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Fawang Liu - One of the best experts on this subject based on the ideXlab platform.
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the analytical solution and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation
Journal of Computational and Applied Mathematics, 2019Co-Authors: Shujun Shen, Fawang Liu, Vo AnhAbstract:In this paper we consider the analytical and numerical solutions for a two-dimensional multi-term time-fractional diffusion and diffusion-wave equation. We derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate Mittag-Leffler function. An implicit Difference Approximation is constructed. Stability and convergence analysis of the numerical scheme are proved by the energy method. Numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis.
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fast finite Difference Approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients
SIAM Journal on Numerical Analysis, 2016Co-Authors: Shanzhen Chen, Ian Turner, Fawang Liu, Xiaoyun Jiang, Kevin BurrageAbstract:In this paper, we consider an inverse problem for identifying the fractional derivative indices in a two-dimensional space-fractional nonlocal model based on a generalization of the two-sided Riemann--Liouville formulation with variable diffusivity coefficients. First, we derive an implicit Difference method (IDM) for the direct problem and the stability and convergence of the IDM are discussed. Second, for the implementation of the IDM, we develop a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) that is superior in computational performance to Gaussian elimination and attains the same accuracy. Third, we utilize the Levenberg--Marquardt (L-M) regularization technique combined with the Armijo rule (the popular inexact line search condition) to solve the modified nonlinear least squares model associated with the parameter identification. Finally, we carry out numerical tests to verify the accuracy and efficiency of the IDM. Numerical investigations are performed with both accurate data and noisy...
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fast finite Difference Approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients
Science & Engineering Faculty, 2016Co-Authors: Shanzhen Chen, Ian Turner, Fawang Liu, Xiaoyun Jiang, Kevin BurrageAbstract:In this paper, we consider an inverse problem for identifying the fractional derivative indices in a two-dimensional space-fractional nonlocal model based on a generalization of the two-sided Riemann--Liouville formulation with variable diffusivity coefficients. First, we derive an implicit Difference method (IDM) for the direct problem and the stability and convergence of the IDM are discussed. Second, for the implementation of the IDM, we develop a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) that is superior in computational performance to Gaussian elimination and attains the same accuracy. Third, we utilize the Levenberg--Marquardt (L-M) regularization technique combined with the Armijo rule (the popular inexact line search condition) to solve the modified nonlinear least squares model associated with the parameter identification. Finally, we carry out numerical tests to verify the accuracy and efficiency of the IDM. Numerical investigations are performed with both accurate data and noisy data to check the effectiveness of the L-M regularization method. The convergence behavior of the L-M for the inverse problem involving the space-fractional diffusion model is shown graphically.
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numerical Approximations and solution techniques for the space time riesz caputo fractional advection diffusion equation
Numerical Algorithms, 2011Co-Authors: Shujun Shen, Fawang Liu, Vo AnhAbstract:In this paper, we consider a space-time Riesz---Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order ????(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β 1???(0,1) and β 2???(1,2], respectively. We present an explicit Difference Approximation and an implicit Difference Approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit Difference Approximation is unconditionally stable and convergent, but the explicit Difference Approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.
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stability and convergence of a new explicit finite Difference Approximation for the variable order nonlinear fractional diffusion equation
Applied Mathematics and Computation, 2009Co-Authors: R Lin, Vo Anh, Fawang Liu, Ian TurnerAbstract:In this paper, we consider the variable-order nonlinear fractional diffusion equation@?u(x,t)@?t=B(x,t)"xR^@a^(^x^,^t^)u(x,t)+f(u,x,t),where "xR^@a^(^x^,^t^) is a generalized Riesz fractional derivative of variable order @a(x,t)(1<@a(x,t)=<2) and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u"1,x,t)-f(u"2,x,t)|=
Akbar Mohebbi - One of the best experts on this subject based on the ideXlab platform.
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high order compact solution of the one dimensional heat and advection diffusion equations
Applied Mathematical Modelling, 2010Co-Authors: Akbar Mohebbi, Mehdi DehghanAbstract:Abstract In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite Difference Approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C 1 -spline collocation method for the resulting linear system of ordinary differential equations. The cubic C 1 -spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O ( h 4 , k 4 ) . Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite Difference Approximation of fourth-order and the cubic C 1 -spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.
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high order solution of one dimensional sine gordon equation using compact finite Difference and dirkn methods
Mathematical and Computer Modelling, 2010Co-Authors: Akbar MohebbiAbstract:In this work we propose a high-order and accurate method for solving the one-dimensional nonlinear sine-Gordon equation. The proposed method is based on applying a compact finite Difference scheme and the diagonally implicit Runge-Kutta-Nystrom (DIRKN) method for spatial and temporal components, respectively. We apply a compact finite Difference Approximation of fourth order for discretizing the spatial derivative and a fourth-order A-stable DIRKN method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables and is unconditionally stable. The results of numerical experiments show that the combination of a compact finite Difference Approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving the one-dimensional sine-Gordon equation.
