ORCID Profile
0000-0002-7133-2884
Current Organisations
University of New South Wales
,
Australian National University
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Numerical Solution of Differential and Integral Equations | Dynamical Systems in Applications | Applied Mathematics |
Expanding Knowledge in the Physical Sciences | Expanding Knowledge in the Biological Sciences | Expanding Knowledge in the Mathematical Sciences
Publisher: American Mathematical Society (AMS)
Date: 1986
DOI: 10.1090/S0025-5718-1986-0856705-2
Abstract: We consider the boundary integral equation which arises when the Dirichlet problem in two dimensions is solved using a single-layer potential. A spectral Galerkin method is analyzed, suitable for the case of a smooth domain and smooth boundary data. The use of trigonometric polynomials rather than splines leads to fast convergence in Sobolev spaces of every order. As a result, there is rapid convergence of the approximate solution to the Dirichlet problem and all its derivatives uniformly up to the boundary.
Publisher: Springer Science and Business Media LLC
Date: 13-02-2021
Publisher: Rocky Mountain Mathematics Consortium
Date: 03-1989
Publisher: Springer Science and Business Media LLC
Date: 2003
Publisher: AIP Publishing
Date: 28-10-2013
DOI: 10.1063/1.4826896
Abstract: This paper presents an accurate and practical mathematical model of time-resolved photoluminescence (PL) response from silicon wafers generated by fast repetitive excitation pulses. The model is valid under low level injection condition and takes into account the depth dependence of carrier generation, diffusion, and surface recombination. Finite element analysis is employed for the carrier density and PL computations. By comparing computational results with results obtained from PC1D (a computer program solving fully coupled nonlinear equations for quasi-one-dimensional carrier transportation in crystalline semiconductor devices, especially focusing on photovoltaic devices), the validity of this method is confirmed. Early stage application and the limitations of this method have been studied, and future work has been proposed.
Publisher: Springer Science and Business Media LLC
Date: 2003
Publisher: Springer Science and Business Media LLC
Date: 16-07-2013
Publisher: Oxford University Press (OUP)
Date: 07-2004
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2018
DOI: 10.1137/17M1125261
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2016
DOI: 10.1137/15M1031734
Publisher: Wiley
Date: 1991
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2013
DOI: 10.1137/120880719
Publisher: Springer Science and Business Media LLC
Date: 03-2016
DOI: 10.1038/NCOMMS10634
Abstract: Chronic stress induces signalling from the sympathetic nervous system (SNS) and drives cancer progression, although the pathways of tumour cell dissemination are unclear. Here we show that chronic stress restructures lymphatic networks within and around tumours to provide pathways for tumour cell escape. We show that VEGFC derived from tumour cells is required for stress to induce lymphatic remodelling and that this depends on COX2 inflammatory signalling from macrophages. Pharmacological inhibition of SNS signalling blocks the effect of chronic stress on lymphatic remodelling in vivo and reduces lymphatic metastasis in preclinical cancer models and in patients with breast cancer. These findings reveal unanticipated communication between stress-induced neural signalling and inflammation, which regulates tumour lymphatic architecture and lymphogenous tumour cell dissemination. These findings suggest that limiting the effects of SNS signalling to prevent tumour cell dissemination through lymphatic routes may provide a strategy to improve cancer outcomes.
Publisher: Elsevier BV
Date: 04-1996
Publisher: Cambridge University Press (CUP)
Date: 04-2020
DOI: 10.1017/S1446181120000152
Abstract: The discontinuous Galerkin ( DG ) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting.
Publisher: Elsevier BV
Date: 07-2015
Publisher: Rocky Mountain Mathematics Consortium
Date: 12-1994
Publisher: Springer Science and Business Media LLC
Date: 13-12-2005
Publisher: Rocky Mountain Mathematics Consortium
Date: 03-1993
Publisher: Australian Mathematical Publishing Association, Inc.
