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Kevin Burrage - One of the best experts on this subject based on the ideXlab platform.

  • a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain Model on an approximate irregular domain
    Journal of Computational Physics, 2015
    Co-Authors: Fawang Liu, Ian Turner, Pinghui Zhuang, Vo Anh, Kevin Burrage
    Abstract:

    A FitzHugh-Nagumo monodomain Model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain Model on an irregular domain. The Model consists of a coupled Riesz space fractional nonlinear reaction-diffusion Model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Second, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion Model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this Model on an approximate irregular domain is proposed. Third, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional FitzHugh-Nagumo Model on both an approximate circular and an approximate irregular domain.

  • a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation
    SIAM Journal on Numerical Analysis, 2014
    Co-Authors: Fanhai Zeng, Kevin Burrage, Changpin Li, Ian Turner
    Abstract:

    In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo Model. Numerical results are provided to verify the theoretical analysis.

  • numerical simulation for two dimensional riesz space fractional diffusion equations with a nonlinear reaction term
    Central European Journal of Physics, 2013
    Co-Authors: Shiping Chen, Kevin Burrage, Ian Turner
    Abstract:

    Fractional differential equations have attracted considerable interest because of their ability to Model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to Model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional FitzHugh-Nagumo Model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

Jungang Wang - One of the best experts on this subject based on the ideXlab platform.

  • finite element method for nonlinear riesz space fractional diffusion equations on irregular domains
    Journal of Computational Physics, 2017
    Co-Authors: Zongze Yang, Zhanbin Yuan, Jungang Wang
    Abstract:

    In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme. A finite element method is developed to solve 2D-space fractional diffusion equations with nonlinear source term.We develop a finite element method for FPDEs on irregular domain.We obtain explicit expressions for fractional derivatives of shape functions.We give the details on how to compute fractional stiffness matrix.Fractional FitzHugh-Nagumo Model is solved on circular domain.

Ian Turner - One of the best experts on this subject based on the ideXlab platform.

  • a fast numerical method for two dimensional riesz space fractional diffusion equations on a convex bounded region
    Applied Numerical Mathematics, 2018
    Co-Authors: Shiping Chen, Ian Turner
    Abstract:

    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.

  • a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain Model on an approximate irregular domain
    Journal of Computational Physics, 2015
    Co-Authors: Fawang Liu, Ian Turner, Pinghui Zhuang, Vo Anh, Kevin Burrage
    Abstract:

    A FitzHugh-Nagumo monodomain Model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain Model on an irregular domain. The Model consists of a coupled Riesz space fractional nonlinear reaction-diffusion Model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Second, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion Model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this Model on an approximate irregular domain is proposed. Third, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional FitzHugh-Nagumo Model on both an approximate circular and an approximate irregular domain.

  • a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation
    SIAM Journal on Numerical Analysis, 2014
    Co-Authors: Fanhai Zeng, Kevin Burrage, Changpin Li, Ian Turner
    Abstract:

    In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo Model. Numerical results are provided to verify the theoretical analysis.

  • numerical simulation for two dimensional riesz space fractional diffusion equations with a nonlinear reaction term
    Central European Journal of Physics, 2013
    Co-Authors: Shiping Chen, Kevin Burrage, Ian Turner
    Abstract:

    Fractional differential equations have attracted considerable interest because of their ability to Model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to Model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional FitzHugh-Nagumo Model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

Zongze Yang - One of the best experts on this subject based on the ideXlab platform.

  • finite element method for nonlinear riesz space fractional diffusion equations on irregular domains
    Journal of Computational Physics, 2017
    Co-Authors: Zongze Yang, Zhanbin Yuan, Jungang Wang
    Abstract:

    In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme. A finite element method is developed to solve 2D-space fractional diffusion equations with nonlinear source term.We develop a finite element method for FPDEs on irregular domain.We obtain explicit expressions for fractional derivatives of shape functions.We give the details on how to compute fractional stiffness matrix.Fractional FitzHugh-Nagumo Model is solved on circular domain.

Zhanbin Yuan - One of the best experts on this subject based on the ideXlab platform.

  • finite element method for nonlinear riesz space fractional diffusion equations on irregular domains
    Journal of Computational Physics, 2017
    Co-Authors: Zongze Yang, Zhanbin Yuan, Jungang Wang
    Abstract:

    In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme. A finite element method is developed to solve 2D-space fractional diffusion equations with nonlinear source term.We develop a finite element method for FPDEs on irregular domain.We obtain explicit expressions for fractional derivatives of shape functions.We give the details on how to compute fractional stiffness matrix.Fractional FitzHugh-Nagumo Model is solved on circular domain.