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numerical methods (15)

fractional differential equations (6)bifurcations (3)integral equations (3)stochastic delay equations (3)multi-term equations (2)qualitative behaviour (2)Bagley-Torvik equation (1)Capute derivative (1)Caputo derivative (1)View MoreJournalJournal of Computational and Applied Mathematics (2)Computers and Fluids (1)Computers and mathematics with applications (1)Fractional calculus and applied analysis (1)AuthorsFord, Neville J. (13)Diogo, Teresa (3)Lima, Pedro M. (3)Morgado, Maria L. (3)Norton, Stewart J. (3)Ford, Judith M. (2)Simpson, A. Charles (2)Connolly, Joseph A. (1)Diethelm, Kai (1)Edwards, John T. (1)View MoreTypesArticle (7)Technical Report (4)Doctoral (2)PhD (2)Preprint (2)View More

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Noise-induced changes to the behaviour of semi-implicit Euler methods for stochastic delay differential equations undergoing bifurcation

Ford, Neville J.; Norton, Stewart J. (Elsevier, 2009-07-15)

This article discusses estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

Numerical approaches to the solution of some fractional differential equations

Ford, Neville J.; Simpson, A. Charles (Lea Press, 2002)

Theory and numerics for multi-term periodic delay differential equations, small solutions and their detection

Ford, Neville J.; Lumb, Patricia M. (University of Chester, 2006)

We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (†) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Diogo, Teresa; Ford, Judith M.; Ford, Neville J.; Lima, Pedro M. (University of Chester, 2006)

We consider the qualitative behaviour of solutions to linear integral equations of the form where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.

Noise-induced changes to the bifurcation behaviour of semi-implicit Euler methods for stochastic delay differential equations

Ford, Neville J.; Norton, Stewart J. (University of Chester, 2007)

We are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there maybe some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

Ford, Neville J.; Connolly, Joseph A. (University of Chester, 2007)

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

Distributed order equations as boundary value problems

Ford, Neville J.; Morgado, Maria L. (Elsevier, 2012-01-20)

This preprint discusses the existence and uniqueness of solutions and proposes a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin.

Higher Order Numerical Methods for Fractional Order Differential Equations

Pal, Kamal (University of Chester, 2015-08)

This thesis explores higher order numerical methods for solving fractional differential equations.

The numerical solution of linear multi-term fractional differential equations: Systems of equations

Edwards, John T.; Ford, Neville J.; Simpson, A. Charles (Elsevier, 2002-11)

This article discusses how the numerical approximation of a linear multi-term fractional differential equation can be calculated by the reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity.

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