# Professor Neville J. Ford

Director of the Mathematical Sciences Research Group & International Research Professor in Mathematics

After graduating from Oxford in 1979 with a BA in Mathematics, Neville took his MA (Oxon) in 1983, an MSc (from Manchester in 1986) and a PhD in 1991 from Liverpool. He was invited to become a Fellow of the Institute of Mathematics and its Applications for services to Mathematics. He is a Fellow of the Higher Education Academy.

## Qualifications

MA MSc PhD FIMA FHEA

## Overview

Neville is the founder and director of the University's Mathematical Sciences Research Group. He is currently Pro Vice Chancellor at the University and has led the University's Research and Knowledge Exchange from 2006 until 2017, and has had responsibility for the University's Academic portfolio for 2015 to 2017. He continues to supervise PhD students and to conduct his own research, in collaboration with members of the Chester group and with researchers elsewhere in Europe.

## Teaching

Having worked at the University since 1986, Neville has taught on a vast range of courses in pure and applied maths. However, these days, his teaching is limited to supervision of PhD students. his students typically work on projects involving functional differential equations, which can include delay differential and integro-differential equations. Fractional differential equations are a popular topic for students.  He is always interested to hear from good potential research students who would like to work with him.

## Research

Neville’s research interests combine theory, computational simulations and modelling applications of functional differential equations. Of great interest are problems with a memory or after-effect. These can arise, for example, in materials science, in the environmental sciences, in the biosciences, or in medicine and immunology. It is always exciting to discover that insights from one area of science can, through the same fundamental mathematical structure of the model, be transferable to an apparently completely unrelated area of science. Fundamental to Neville’s work is the quality assurance of the modelling results. One way of describing this is as follows: the mathematical model is an imprecise representation of the real world problem; the computational simulation is usually an imprecise representation of the mathematical equations; we solve the equations approximately. Therefore there are at least three potential sources of error. We work to try to ensure that the solutions from the simulation can be relied upon when they are used to predict the real-world outcomes of the system.

Neville and his collaborators analyse problems that are formulated as differential equations (with or without delay and advanced terms), integral and integro-differential equations, difference equations, and fractional differential equations. For those interested, his work can most easily be found using either Chesterrep or Researchgate.

