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Fawang Liu - One of the best experts on this subject based on the ideXlab platform.
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a numerical method for solving the two dimensional distributed order space Fractional Diffusion Equation on an irregular convex domain
Applied Mathematics Letters, 2018Co-Authors: Wenping Fan, Fawang LiuAbstract:In this paper, the two-dimensional distributed order space-Fractional Diffusion Equation on an irregular convex domain is considered. The finite element method using unstructured mesh adapted to the irregular domain is proposed to solve the considered Equation. To testify the efficiency of the proposed method, two numerical examples are given. By the error analysis and the comparison between the numerical solution and the exact solution, the finite element method applied in this paper is shown to be valid in solving the two-dimensional distributed order space-Fractional Diffusion Equation on an irregular convex domain.
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Time-Fractional Diffusion Equation for signal smoothing
Applied Mathematics and Computation, 2018Co-Authors: Fawang Liu, Ian TurnerAbstract:The time-Fractional Diffusion Equation is used for signal smoothing. Compared to the classical Diffusion Equation, the time-Fractional Diffusion Equation has another adjustable time-Fractional derivative order to control the Diffusion process. Therefore, some simulated signals are used to compare the smoothing performance between the time-Fractional Diffusion Equation and the classical Diffusion Equation as well as between classical smoothing methods (regularization method, Savitzky–Golay method and wavelet method). In the end, the time-Fractional Diffusion filtering is applied in an NMR spectrum smoothing. Results indicate that the time-Fractional Diffusion filtering is advantage over the classical Diffusion filtering and their smoothing performance is better than that of classical smoothing methods.
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stability and convergence of a new finite volume method for a two sided space Fractional Diffusion Equation
Applied Mathematics and Computation, 2015Co-Authors: Libo Feng, P. Zhuang, Fawang Liu, Ian TurnerAbstract:In this paper, we consider a two-sided space-Fractional Diffusion Equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new Fractional finite volume method for the two-sided space-Fractional Diffusion Equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit Fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.
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a meshless method based on point interpolation method pim for the space Fractional Diffusion Equation
ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS); Science & Engineering Faculty, 2015Co-Authors: Qingxia Liu, P. Zhuang, Fawang Liu, Junhao Chen, Ian TurnerAbstract:This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space Fractional Diffusion Equation. Two fully-discrete schemes for the one-dimensional space Fractional Diffusion Equation are obtained by using the PIM and the strong-forms of the space Diffusion Equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.
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numerical techniques for the variable order time Fractional Diffusion Equation
Applied Mathematics and Computation, 2012Co-Authors: Shujun Shen, Ian Turner, Fawang Liu, Junhao Chen, Vo AnhAbstract:Abstract In this paper we consider the variable order time Fractional Diffusion Equation. We adopt the Coimbra variable order (VO) time Fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order Fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for Fractional partial differential Equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient.
P. Zhuang - One of the best experts on this subject based on the ideXlab platform.
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stability and convergence of a new finite volume method for a two sided space Fractional Diffusion Equation
Applied Mathematics and Computation, 2015Co-Authors: Libo Feng, P. Zhuang, Fawang Liu, Ian TurnerAbstract:In this paper, we consider a two-sided space-Fractional Diffusion Equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new Fractional finite volume method for the two-sided space-Fractional Diffusion Equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit Fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.
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a meshless method based on point interpolation method pim for the space Fractional Diffusion Equation
ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS); Science & Engineering Faculty, 2015Co-Authors: Qingxia Liu, P. Zhuang, Fawang Liu, Junhao Chen, Ian TurnerAbstract:This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space Fractional Diffusion Equation. Two fully-discrete schemes for the one-dimensional space Fractional Diffusion Equation are obtained by using the PIM and the strong-forms of the space Diffusion Equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.
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finite difference approximation for two dimensional time Fractional Diffusion Equation
Journal of Algorithms & Computational Technology, 2007Co-Authors: P. ZhuangAbstract:Fractional Diffusion Equations have recently been used to model problems in physics, hydrology, biology and other areas of application. In this paper, we consider a two-dimensional time Fractional Diffusion Equation (2D-TFDE) on a finite domain. An implicit difference approximation for the 2D-TFDE is presented. Stability and convergence of the method are discussed using mathematical induction. Finally, a numerical example is given. The numerical result is in excellent agreement with our theoretical analysis.