Date: 13-01-2021
DOI: 10.21914/ANZIAMJ.V62.15275
Abstract: The discontinuous Galerkin (DG) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting. doi: 10.1017/S1446181120000152
Publisher: Oxford University Press (OUP)
Date: 1994
Publisher: Springer Science and Business Media LLC
Date: 22-04-2010
Publisher: Rocky Mountain Mathematics Consortium
Date: 03-2010
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2012
DOI: 10.1137/120870505
Publisher: Cambridge University Press (CUP)
Date: 12-1988
DOI: 10.1017/S0004972700027799
Abstract: The p -norm of the Hilbert transform is the same as the p -norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.
Publisher: Cambridge University Press (CUP)
Date: 05-04-2017
DOI: 10.1017/S1446181116000365
Abstract: An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer $H^{2}$ -regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation.
Publisher: Springer Science and Business Media LLC
Date: 12-1992
DOI: 10.1007/BF01396241
Publisher: Oxford University Press (OUP)
Date: 19-06-2009
Publisher: Oxford University Press (OUP)
Date: 26-09-2011
Publisher: Oxford University Press (OUP)
Date: 1989
Publisher: Elsevier BV
Date: 02-2020
Publisher: Cambridge University Press (CUP)
Date: 10-2010
DOI: 10.1017/S1446181111000617
Abstract: We prove estimates for the partial derivatives of the solution to a time-fractional diffusion equation posed over a bounded spatial domain. Such estimates are needed for the analysis of effective numerical methods, particularly since the solution is typically less regular than in the familiar case of classical diffusion.
Publisher: Springer Science and Business Media LLC
Date: 19-12-2008
Publisher: American Mathematical Society (AMS)
Date: 23-02-2009
DOI: 10.1090/S0025-5718-09-02234-0
Abstract: We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by C ( t − s ) α − 1 C(t-s)^{\\alpha -1} , where 0 α 1 0 \\alpha . For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most q − 1 q-1 , for q = 1 q=1 or 2 2 . For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order k q + h 2 ℓ ( k ) k^q+h^2\\ell (k) , where k k and h h are the mesh sizes in time and space, respectively, and ℓ ( k ) = max ( 1 , log k − 1 ) \\ell (k)=\\max (1,\\log k^{-1}) . In the case q = 2 q=2 , we prove a higher convergence rate of order k 3 + h 2 ℓ ( k ) k^3+h^2\\ell (k) at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at t = 0 t=0 , necessitating the use of non-uniform time steps. We compare our theoretical error bounds with the results of numerical computations.
Publisher: Australian Mathematical Publishing Association, Inc.
Date: 05-09-2017
Publisher: Springer International Publishing
Date: 2018
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2017
DOI: 10.1137/16M1091861
Publisher: Springer Science and Business Media LLC
Date: 07-07-2021
Publisher: Springer Science and Business Media LLC
Date: 14-11-2006
Publisher: Cambridge University Press (CUP)
Date: 10-2011
DOI: 10.1017/S1446181112000107
Abstract: In earlier work we have studied a method for discretization in time of a parabolic problem, which consists of representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In application to a spatially semidiscrete finite-element version of the parabolic problem, at each quadrature point one then needs to solve a linear algebraic system having a positive-definite matrix with a complex shift. We study iterative methods for such systems, considering the basic and preconditioned versions of first the Richardson algorithm and then a conjugate gradient method.
Publisher: Rocky Mountain Mathematics Consortium
Date: 09-1988
Publisher: American Mathematical Society (AMS)
Date: 1994
DOI: 10.1090/S0025-5718-1994-1240661-6
Abstract: We present a method for the accurate numerical evaluation of a family of trigonometric series arising in the design of special-purpose quadrature rules for boundary element methods. The series converge rather slowly, but can be expressed in terms of Fourier-Chebyshev series that converge rapidly.
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2019
DOI: 10.1137/16M1175742
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2006
DOI: 10.1137/040621247
Start Date: 2014
End Date: 10-2018
Amount: $345,000.00
Funder: Australian Research Council
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