimensional multi-term time-space fractional Bloch-Torrey models are decoupled; the problem is then equivalent to solving the three-dimensional time-space fractional diffusion equations (Model I and Model II). Secondly, we propose the finite element method for Model I and finite difference method for Model II, respectively. These methods can be directly used to simulate three-dimensional multi-term time-space fractional Bloch-Torrey models. Finally, some numerical examples are given to demonstrate the versatility and application of the models.

关键词： mathematical model Springs Brain modeling Solid modeling Numerical models Magnetic resonance imaging Computational modeling

Symmetric integrators based on continuous-stage Runge-Kutta-Nyström methods for reversible systems

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arXiv 2019年

作者： Tang, Wensheng Zhang, Jingjing College of Mathematics and Statistics Changsha University of Science and Technology Changsha410114 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha410114 China School of Science East China Jiaotong University Nanchang330013 China

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nyström methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported. Copyright © 2019, The Authors. All rights reserved.

关键词： Ordinary differential equations

New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell-Whitehead Equation

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Advances in mathematical Physics 2020年第1期2020卷

作者： Chu, Yu-Ming Javeed, Shumaila Baleanu, Dumitru Riaz, Sidra Rezazadeh, Hadi Department of Mathematics Huzhou University Huzhou 313000 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and Technology Changsha 410114 China Department of Mathematics Comsats University Islamabad Park Road Chak Shahzad Islamabad Pakistan Department of Mathematics Cankaya University Ankara Turkey Institute of Space Sciences Magurele-Bucharest Romania Department of Mathematics Riphah International University Sector I-14 Islamabad Pakistan Faculty of Engineering Technology Amol University of Special Modern Technologies Amol Iran

This work presents the new exact solutions of nonlinear partial differential equations (PDEs). The solutions are acquired by using an effectual approach, the first integral method (FIM). The suggested technique is implemented to obtain the solutions of space-time Kolmogorov Petrovskii Piskunov (KPP) equation and its derived equations, namely, Fitzhugh Nagumo (FHN) equation and Newell-Whitehead (NW) equation. The considered models are significant in biology. The KPP equation describes genetic model for spread of dominant gene through population. The FHN equation is imperative in the study of intercellular trigger waves. Similarly, the NW equation is applied for chemical reactions, Faraday instability, and Rayleigh-Benard convection. The proposed technique FIM can be applied to find the exact solutions of PDEs. © 2020 Yu-Ming Chu et al.

关键词： Integrable system Differential equation

On self-duality and hulls of cyclic codes over 2m[u]huki with oddly even length

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arXiv 2019年

作者： Cao, Yonglin Cao, Yuan Fu, Fang-Wei School of Mathematics and Statistics Shandong University of Technology Zibo Shandong255091 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and Technology Changsha Hunan410114 Hubei Key Laboratory of Applied Mathematics Faculty of Mathematics and Statistics Hubei University Wuhan430062 Chern Institute of Mathematics and Lpmc Tianjin Key Laboratory of Network and Data Security Technology Nankai University Tianjin300071 China

Let F2m be a finite field of 2m elements, and R = F2m[u]/huki = 2m + u2m + . . . + uk-1F2m (uk = 0) where k is an integer satisfying k ≥ 2. For any odd positive integer n, an explicit representation for every self-dual cyclic code over R of length 2n and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and 2-quasi-cyclic code of length 4n over F2m derived by every self-dual cyclic code of length 2n over 2m + u2m and a Gray map from 2m + u2m onto F22 m. Finally, the hull of each cyclic code with length 2n over F2m +uF2m is determined and all distinct self-orthogonal cyclic codes of length 2n over 2m + u2m are listed. Copyright © 2019, The Authors. All rights reserved.

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Two types of variational integrators and their equivalence

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arXiv 2018年

作者： Tang, Wensheng College of Mathematics and Statistics Changsha University of Science and Technology Changsha410114 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha410114 China

In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton’s principle while the other from Galerkin variational approach. It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function L(q,q) = 21qTMq−U(q) (M is an invertible symmetric constant matrix). They are symplectic, symmetric, possess super-convergence order 2s (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods. Copyright © 2018, The Authors. All rights reserved.

关键词： Galerkin methods

Chebyshev symplectic methods based on continuous-stage Runge-Kutta methods

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arXiv 2018年

We develop Chebyshev symplectic methods based on Chebyshev orthogonal polynomials of the first and second kind separately in this paper. Such type of symplectic methods can be conveniently constructed with the newly-built theory of weighted continuous-stage Runge-Kutta methods. A few numerical experiments are well performed to verify the efficiency of our new methods. Copyright © 2018, The Authors. All rights reserved.

关键词： Runge Kutta methods

Energy-preserving continuous-stage Runge-Kutta-Nyström methods

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arXiv 2018年

Many practical problems can be described by second-order system q¨ = −M∇U(q), in which people give special emphasis to some invariants with explicit physical meaning, such as energy, momentum, angular momentum, etc. However, conventional numerical integrators for such systems will fail to preserve any of these quantities which may lead to qualitatively incorrect numerical solutions. This paper is concerned with the development of energy-preserving continuous-stage Runge-Kutta-Nyström (csRKN) methods for solving second-order systems. Sufficient conditions for csRKN methods to be energy-preserving are presented and it is proved that all the energy-preserving csRKN methods satisfying these sufficient conditions can be essentially induced by energy-preserving continuous-stage partitioned Runge-Kutta methods. Some illustrative examples are given and relevant numerical results are reported. Copyright © 2018, The Authors. All rights reserved.

关键词： Hamiltonians

Energy-preserving continuous-stage partitioned Runge-Kutta methods

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arXiv 2018年

作者： Tan, Wensheng College of Mathematics and Statistics Changsha University of Science and Technology Changsha410114 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha410114 China

In this paper, we present continuous-stage partitioned Runge-Kutta (csPRK) methods for energy-preserving integration of Hamiltonian systems. A sufficient condition for the energy preservation of the csPRK methods is derived. It is shown that the presented condition contains the existing condition for energy-preserving continuous-stage Runge-Kutta methods as a special case. A noticeable and interesting result is that when we use the simplifying assumptions of order conditions and the normalized shifted Legendre polynomials for constructing high-order energy-preserving csPRK methods, both the Butcher "weight" coefficients Bτ and Bbτ must be equal to 1. As illustrative examples, new energy-preserving integrators are acquired by virtue of the presented condition, and for the sake of verifying our theoretical results, some numerical experiments are reported. Copyright © 2018, The Authors. All rights reserved.

关键词： Runge Kutta methods

Continuous-stage Runge-Kutta-Nyström methods

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arXiv 2018年

We develop continuous-stage Runge-Kutta-Nyström (csRKN) methods in this paper. By leading weight function into the formalism of csRKN methods and modifying the original pattern of continuous-stage methods, we establish a new and larger framework for csRKN methods and it enables us to derive more effective RKN-type methods. Particularly, a variety of classical weighted orthogonal polynomials can be used in the construction of RKN-type methods. As an important application, new families of symmetric and symplectic integrators can be easily acquired in such framework. Numerical experiments have verified the effectiveness of the new integrators presented in this paper. Copyright © 2018, The Authors. All rights reserved.

关键词： Orthogonal functions