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implicit difference approximation for the time Fractional Diffusion Equation
Journal of Applied Mathematics and Computing, 2006Co-Authors: P. Zhuang, Fawang LiuAbstract:In this paper, we consider a time Fractional Diffusion Equation on a finite domain. The Equation is obtained from the standard Diffusion Equation by replacing the first-order time derivative by a Fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time Fractional Diffusion Equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.
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a Fractional order implicit difference approximation for the space time Fractional Diffusion Equation
Anziam Journal, 2006Co-Authors: P. Zhuang, Ian TurnerAbstract:We consider a space-time Fractional Diffusion Equation on a finite domain. The Equation is obtained from the standard Diffusion Equation by replacing the second order space derivative by a Riemann--Liouville Fractional derivative of order between one and two, and the first order time derivative by a Caputo Fractional derivative of order between zero and one. A Fractional order implicit finite difference approximation for the space-time Fractional Diffusion Equation with initial and boundary values is investigated. Stability and convergence results for the method are discussed, and finally, some numerical results show the system exhibits diffusive behaviour.
Yuri Luchko - One of the best experts on this subject based on the ideXlab platform.
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analyticity of solutions to a distributed order time Fractional Diffusion Equation and its application to an inverse problem
Computers & Mathematics With Applications, 2017Co-Authors: Yuri Luchko, Masahiro YamamotoAbstract:Abstract This paper deals with an initial–boundary value problem for a distributed order time-Fractional Diffusion Equation with the Fractional derivative in the Caputo sense. The method of the eigenfunctions expansion in combination with the Laplace transform is first employed to prove the uniqueness and existence of the solution to the initial–boundary value problem and then to show its analyticity in time. As an application of the analyticity of the solution, a uniqueness result for an important inverse problem of determination of the weight function in the distributed order derivative contained in the time-Fractional Diffusion Equation from one interior point observation of its solution is obtained.
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on the maximum principle for a time Fractional Diffusion Equation
arXiv: Analysis of PDEs, 2017Co-Authors: Yuri Luchko, Masahiro YamamotoAbstract:In this paper, we discuss the maximum principle for a time-Fractional Diffusion Equation $$ \partial_t^\alpha u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t)) + c(x)u(x,t) + F(x,t),\ t>0,\ x \in \Omega \subset {\mathbb R}^n$$ with the Caputo time-derivative of the order $\alpha \in (0,1)$ in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the Fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions $F = F(x,t)$ we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient $c=c(x)$ by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient $c=c(x)$.
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initial boundary value problems for the one dimensional time Fractional Diffusion Equation
Fractional Calculus and Applied Analysis, 2012Co-Authors: Yuri LuchkoAbstract:In this paper, some initial-boundary-value problems for the time-Fractional Diffusion Equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-Fractional Diffusion Equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-Fractional Diffusion Equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-Fractional Diffusion Equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.
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initial boundary value problems for the one dimensional time Fractional Diffusion Equation
arXiv: Analysis of PDEs, 2011Co-Authors: Yuri LuchkoAbstract:In this paper, some initial-boundary-value problems for the time-Fractional Diffusion Equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-Fractional Diffusion Equation. In its turn, the maximum principle is used to show uniqueness of solution to the initial-boundary-value problems for the time-Fractional Diffusion Equation. The generalized solution in sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the one-dimensional time-Fractional Diffusion Equation $$ (D_t^{\alpha} u)(t) = \frac{\partial}{\partial x}(p(x) \frac{\partial u}{\partial x}) -q(x)\, u + F(x,t),\ \ x\in (0,l),\ t\in (0,T) $$ the generalized solution to the initial-boundary-value problem with the Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of the solution are investigated including its smoothness and asymptotics for some special cases of the source function.
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some uniqueness and existence results for the initial boundary value problems for the generalized time Fractional Diffusion Equation
Computers & Mathematics With Applications, 2010Co-Authors: Yuri LuchkoAbstract:In this paper, some uniqueness and existence results for the solutions of the initial-boundary-value problems for the generalized time-Fractional Diffusion Equation over an open bounded domain Gx(0,T),[email protected]?R^n are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-Fractional Diffusion Equation is used. In turn, the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan Fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-Fractional Diffusion Equation that turns out to be a classical solution under some additional conditions.
Ting Wei - One of the best experts on this subject based on the ideXlab platform.
