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SubjectsEnergy harvesting (18)XPS (18)Ageing (10)Error estimates (9)fractional differential equations (8)stability (8)numerical methods (7)Virtual Reality (7)Atmospheric chemistry (6)Caputo derivative (6)View MoreJournalJournal of Computational and Applied Mathematics (11)Applied Numerical Mathematics (8)Journal of Physics: Conference Series (8)Applied Surface Science (5)Atmospheric Chemistry and Physics (5)View MoreAuthorsFord, Neville J. (49)Smith, Graham C. (39)Yan, Yubin (30)Jia, Yu (20)Mc Auley, Mark T. (20)Du, Sijun (17)Seshia, Ashwin A. (17)Banks, Craig E. (16)Yang, Bin (16)Baker, Christopher T. H. (15)View MoreTypes

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Numerical solution methods for distributed order differential equations

Diethelm, Kai; Ford, Neville J. (Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, 2005)

This article discusses a basic framework for the numerical solution of distributed order differential equations.

Superfast solution of linear convolutional Volterra equations using QTT approximation

Roberts, Jason A.; Savostyanov, Dmitry V.; Tyrtyshnikov, Eugene E. (Elsevier, 2014-04)

This article address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log^2n). The results of the paper are illustrated by numerical examples.

An analytic approach to the normalized Ricci flow-like equation: Revisited

Kavallaris, Nikos I.; Suzuki, Takashi (Elsevier, 2015-01-07)

In this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence.

Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances

Ferreira, José M.; Ford, Neville J.; Malique, Md A.; Pinelas, Sandra; Yan, Yubin (Elsevier, 2011-04-12)

This article discusses the oscillatory behaviour of the differential equation of mixed type.

Numerical treatment of oscillary functional differential equations

Ford, Neville J.; Yan, Yubin; Malique, Md A. (Elsevier, 2010-09)

This preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear v-methods will also be oscillatory. We conclude with some general theory

A nonpolynomial collocation method for fractional terminal value problems

Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S. (Elsevier, 2014-06-24)

In this paper we propose a non-polynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a non-polynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.

Algorithms for the fractional calculus: A selection of numerical methods

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Luchko, Yury (Elsevier Science, 2005-02)

This article discusses how numerical algorithms can help engineers work with fractional models in an efficient way.

Crank-Nicolson finite element discretizations for a two-dimenional linear Schroedinger-type equation posed in noncylindrical domain

Antonopoulou, Dimitra; Karali, Georgia D.; Plexousakis, Michael; Zouraris, Georgios (AMS, 2014-11-05)

Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundary-value problem for a general Schr¨odinger-type equation posed on a two space-dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain, and we approximate its solution by a Crank–Nicolson finite element method. For the proposed numerical method, we derive an optimal order error estimate in the L2 norm, and to support the error analysis we prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments are presented which verify the optimal order of convergence of the method.

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

Liu, Fang; Liang, Zongqi; Yan, Yubin (Elsevier, 2018-12-17)

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Numerical methods for a Volterra integral equation with non-smooth solutions

Diogo, Teresa; Ford, Neville J.; Lima, Pedro M.; Valtchev, Svilen (Elsevier Science, 2006-05)

This article discusses the numerical treatment of a singular Volterra integral equation with an infinite set of solutions.

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