The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Ian Turner - One of the best experts on this subject based on the ideXlab platform.
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high order numerical methods for the Riesz Space fractional advection dispersion equations
arXiv: Numerical Analysis, 2020Co-Authors: Libo Feng, Ian Turner, Pinghui Zhuang, J LiAbstract:In this paper, we propose high-order numerical methods for the Riesz Space fractional advection-dispersion equations (RSFADE) on a {f}inite domain. The RSFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order Space derivative with the Riesz fractional derivatives of order $\alpha\in(0,1)$ and $\beta\in(1,2]$, respectively. Firstly, we utilize the weighted and shifted Grunwald difference operators to approximate the Riesz fractional derivative and present the {f}inite difference method for the RSFADE. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of $\mathcal {O}(\tau^2+h^2)$. Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be $\mathcal {O}(\tau^4+h^4)$. Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are excellent with the theoretical analysis.
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an unstructured mesh finite element method for solving the multi term time fractional and Riesz Space distributed order wave equation on an irregular convex domain
Applied Mathematical Modelling, 2019Co-Authors: Y M Zhao, F L Wang, Ian TurnerAbstract:Abstract In this paper, the numerical analysis for a multi-term time fracstional and Riesz Space distributed-order wave equation is discussed on an irregular convex domain. Firstly, the equation is transformed into a multi-term time-Space fractional wave equation using the mid-point quadrature rule to approximate the distributed-order Riesz Space derivative. Next, the equation is solved by discretising in time using a Crank–Nicolson scheme and in Space using the finite element method (FEM) with an unstructured mesh, respectively. Furthermore, stability and convergence are investigated by introducing some important lemmas on irregular convex domain. Finally, some examples are provided to show the effectiveness and correctness of the proposed numerical method.
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a fast numerical method for two dimensional Riesz Space fractional diffusion equations on a convex bounded region
Applied Numerical Mathematics, 2018Co-Authors: Shiping Chen, Ian TurnerAbstract:Fractional differential equations have attracted considerable attention because of their many applications in physics, geology, biology, chemistry, and finance. In this paper, a two-dimensional Riesz Space fractional diffusion equation on a convex bounded region (2D-RSFDE-CBR) is considered. These regions are more general than rectangle or circular domains. A novel alternating direction implicit method for the 2D-RSFDE-CBR with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the method are discussed. The resulting linear systems are Toeplitz-like and are solved by the preconditioned conjugate gradient method with a suitable circulant preconditioner. By the fast Fourier transform, the method only requires a computational cost of per time step. These numerical techniques are used for simulating a two-dimensional Riesz Space fractional FitzHugh-Nagumo model. The numerical results demonstrate the effectiveness of the method. These techniques can be extended to three spatial dimensions, which will be the topic of our future research.
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a crank nicolson adi galerkin legendre spectral method for the two dimensional Riesz Space distributed order advection diffusion equation
Computers & Mathematics With Applications, 2018Co-Authors: Hui Zhang, Xiaoyun Jiang, Fanhai Zeng, Ian TurnerAbstract:Abstract In the paper, a Crank–Nicolson alternating direction implicit (ADI) Galerkin–Legendre spectral scheme is presented for the two-dimensional Riesz Space distributed-order advection–diffusion equation. The Gauss quadrature has a higher computational accuracy than the mid-point quadrature rule, which is proposed to approximate the distributed order Riesz Space derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in Space by the ADI Galerkin–Legendre spectral scheme and in time using the Crank–Nicolson difference method. Stability and convergence analysis are verified for the numerical approximation. A lot of numerical results are demonstrated to justify the theoretical analysis.
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high order numerical methods for the Riesz Space fractional advection dispersion equations
Computers & Mathematics With Applications, 2016Co-Authors: Libo Feng, Ian Turner, Pinghui Zhuang, J LiAbstract:Abstract In this paper, we propose high-order numerical methods for the Riesz Space fractional advection–dispersion equations (RSFADE) on a finite domain. The RSFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order Space derivative with the Riesz fractional derivatives of order α ∈ ( 0 , 1 ) and β ∈ ( 1 , 2 ] , respectively. Firstly, we utilize the weighted and shifted Grunwald difference operators to approximate the Riesz fractional derivative and present the finite difference method for the RSFADE. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of O ( τ 2 + h 2 ) . Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be O ( τ 4 + h 4 ) . Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent with theoretical analysis.
Jun Kawabe - One of the best experts on this subject based on the ideXlab platform.