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an inverse time dependent source problem for a time Fractional Diffusion Equation
Inverse Problems, 2016Co-Authors: Ting WeiAbstract:This paper is devoted to identifying a time-dependent source term in a multi-dimensional time-Fractional Diffusion Equation from boundary Cauchy data. The existence and uniqueness of a strong solution for the corresponding direct problem with homogeneous Neumann boundary condition are firstly proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Then we use the Tikhonov regularization method to solve the inverse source problem and propose a conjugate gradient algorithm to find a good approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.
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optimal error bound and simplified tikhonov regularization method for a backward problem for the time Fractional Diffusion Equation
Journal of Computational and Applied Mathematics, 2015Co-Authors: Jungang Wang, Ting Wei, Yubin ZhouAbstract:In this paper, we consider a backward problem for a time-Fractional Diffusion Equation. Such a problem is ill-posed. The optimal error bound for the problem under a source condition is analyzed. A simplified Tikhonov regularization method is utilized to solve the problem, and its convergence rates are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule, respectively. Numerical examples show that the proposed regularization method is effective and stable, and both parameter choice rules work well.
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a modified quasi boundary value method for an inverse source problem of the time Fractional Diffusion Equation
Applied Numerical Mathematics, 2014Co-Authors: Ting Wei, Jungang WangAbstract:In this paper, we consider an inverse source problem for a time-Fractional Diffusion Equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-Fractional Diffusion Equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral Equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.
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two regularization methods to identify a space dependent source for the time Fractional Diffusion Equation
Applied Numerical Mathematics, 2013Co-Authors: Jungang Wang, Yubin Zhou, Ting WeiAbstract:In this paper, the inverse problem of identifying a space-dependent source for the time-Fractional Diffusion Equation is investigated. Such a problem is obtained from the classical Diffusion Equation in which the time derivative is replaced with a Caputo derivative of order @[email protected]?(0,1]. We show that such a problem is ill-posed and apply the Tikhonov regularization method and a simplified Tikhonov regularization method to solve it based on the solution given by the separation of variables. Convergence estimates are presented under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. Finally, numerical examples are given to show that the regularization methods are effective and stable.
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Tikhonov regularization method for a backward problem for the time-Fractional Diffusion Equation
Applied Mathematical Modelling, 2013Co-Authors: Jungang Wang, Ting Wei, Yu Bin ZhouAbstract:Abstract This paper is devoted to solve a backward problem for a time-Fractional Diffusion Equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral Equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral Equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.
Ian Turner - One of the best experts on this subject based on the ideXlab platform.
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Time-Fractional Diffusion Equation for signal smoothing
Applied Mathematics and Computation, 2018Co-Authors: Fawang Liu, Ian TurnerAbstract:The time-Fractional Diffusion Equation is used for signal smoothing. Compared to the classical Diffusion Equation, the time-Fractional Diffusion Equation has another adjustable time-Fractional derivative order to control the Diffusion process. Therefore, some simulated signals are used to compare the smoothing performance between the time-Fractional Diffusion Equation and the classical Diffusion Equation as well as between classical smoothing methods (regularization method, Savitzky–Golay method and wavelet method). In the end, the time-Fractional Diffusion filtering is applied in an NMR spectrum smoothing. Results indicate that the time-Fractional Diffusion filtering is advantage over the classical Diffusion filtering and their smoothing performance is better than that of classical smoothing methods.
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stability and convergence of a new finite volume method for a two sided space Fractional Diffusion Equation
Applied Mathematics and Computation, 2015Co-Authors: Libo Feng, P. Zhuang, Fawang Liu, Ian TurnerAbstract:In this paper, we consider a two-sided space-Fractional Diffusion Equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new Fractional finite volume method for the two-sided space-Fractional Diffusion Equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit Fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.
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a meshless method based on point interpolation method pim for the space Fractional Diffusion Equation
ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS); Science & Engineering Faculty, 2015Co-Authors: Qingxia Liu, P. Zhuang, Fawang Liu, Junhao Chen, Ian TurnerAbstract:This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space Fractional Diffusion Equation. Two fully-discrete schemes for the one-dimensional space Fractional Diffusion Equation are obtained by using the PIM and the strong-forms of the space Diffusion Equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.
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numerical techniques for the variable order time Fractional Diffusion Equation
Applied Mathematics and Computation, 2012Co-Authors: Shujun Shen, Ian Turner, Fawang Liu, Junhao Chen, Vo AnhAbstract:Abstract In this paper we consider the variable order time Fractional Diffusion Equation. We adopt the Coimbra variable order (VO) time Fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order Fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for Fractional partial differential Equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient.
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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)|=