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fourth order compact solution of the nonlinear klein gordon equation
Numerical Algorithms, 2009Co-Authors: Akbar Mohebbi, Zohreh AsgariAbstract:In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite Difference Approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nystrom (DIRKN) method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite Difference Approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.
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high order implicit collocation method for the solution of two dimensional linear hyperbolic equation
Numerical Methods for Partial Differential Equations, 2009Co-Authors: Akbar MohebbiAbstract:In this article, we introduce a high-order accurate method for solving the two dimensional linear hyperbolic equation. We apply a compact finite Difference Approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component. The resulted method is unconditionally stable and solves the two-dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite Difference Approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009
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high order compact boundary value method for the solution of unsteady convection diffusion problems
Mathematics and Computers in Simulation, 2008Co-Authors: Mehdi Dehghan, Akbar MohebbiAbstract:In this paper, we propose a new class of high-order accurate methods for solving the two-dimensional unsteady convection-diffusion equation. These techniques are based on the method of lines approach. We apply a compact finite Difference Approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulted linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Also this method is unconditionally stable due to the favorable stability property of boundary value methods. Numerical results obtained from solving several problems include problems encounter in many transport phenomena, problems with Gaussian pulse initial condition and problems with sharp discontinuity near the boundary, show that the compact finite Difference Approximation of fourth order and a boundary value method of fourth order give an efficient algorithm for solving such problems.
Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.
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numerical solution of 2d navier stokes equation discretized via boundary elements method and finite Difference Approximation
Engineering Analysis With Boundary Elements, 2018Co-Authors: Zeynab Sedaghatjoo, Mehdi Dehghan, Hossein HosseinzadehAbstract:Abstract In this paper boundary elements method (BEM) is equipped with finite Difference Approximation (FDA) to solve two-dimensional Navier–Stokes (N–S) equation. The N–S equation is converted to a system of ordinary differential equations (ODEs) respect to time in streamfunction-vorticity formulation. The constant direct BEM and 9-point stencil FDA are utilized to handle spatial derivatives, and the final system of ODEs is solved via three numerical schemes, forward Euler, Runge–Kutta and Newton’s methods to find a fast ODE solver. Numerical investigations presented in this article show Newton’s method is faster than the others and it is able to solve the N–S equation for Reynolds number up to 40,000 when grid points are at most 141 × 141. Thanks to BEM and boundary conditions of lid-driven cavity flow, the final system of ODEs is stable also for higher Reynolds numbers. A new technique is proposed in this article which converts BEM’s two-dimensional singular integrals to one-dimensional non-singular ones. The proposed technique reduces computational cost of BEM, significantly, when more accurate results are requested. Numerical experiments show the numerical results are fairly agree with that accurate ones available in the literature.
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high order compact solution of the one dimensional heat and advection diffusion equations
Applied Mathematical Modelling, 2010Co-Authors: Akbar Mohebbi, Mehdi DehghanAbstract:Abstract In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite Difference Approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C 1 -spline collocation method for the resulting linear system of ordinary differential equations. The cubic C 1 -spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O ( h 4 , k 4 ) . Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite Difference Approximation of fourth-order and the cubic C 1 -spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.
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high order compact boundary value method for the solution of unsteady convection diffusion problems
Mathematics and Computers in Simulation, 2008Co-Authors: Mehdi Dehghan, Akbar MohebbiAbstract:In this paper, we propose a new class of high-order accurate methods for solving the two-dimensional unsteady convection-diffusion equation. These techniques are based on the method of lines approach. We apply a compact finite Difference Approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulted linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Also this method is unconditionally stable due to the favorable stability property of boundary value methods. Numerical results obtained from solving several problems include problems encounter in many transport phenomena, problems with Gaussian pulse initial condition and problems with sharp discontinuity near the boundary, show that the compact finite Difference Approximation of fourth order and a boundary value method of fourth order give an efficient algorithm for solving such problems.
Ian Turner - One of the best experts on this subject based on the ideXlab platform.
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fast finite Difference Approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients
SIAM Journal on Numerical Analysis, 2016Co-Authors: Shanzhen Chen, Ian Turner, Fawang Liu, Xiaoyun Jiang, Kevin BurrageAbstract:In this paper, we consider an inverse problem for identifying the fractional derivative indices in a two-dimensional space-fractional nonlocal model based on a generalization of the two-sided Riemann--Liouville formulation with variable diffusivity coefficients. First, we derive an implicit Difference method (IDM) for the direct problem and the stability and convergence of the IDM are discussed. Second, for the implementation of the IDM, we develop a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) that is superior in computational performance to Gaussian elimination and attains the same accuracy. Third, we utilize the Levenberg--Marquardt (L-M) regularization technique combined with the Armijo rule (the popular inexact line search condition) to solve the modified nonlinear least squares model associated with the parameter identification. Finally, we carry out numerical tests to verify the accuracy and efficiency of the IDM. Numerical investigations are performed with both accurate data and noisy...