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Uniformity for weak order convergence of Riesz Space-valued measures
Bulletin of the Australian Mathematical Society, 2020Co-Authors: Jun KawabeAbstract:The purpose of the paper is to show that weak order convergence of a net of Dedekind complete Riesz Space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions. The paper ends with generalizing Ulam's theorem for tightness of positive, finite Borel measures to Riesz Space-valued σ-measures.
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regularities of Riesz Space valued non additive measures with applications to convergence theorems for choquet integrals
Fuzzy Sets and Systems, 2010Co-Authors: Jun KawabeAbstract:A deeper investigation of Radonness and @t-smoothness properties of Riesz Space-valued Borel non-additive measures is carried out. To this end, due to a lack of @e-argument in a Riesz Space, the multiple Egoroff property is introduced and enforced on the involved Riesz Space. The established regularity properties of Borel non-additive measures are instrumental when formulating certain types of monotone convergence theorems for Choquet integrals.
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The Choquet integral in Riesz Space
Fuzzy Sets and Systems, 2008Co-Authors: Jun KawabeAbstract:A comprehensive discussion of the theory of Choquet integration in a Riesz Space is given. In particular, it is proved that the monotone convergence theorem, the Fatou lemma, and the dominated convergence theorem are still valid for Riesz Space-valued non-additive measures if we assume that the Riesz Space has a new property concerning the cardinality of the set of points of discontinuity of a monotone function.
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the alexandroff theorem for Riesz Space valued non additive measures
Fuzzy Sets and Systems, 2007Co-Authors: Jun KawabeAbstract:The Alexandroff theorem for a compact non-additive measure with values in a Riesz Space is still valid for the following two cases: one is the case that the measure is autocontinuous and the Riesz Space has the weak asymptotic Egoroff property and the other is the case that the measure is uniformly autocontinuous and the Riesz Space is weakly σ-distributive. A close connection between regularity and continuity of non-additive measures is also given.
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regularity and lusin s theorem for Riesz Space valued fuzzy measures
Fuzzy Sets and Systems, 2007Co-Authors: Jun KawabeAbstract:A smoothness condition (the multiple Egoroff property) is introduced and imposed on a Riesz Space to show that every weakly null-additive, Riesz Space-valued fuzzy Borel measure on any metric Space is regular. It is also proved that Lusin's theorem remains valid for such Riesz Space-valued non-additive measures.
J E Maciasdiaz - One of the best experts on this subject based on the ideXlab platform.
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an explicit dissipation preserving method for Riesz Space fractional nonlinear wave equations in multiple dimensions
Communications in Nonlinear Science and Numerical Simulation, 2018Co-Authors: J E MaciasdiazAbstract:Abstract In this work, we investigate numerically a model governed by a multidimensional nonlinear wave equation with damping and fractional diffusion. The governing partial differential equation considers the presence of Riesz Space-fractional derivatives of orders in (1, 2], and homogeneous Dirichlet boundary data are imposed on a closed and bounded spatial domain. The model under investigation possesses an energy function which is preserved in the undamped regime. In the damped case, we establish the property of energy dissipation of the model using arguments from functional analysis. Motivated by these results, we propose an explicit finite-difference discretization of our fractional model based on the use of fractional centered differences. Associated to our discrete model, we also propose discretizations of the energy quantities. We establish that the discrete energy is conserved in the undamped regime, and that it dissipates in the damped scenario. Among the most important numerical features of our scheme, we show that the method has a consistency of second order, that it is stable and that it has a quadratic order of convergence. Some one- and two-dimensional simulations are shown in this work to illustrate the fact that the technique is capable of preserving the discrete energy in the undamped regime. For the sake of convenience, we provide a Matlab implementation of our method for the one-dimensional scenario.
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a pseudo energy invariant method for relativistic wave equations with Riesz Space fractional derivatives
Computer Physics Communications, 2017Co-Authors: J E Maciasdiaz, Ahmed S Hendy, R H De StaelenAbstract:Abstract In this work, we investigate a general nonlinear wave equation with Riesz Space-fractional derivatives that generalizes various classical hyperbolic models, including the sine-Gordon and the Klein–Gordon equations from relativistic quantum mechanics. A finite-difference discretization of the model is provided using fractional centered differences. The method is a technique that is capable of preserving an energy-like quantity at each iteration. Some computational comparisons against solutions available in the literature are performed in order to assess the capability of the method to preserve the invariant. Our experiments confirm that the technique yields good approximations to the solutions considered. As an application of our scheme, we provide simulations that confirm, for the first time in the literature, the presence of the phenomenon of nonlinear supratransmission in Riesz Space-fractional Klein–Gordon equations driven by a harmonic perturbation at the boundary.
