The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Fawang Liu - 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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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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analysis of a discrete non markovian random walk approximation for the time fractional Diffusion Equation
Anziam Journal, 2005Co-Authors: Fawang Liu, Vo Anh, Shujun Shen, Ian TurnerAbstract:The time fractional Diffusion Equation (TFDE) is obtained from the standard Diffusion Equation by replacing the first-order time derivative with a fractional derivative of order in (0,1). In this work, an explicit finite-difference scheme for TFDE is presented. Discrete models of a non-Markovian random walk are generated for simulating random processes whose spatial probability density evolves in time according to this fractional Diffusion Equation. We derive the scaling restriction of the stability and convergence of the discrete non-Markovian random walk approximation for TFDE in a bounded domain. Finally, some numerical examples are presented to show the application of the present technique.
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THE SPACE-TIME FRACTIONAL Diffusion Equation WITH CAPUTO DERIVATIVES
Journal of Applied Mathematics and Computing, 2005Co-Authors: Feng-hui Huang, Fawang LiuAbstract:We deal with the Cauchy problem for the space-time fractional Diffusion Equation, which is obtained from standard Diffusion Equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional Diffusion Equation.
Yuri Luchko - One of the best experts on this subject based on the ideXlab platform.
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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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maximum principle for the generalized time fractional Diffusion Equation
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Yuri LuchkoAbstract:Abstract In the paper, a maximum principle for the generalized time-fractional Diffusion Equation over an open bounded domain G × ( 0 , T ) , G ⊂ R n is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo–Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional Diffusion Equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.
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the fundamental solution of the space time fractional Diffusion Equation
arXiv: Statistical Mechanics, 2007Co-Authors: Francesco Mainardi, Yuri Luchko, Gianni PagniniAbstract:We deal with the Cauchy problem for the space-time fractional Diffusion Equation, which is obtained from the standard Diffusion Equation by replacing the second-order space derivative with a Riesz-Feller derivative of order � ∈ (0,2] and skewness � (|�| ≤ min {�,2 − �}), and the first-order time derivative with a Caputo derivative of order � ∈ (0,2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional Diffusion {0 < � ≤ 2, � = 1}, time-fractional Diffusion {� = 2, 0 < � ≤ 2}, and neutral-fractional Diffusion {0 < � = � ≤ 2}, for which the fundamental solution can be interpreted as a spatial probability density function evolving
Mohammad Danish Khan - One of the best experts on this subject based on the ideXlab platform.
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Symmetry classification of time-fractional Diffusion Equation
Communications in Nonlinear Science and Numerical Simulation, 2017Co-Authors: I. Naeem, Mohammad Danish KhanAbstract:Abstract In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential Equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional Diffusion Equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional Diffusion Equation transforms to a partial differential Equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional Diffusion Equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α . This study provides a new way of computing symmetries for a class of fractional differential Equations.
Asato Tsuchiya - One of the best experts on this subject based on the ideXlab platform.
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Renormalization group and Diffusion Equation
Progress of Theoretical and Experimental Physics, 2020Co-Authors: Masami Matsumoto, Gota Tanaka, Asato TsuchiyaAbstract:Abstract We study the relationship between the renormalization group and the Diffusion Equation. We consider the exact renormalization group Equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized Diffusion Equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.
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Renormalization group and Diffusion Equation
arXiv: High Energy Physics - Theory, 2020Co-Authors: Masami Matsumoto, Gota Tanaka, Asato TsuchiyaAbstract:We study relationship between renormalization group and Diffusion Equation. We consider the exact renormalization group Equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized Diffusion Equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.
P. Zhuang - One of the best experts on this subject based on the ideXlab platform.
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Finite element method for space-time fractional Diffusion Equation
Numerical Algorithms, 2015Co-Authors: Libo Feng, P. Zhuang, Ian Turner, Yuantong GuAbstract:In this paper, we consider two types of space-time fractional Diffusion Equations(STFDE) on a finite domain. The Equation can be obtained from the standard Diffusion Equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order s (1
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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.