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fast finite Difference Approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients
Science & Engineering Faculty, 2016Co-Authors: Shanzhen Chen, Ian Turner, Fawang Liu, Xiaoyun Jiang, Kevin BurrageAbstract:In this paper, we consider an inverse problem for identifying the fractional derivative indices in a two-dimensional space-fractional nonlocal model based on a generalization of the two-sided Riemann--Liouville formulation with variable diffusivity coefficients. First, we derive an implicit Difference method (IDM) for the direct problem and the stability and convergence of the IDM are discussed. Second, for the implementation of the IDM, we develop a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) that is superior in computational performance to Gaussian elimination and attains the same accuracy. Third, we utilize the Levenberg--Marquardt (L-M) regularization technique combined with the Armijo rule (the popular inexact line search condition) to solve the modified nonlinear least squares model associated with the parameter identification. Finally, we carry out numerical tests to verify the accuracy and efficiency of the IDM. Numerical investigations are performed with both accurate data and noisy data to check the effectiveness of the L-M regularization method. The convergence behavior of the L-M for the inverse problem involving the space-fractional diffusion model is shown graphically.
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stability and convergence of a new explicit finite Difference Approximation for the variable order nonlinear fractional diffusion equation
Applied Mathematics and Computation, 2009Co-Authors: R Lin, Vo Anh, Fawang Liu, Ian TurnerAbstract:In this paper, we consider the variable-order nonlinear fractional diffusion equation@?u(x,t)@?t=B(x,t)"xR^@a^(^x^,^t^)u(x,t)+f(u,x,t),where "xR^@a^(^x^,^t^) is a generalized Riesz fractional derivative of variable order @a(x,t)(1<@a(x,t)=<2) and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u"1,x,t)-f(u"2,x,t)|=
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the fundamental and numerical solutions of the riesz space fractional reaction dispersion equation
Anziam Journal, 2008Co-Authors: J. Chen, Ian TurnerAbstract:A Riesz space-fractional reaction–dispersion equation (RSFRDE) is obtained from the classical reaction–dispersion equation (RDE) by replacing the second-order space derivative with a Riesz derivative of order beta in (1,2]. In this paper, using Laplace and Fourier transforms, we obtain the fundamental solution for a RSFRDE. We propose an explicit finite-Difference Approximation for a RSFRDE in a bounded spatial domain, and analyse its stability and convergence. Some numerical examples are presented. doi:10.1017/S1446181108000333
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a fourier method for the fractional diffusion equation describing sub diffusion
Journal of Computational Physics, 2007Co-Authors: Changming Chen, Ian Turner, Fawang Liu, Vo AnhAbstract:In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit Difference Approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.
Vo Anh - One of the best experts on this subject based on the ideXlab platform.
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the analytical solution and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation
Journal of Computational and Applied Mathematics, 2019Co-Authors: Shujun Shen, Fawang Liu, Vo AnhAbstract:In this paper we consider the analytical and numerical solutions for a two-dimensional multi-term time-fractional diffusion and diffusion-wave equation. We derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate Mittag-Leffler function. An implicit Difference Approximation is constructed. Stability and convergence analysis of the numerical scheme are proved by the energy method. Numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis.
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numerical Approximations and solution techniques for the space time riesz caputo fractional advection diffusion equation
Numerical Algorithms, 2011Co-Authors: Shujun Shen, Fawang Liu, Vo AnhAbstract:In this paper, we consider a space-time Riesz---Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order ????(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β 1???(0,1) and β 2???(1,2], respectively. We present an explicit Difference Approximation and an implicit Difference Approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit Difference Approximation is unconditionally stable and convergent, but the explicit Difference Approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.
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stability and convergence of a new explicit finite Difference Approximation for the variable order nonlinear fractional diffusion equation
Applied Mathematics and Computation, 2009Co-Authors: R Lin, Vo Anh, Fawang Liu, Ian TurnerAbstract:In this paper, we consider the variable-order nonlinear fractional diffusion equation@?u(x,t)@?t=B(x,t)"xR^@a^(^x^,^t^)u(x,t)+f(u,x,t),where "xR^@a^(^x^,^t^) is a generalized Riesz fractional derivative of variable order @a(x,t)(1<@a(x,t)=<2) and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u"1,x,t)-f(u"2,x,t)|=
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a fourier method for the fractional diffusion equation describing sub diffusion
Journal of Computational Physics, 2007Co-Authors: Changming Chen, Ian Turner, Fawang Liu, Vo AnhAbstract:In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit Difference Approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.