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numerical study of the process of nonlinear supratransmission in Riesz Space fractional sine gordon equations
Communications in Nonlinear Science and Numerical Simulation, 2017Co-Authors: J E MaciasdiazAbstract:Abstract In this work, we consider a ( 1 + 1 ) -dimensional Riesz Space-fractional damped sine-Gordon equation defined on a bounded spatial interval. Sinusoidal Dirichlet boundary data are imposed at one end of the interval and homogeneous Neumann conditions at the other. The system is initially at rest in the equilibrium position, and is discretized to simulate its complex dynamics. The method employed in this work is a finite-difference discretization of the mathematical model of interest. Our scheme is throughly validated against simulations on the dynamics of the classical and the Space-fractional sine-Gordon equations, which are available in the literature. As the main result of this manuscript, we have found numerical evidence on the presence of the phenomenon of nonlinear supratransmission in Riesz Space-fractional sine-Gordon systems. Simulations have been conducted in order to predict its occurrence for some values of the fractional order of the spatial derivative, and a wide range of values of the frequency of the sinusoidal perturbation at the boundary. As far as the author knows, this may be one of the first numerical reports on the existence of nonlinear supratransmission in sine-Gordon systems of Riesz Space-fractional order.
Yang Xu - One of the best experts on this subject based on the ideXlab platform.
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implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear Riesz Space fractional diffusion equation
Applied Mathematics and Computation, 2020Co-Authors: Jingjun Zhao, Yanming Zhang, Yang XuAbstract:Abstract A numerical method with high accuracy both in time and in Space is proposed for the two-dimensional nonlinear Riesz Space fractional diffusion equation. The main idea is based on a spectral Galerkin method in spatial direction and an s-stage implicit Runge-Kutta method in temporal direction. A rigorous stability and error analysis is performed for the proposed method. It is shown that the proposed method is stable and convergent. The optimal spatial error estimate is also derived. Numerical experiments are provided to illustrate the theoretical results.
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implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear Riesz Space distributed order diffusion equation
Applied Numerical Mathematics, 2020Co-Authors: Jingjun Zhao, Yanming Zhang, Yang XuAbstract:Abstract To discretize the distributed-order term of two-dimensional nonlinear Riesz Space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s-stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies the local Lipschitz condition, the s-stage implicit Runge-Kutta method with order p ( p ≥ s + 1 ) is coercive and algebraically stable, then we can prove that the proposed method is stable and convergent of order s + 1 in time. In addition, we also derive the optimal error estimate for the discretization of distributed-order term and spatial term. Finally, numerical experiments are presented to verify the theoretical results.
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implicit runge kutta and spectral galerkin methods for Riesz Space fractional distributed order diffusion equation
Computational & Applied Mathematics, 2020Co-Authors: Jingjun Zhao, Yanming Zhang, Yang XuAbstract:A numerical method with high accuracy both in time and in Space is constructed for the Riesz Space fractional diffusion equation, in which the temporal component is discretized by an s-stage implicit Runge–Kutta method and the spatial component is approximated by a spectral Galerkin method. For an algebraically stable Runge–Kutta method of order p $$(p\ge s+1)$$ , the unconditional stability of the full discretization is proven and the convergence order of $$s+1$$ in time is obtained. The optimal error estimate in Space, with convergence order only depending on the regularity of initial value and f, is also derived. Meanwhile, this kind of method is applied to the Riesz Space distributed-order diffusion equation, and similar stability and convergence results are obtained. Finally, numerical experiments are provided to illustrate the theoretical results.
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general linear and spectral galerkin methods for the Riesz Space fractional diffusion equation
Applied Mathematics and Computation, 2020Co-Authors: Yang Xu, Yanming Zhang, Jingjun ZhaoAbstract:Abstract The general linear method is considered to discretize the temporal term of Riesz Space fractional diffusion equation. Combined with a spectral Galerkin method in the spatial direction, a method with high global accuracy is constructed. If the general linear method is algebraically stable, the stability is proven for the full discretization. Furthermore, under some conditions, the convergence order in time and the optimal error estimate in Space are also obtained. Meanwhile, numerical examples are given to confirm the theoretical results.