## Published work

Articles

• (with C.T.H Baker), “Some applications of the boundary-locus method and the method of D-partitions”, IMA Journal of Numerical Analysis, Vol. 11(2), 1991.
• (with V Wulf), “The use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations”, Journal of Computational and Applied Mathematics, Vol. 111(1-2), 1999.
• (with V Wulf), “Numerical Hopf bifurcation for a class of delay differential equations”, Journal of Computational and Applied Mathematics, Vol. 115(1-2), 2000.
• (with V Wulf), “How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?”, Vol. 125(1-2), 2000.
• (with A.C Simpson), “The numerical solution of fractional differential equations: Speed versus accuracy”, Numerical Algorithms, Vol. 26(4), 2001.
• (with K Diethelm & A.D Freed), “A predictor-corrector approach for the numerical solution of fractional differential equations”, Nonlinear Dynamics, Vol. 29(1-4), 2002.
• (with K Diethelm), “Analysis of fractional differential equations”, Journal of Mathematical Analysis and Applications, Vol. 265(2), 2002.
• (with K Diethelm), “Numerical solution of the Bagley-Torvik equation”, BIT, Vol. 42(3), 2002.
• (with J.T Edwards & A.C Simpson), “The numerical solution of linear multi-term fractional differential equations: systems of equations”,  Journal of Computational and Applied Mathematics, Vol. 148 (2), 2002.
• (with S.M.V Lunel), “Characterising small solutions in delay differential equations through numerical approximations”, Applied Mathematics and Computation, Vol. 131(2-3), 2002.
• (with J.T Edwards & J.A Roberts), “Bifurcations in numerical methods for Volterra integro-differential equations”, International Journal of Bifurcation and Chaos, Vol. 13(11), 2003.
• (with K Diethelm & A.D Freed), “Detailed error analysis for a fractional Adams method”, Numerical Algorithms, Vol. 36(1), 2004.
• (with K Diethelm), “Multi-order fractional differential equations and their numerical solution”, Applied Mathematics and Computation, Vol. 154(3), 2004.
• (with R.M Anderson, G Bocharov, T Junt, P Krebs, B Ludewig & H Metters), “Determining control parameters for dendritic cell-cytotoxic T lymphocyte interaction”, European Journal of Immunology, Vol. 34(9), 2004.
• (with J.M Ford & J Wheeler), “Simulation of grain-boundary diffusion creep: analysis of some new numerical techniques”, Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, Vol. 460(2048), 2004.
• (with K Diethelm, A.D Freed & Y Luchko), “Algorithms for the fractional calculus: A selection of numerical methods”, Computer Methods in Applied Mechanics and Engineering, Vol. 194(6-8), 2005.
• (with G Bocharov, T Breinig, J Edwards, A Meyerhans & S Wain-Hobson), “A genetic-algorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombination”, Journal of General Virology, Vol. 86, 2005.
• (with T Diogo, J.T Edwards & S.M Thomas), “Numerical analysis of a singular integral equation”, Applied Mathematics and Computation, Vol. 167(1), 2005.
• (with K Diethelm, J.M Ford & M Weilbeer), “Pitfalls in fast numerical solvers for fractional differential equations”, Journal of Computational and Applied Mathematics, Vol. 186(2), 2006.
• (with J.A Connolly), “Comparison of numerical methods for fractional differential equations”, Communications on Pure and Applied Analysis, Vol. 5(2), 2006.
• (with P.A Lumb), “An algorithm to detect small solutions in linear delay differential equations”, Journal of Computational and Applied Mathematics, Vol. 193(1), 2006.
• (with S.J Norton) “Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations”, Communications on Pure and Applied Analysis, Vol. 5(2), 2006.
• (with T Diogo, P.M Lima & S.M Thomas), “Solution of equation by a split-interval method”, International Journal of Numerical Analysis and Modelling, Vol. 4(1), 2007.
• (with T Diogo, J.M Ford & P.M Lima), “Numerical modelling of qualitative behaviour of solutions to convolution integral equations”, Journal of Computational and Applied Mathematics, Vol. 205(2), 2007.
• (with P.M Lumb), “Theory and numerics for multi-term periodic delay differential equations: Small solutions and their detection”, Electronic Transactions on Numerical Analysis, Vol. 26, 2007.
• (with C.T.H Baker & K Burrage), “Special issue on evolutionary problems – Preface”, Journal of Computational and Applied Mathematics, Vol. 205(2), 2007.
• (with K Diethelm), “Numerical analysis for distributed-order differential equations”, Journal of Computational and Applied Mathematics, Vol. 225(1), 2009.
• (with P.M Lima, P.M Lumb & F.M Teodoro), “Analytical and numerical investigation of mixed-type functional differential equations”, Journal of Computational and Applied Mathematics, Vol. 234(9), 2010.
• (with P.M Lima, P.M Lumb & F.M Teodoro), “The numerical solution of forward-backward differential equations: Decomposition and related issues”, Journal of Computational and Applied Mathematics, Vol. 234(9), 2010.
• (with E Ekaka-a & P.M Lumb), “Mathematical modelling of plant species interactions in a harsh climate”, Journal of Computational and Applied Mathematics, Vol. 234(9), 2010.
• (with A Malique & Y Yan), “Numerical treatment of oscillatory functional differential equations”, Journal of Computational and Applied Mathematics, Vol. 234(9), 2010.
• (with C.T.H Baker, T Diogo, P Lima & A Sequeira),“Special Issue: Third International Workshop on Analysis and Numerical Approximation of Singular Problems [IWANASP08] Preface”, Journal of Computational and Applied Mathematics, Vol. 234(9), 2010.
• (with M Luisa Morgado), “Fractional boundary value problems: Analysis and numerical methods”, Fractional Calculus and Applied Analysis, Vol. 14(4), 2011.
• (with J Xiao & Y Yan), “A finite element method for time fractional partial differential equations”, Fractional Calculus and Applied Analysis, Vol. 14(3),  2011.
• (with J.M Ferreira, A Malique, S Pinelas & Y Yan), “Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances”, Journal of Computational and Applied Mathematics, Vol. 235(17), 2011.
• (with M.L Morgado), “Distributed order equations as boundary value problems”, Computers & Mathematics with Applications, Vol. 64(10), 2012.
• (with K Diethelm), “Volterra integral equations and fractional calculus: Do neighboring solutions intersect?”, Journal of Integral Equations and Applications, Vol. 24(1), 2012.
• (with M Luisa Morgado & M Rebelo), “Nonpolynomial collocation approximation of solutions to fractional differential equations”, Fractional Calculus and Applied Analysis, Vol. 16(4), 2013.
• (with P.M Lima & M.L Morgado), “Analysis and numerical methods for fractional differential equations with delay”, Journal of Computational and Applied Mathematics, Vol. 252, 2013.
• (with M Manuela Rodrigues, J Xiao & Y Yan), “Numerical analysis of a two-parameter fractional telegraph equation”, Journal of Computational and Applied Mathematics, Vol. 249, 2013.
• (with M Manuela Rodrigues & N Vieira), “A numerical method for the fractional Schrodinger type equation of spatial dimension two”, Fractional Calculus and Applied Analysis, Vol. 16(2), 2013.
• (with P.M Lima & P.M Lumb), “Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons”, Applied Numerical Mathematics, Vol. 85, 2014.
• (with K Pal & Y Yan), “Higher order numerical methods for solving fractional differential equations”, BIT Numerical Mathematics, Vol. 54(2), 2014.
• (with J Xiao & J Zhao), “Collocation methods for fractional integro-differential equations with weakly singular kernels”, Numerical Algorithms, Vol. 65(4), 2014.
• (with Morgado, ML, Rebelo, M) “High Order Numerical Methods for Fractional Terminal Value Problems” COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, 14.1, 2014
• (with Savostyanov, DV, Zamarashkin, NL) “ON THE DECAY OF THE ELEMENTS OF INVERSE TRIANGULAR TOEPLITZ MATRICES” SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 35.4, 2014
• (with Ferras, LL, Morgado, ML, Rebelo, M) “A Numerical Method for the Solution of the Time-Fractional Diffusion Equation” COMPUTATIONAL SCIENCE AND ITS APPLICATIONS - ICCSA 2014, PT 1 Lecture Notes in Computer Science, 8579, 2014
• (with Pal, K, Yan, YB) “An Algorithm for the Numerical Solution of Two-Sided Space-Fractional
Partial Differential Equations” COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, 15.4, 2015
• (with Ferras, LL, Morgado, ML, Nobrega, JAM, Rebelo, MS) “FRACTIONAL PENNES' BIOHEAT EQUATION: THEORETICAL AND NUMERICAL STUDIES” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 18.4, 2015
• (with Morgado, ML, Rebelo, M) “A nonpolynomial collocation method for fractional terminal value
problems” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 275, 2015
• (with Morgado, ML, Rebelo, M) “AN IMPLICIT FINITE DIFFERENCE APPROXIMATION FOR THE SOLUTION OF THE DIFFUSION EQUATION WITH DISTRIBUTED ORDER IN TIME” ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS, 44, 2015
• (with Diogo, T, Jackiewicz, Z, Lima, P, Morgado, L, Rebelo, M) “Special Issue: The Fifth International Workshop on Analysis and Numerical Approximation of Singular Problems (IWANASP 2015), October
22-24, 2015, Lagos, Algarve, Portugal, Preface”, APPLIED NUMERICAL MATHEMATICS 114, 2017
• (with Morgado, ML, Rebelo, M, Ferras, LL) “Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method” APPLIED NUMERICAL MATHEMATICS, 114, 2017
• (with Li, ZQ, Yan, YB) “Error estimates of a high order numerical method for solving linear
fractional differential equations” APPLIED NUMERICAL MATHEMATICS, 114, 2017.
• (with Lima, PM, Lumb, PM) “Numerical investigation of noise induced changes to the solution
behaviour of the discrete FitzHugh-Nagumo equation” APPLIED MATHEMATICS AND COMPUTATION, 293, 2017
• (with Moayyed, H, Rodrigues, MM) “Orthogonality for a class of generalised Jacobi polynomial $P^{\alpha,\beta}_{\nu}(x)$” Fractional Differential Calculus, to appear, 2017.
• (with Diethelm, K) “A Note on the Well-Posedness of Terminal Value Problems for Fractional Differential Equations” Journal of Integral Equations and Applications, to appear, 2018.

