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From standard alpha-stable Lévy motions to horizontal visibility networks: dependence of multifractal and Laplacian spectrum

In recent years, researchers have proposed several methods to transform time series (such as those of fractional Brownian motion) into complex networks. In this paper, we construct horizontal visibility networks (HVNs) based on the α-stable Levy motion. We aim to study the relations of multifractal and Laplacian spectrum of transformed networks on the parameters 1.1 ≤ α ≤ 2 and |β| ≤ 1 of the α-stable Levy motion. First, we employ the sandbox algorithm to compute the mass exponents and multifractal spectrum to investigate the multifractality of these HVNs. Then we perform least squares fits to find possible relations of the average fractal dimension 〉D(0)〈, the average information dimension 〉D(1)〈 and the average correlation dimension 〉D(2)〈 against α, β using several methods of model selection. We also investigate possible dependence relations of eigenvalues and energy on α, β, calculated from the Laplacian and normalized Laplacian operators of the constructed HVNs. All of these constructions and estimates w…

Author: Zou, Hai-Long, Yu, Zu-Guo, Anh, Vo, Ma, Yuan-Lin
Publication year: 2018
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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On approximation for fractional stochastic partial differential equations on the sphere

This paper gives the exact solution in terms of the Karhunen–Loève expansion to a fractional stochastic partial differential equation on the unit sphere S2⊂ R3 with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen–Loève expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background are given to illustrate the theoretical results.

Author: Anh, Vo V., Broadbridge, Philip, Olenko, Andriy, Wang, Yu Guang
Publication year: 2018
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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The unstructured mesh finite element method for the two-dimensional multi-term time–space fractional diffusion-wave equation on an irregular convex domain

In this paper, the two-dimensional multi-term time-space fractional diffusion-wave equation on an irregular convex domain is considered as a much more general case for wider applications in fluid mechanics. A novel unstructured mesh finite element method is proposed for the considered equation. In most existing works, the finite element method is applied on regular domains using uniform meshes. The case of irregular convex domains, which would require subdivision using unstructured meshes, is mostly still open. Furthermore, the orders of the multi-term time-fractional derivatives have been considered to belong to (0, 1] or (1, 2] separately in existing models. In this paper, we consider two-dimensional multi-term time-space fractional diffusion-wave equations with the time fractional orders belonging to the whole interval (0, 2) on an irregular convex domain. We propose to use a mixed difference scheme in time and an unstructured mesh finite element method in space. Detailed implementation and the stability a…

Author: Fan, Wenping, Jiang, Xiaoyun, Liu, Fawang, Anh, Vo
Publication year: 2018
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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A fast numerical method for two-dimensional Riesz space fractional diffusion equations on a convex bounded region

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 rectangular 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 O(nlog⁡n) 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 t…

Author: Chen, S., Liu, F., Turner, I., Anh, V.
Publication year: 2018
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations

Using finite element method in spatial direction and classical L1 approximation in temporal direction, a fully-discrete scheme is established for a class of two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivatives. The stability analysis of the approximate scheme is proposed. The spatial global superconvergence and temporal convergence of order O(h2+τ2−α) for the original variable in H1-norm is presented by means of properties of bilinear element and interpolation postprocessing technique, where h and τ are the step sizes in space and time, respectively. Finally, several numerical examples are implemented to evaluate the efficiency of the theoretical results.

Author: Zhao, Yanmin, Zhang, Yadong, Liu, F., Turner, I., Tang, Yifa, Anh, V.
Publication year: 2017
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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Governing the Murray: Who are its people, and what are their responsibilities?

Jonathan Mann Memorial Lecture on the broad theme of the Murray and its People. Public lecture at La Trobe University Albury-Wodonga Campus, Victoria, Australia.

Author: Coghill, Ken
Publication year: 2018
Publication type: Seminar, speech or other presentation
Status: Live|Last updated:18 October 2018 4:32 PM
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A fast second-order accurate method for a two-sided space-fractional diffusion equation with variable coefficients

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficients on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann–Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second-order accuracy of O(τ2+h2). Furthermore, we develop a fast accurate iterative method for the Crank–Nicolson scheme, which only requires storage of O(m) and computational cost of O(mlogm) while retaining the same accuracy and approximation property as Gauss elimination, where m=1/h is the partition number in space direction. Finally, several numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis.

Author: Feng, L.B., Zhuang, P., Liu, F., Turner, I., Anh, V., Li, J.
Publication year: 2017
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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Multifractal temporally weighted detrended cross-correlation analysis to quantify power-law cross-correlation and its application to stock markets

A new method-multifractal temporally weighted detrended cross-correlation analysis (MF-TWXDFA)-is proposed to investigate multifractal cross-correlations in this paper. This new method is based on multifractal temporally weighted detrended fluctuation analysis and multifractal cross-correlation analysis (MFCCA). An innovation of the method is applying geographically weighted regression to estimate local trends in the nonstationary time series. We also take into consideration the sign of the fluctuations in computing the corresponding detrended cross-covariance function. To test the performance of the MF-TWXDFA algorithm, we apply it and the MFCCA method on simulated and actual series. Numerical tests on artificially simulated series demonstrate that our method can accurately detect long-range cross-correlations for two simultaneously recorded series. To further show the utility of MF-TWXDFA, we apply it on time series from stock markets and find that power-law cross-correlation between stock returns is signif…

Author: Wei, Yun-Lan, Yu, Zu-Guo, Zou, Hai-Long, Anh, Vo
Publication year: 2017
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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A novel genome signature based on inter-nucleotide distances profiles for visualization of metagenomic data

There has been a growing interest in visualization of metagenomic data. The present study focuses on the visualization of metagenomic data using inter-nucleotide distances profile. We first convert the fragment sequences into inter-nucleotide distances profiles. Then we analyze these profiles by principal component analysis. Finally the principal components are used to obtain the 2-D scattered plot according to their source of species. We name our method as inter-nucleotide distances profiles (INP) method. Our method is evaluated on three benchmark data sets used in previous published papers. Our results demonstrate that the INP method is good, alternative and efficient for visualization of metagenomic data.

Author: Xie, Xian-Hua, Yu, Zu-Guo, Ma, Yuan-Lin, Han, Guo-Sheng, Anh, Vo
Publication year: 2017
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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Numerical methods for the two-dimensional multi-term time-fractional diffusion equations

In this paper, we consider a numerical approach based on the matrix transfer method for numerical solution of multi-term time-fractional diffusion equations (MT-TFDEs). The semi- and fully-discrete schemes are developed by using the classical finite difference method and the matrix transfer technique. The unconditional stability and convergence of these two schemes are discussed and theoretically proved. The technique is then extended to MT-TFDEs with fractional Laplace operator. Numerical examples are given to validate and investigate the efficiency and the accuracy of the developed schemes. The results indicate that the present schemes are very effective for modeling and simulation of the MT-TFDEs with integral or fractional Laplacians.

Author: Zhao, Linlin, Liu, Fawang, Anh, Vo V.
Publication year: 2017
Publication type: Journal article
Status: Live|Last updated:18 October 2018 4:32 PM
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