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backward difference formulae and spectral galerkin methods for the Riesz Space fractional diffusion equation
Mathematics and Computers in Simulation, 2019Co-Authors: Yang Xu, Yanming Zhang, Jingjun ZhaoAbstract:Abstract Approximating Riesz Space fractional diffusion equation in time by k -step backward difference formula and in Space by spectral Galerkin method, we establish a fully discrete scheme with high order both in time and in Space. For k ≤ 5 , we prove the stability of full discretization and obtain the error estimate with order O ( τ k + N α 2 − m ) , which depends only on the regularity of initial value and right-hand function. Moreover, we extend the proposed method to two dimensional case and derive similar results. Finally, we illustrate the theoretical estimates by numerical examples.
Jingjun Zhao - One of the best experts on this subject based on the ideXlab platform.
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implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear Riesz Space fractional diffusion equation
Applied Mathematics and Computation, 2020Co-Authors: Jingjun Zhao, Yanming Zhang, Yang XuAbstract:Abstract A numerical method with high accuracy both in time and in Space is proposed for the two-dimensional nonlinear Riesz Space fractional diffusion equation. The main idea is based on a spectral Galerkin method in spatial direction and an s-stage implicit Runge-Kutta method in temporal direction. A rigorous stability and error analysis is performed for the proposed method. It is shown that the proposed method is stable and convergent. The optimal spatial error estimate is also derived. Numerical experiments are provided to illustrate the theoretical results.
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implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear Riesz Space distributed order diffusion equation
Applied Numerical Mathematics, 2020Co-Authors: Jingjun Zhao, Yanming Zhang, Yang XuAbstract:Abstract To discretize the distributed-order term of two-dimensional nonlinear Riesz Space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s-stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies the local Lipschitz condition, the s-stage implicit Runge-Kutta method with order p ( p ≥ s + 1 ) is coercive and algebraically stable, then we can prove that the proposed method is stable and convergent of order s + 1 in time. In addition, we also derive the optimal error estimate for the discretization of distributed-order term and spatial term. Finally, numerical experiments are presented to verify the theoretical results.
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implicit runge kutta and spectral galerkin methods for Riesz Space fractional distributed order diffusion equation
Computational & Applied Mathematics, 2020Co-Authors: Jingjun Zhao, Yanming Zhang, Yang XuAbstract:A numerical method with high accuracy both in time and in Space is constructed for the Riesz Space fractional diffusion equation, in which the temporal component is discretized by an s-stage implicit Runge–Kutta method and the spatial component is approximated by a spectral Galerkin method. For an algebraically stable Runge–Kutta method of order p $$(p\ge s+1)$$ , the unconditional stability of the full discretization is proven and the convergence order of $$s+1$$ in time is obtained. The optimal error estimate in Space, with convergence order only depending on the regularity of initial value and f, is also derived. Meanwhile, this kind of method is applied to the Riesz Space distributed-order diffusion equation, and similar stability and convergence results are obtained. Finally, numerical experiments are provided to illustrate the theoretical results.
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general linear and spectral galerkin methods for the Riesz Space fractional diffusion equation
Applied Mathematics and Computation, 2020Co-Authors: Yang Xu, Yanming Zhang, Jingjun ZhaoAbstract:Abstract The general linear method is considered to discretize the temporal term of Riesz Space fractional diffusion equation. Combined with a spectral Galerkin method in the spatial direction, a method with high global accuracy is constructed. If the general linear method is algebraically stable, the stability is proven for the full discretization. Furthermore, under some conditions, the convergence order in time and the optimal error estimate in Space are also obtained. Meanwhile, numerical examples are given to confirm the theoretical results.
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backward difference formulae and spectral galerkin methods for the Riesz Space fractional diffusion equation
Mathematics and Computers in Simulation, 2019Co-Authors: Yang Xu, Yanming Zhang, Jingjun ZhaoAbstract:Abstract Approximating Riesz Space fractional diffusion equation in time by k -step backward difference formula and in Space by spectral Galerkin method, we establish a fully discrete scheme with high order both in time and in Space. For k ≤ 5 , we prove the stability of full discretization and obtain the error estimate with order O ( τ k + N α 2 − m ) , which depends only on the regularity of initial value and right-hand function. Moreover, we extend the proposed method to two dimensional case and derive similar results. Finally, we illustrate the theoretical estimates by numerical examples.