Conference Presentations

• (with C.T.H Baker), “Stability properties of a scheme for the approximate solution of a delay-integro-differential equation”, International Conference on the Numerical Solution of Volterra and Delay Equations, May 1990.
• (with C.T.H Baker), “Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type”, 6th International Congress on Computational and Applied Mathematics, July 1994.
• (with C.T.H Baker), “Stability of a numerical solution of a non-linear Volterra integral equation of convolution type”,3rd International Congress on Industrial and Applied Mathematics (ICIAM 95), July 1995.
• (with J.T Edwards), “Boundedness and stability of solutions to difference equations”, 9th International Congress on Computational and Applied Mathematics, July 2000.
• (with S.J Norton), “Numerical investigation of D-bifurcations for a stochastic delay logistic equation”, Workshop on Stochastic Dynamics with Delay and Memory, February 2004.
• (with T Diogo, P Lima & S Valtchev), “Numerical methods for a Volterra integral equation with non-smooth solutions”, 11th International Congress on Computational and Applied Mathematics, July 2004.
• (with P.M Lumb), “Scalar periodic complex delay differential equations: Small solutions and their detection”, 5th International Conference on Algorithms for Approximation, July 2005.
• (with S.J Norton), “Using approximations to Lyapunov exponents to predict changes in dynamical behaviour in numerical solutions to stochastic delay differential equations”, 5th International Conference on Algorithms for Approximation, July 2005.
• (with J.A Connolly), “Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations”, 2nd International Workshop on Analysis and Numerical Approximation of Singular Pro, September 2006.
• (with P.M Lumb), “Mixed-type functional differential equations: A numerical approach”, 2nd International Workshop on Analysis and Numerical Approximation of Singular Pro, September 2006.
• (with S.J Norton), “Noise-induced changes to the behaviour of semi-implicit Euler methods for stochastic delay differential equations undergoing bifurcation”, 2nd International Workshop on Analysis and Numerical Approximation of Singular Pro, September 2006.
• (with P.M Lima, P.M Lumb & F Teodoro), “New approach to the numerical solution of forward-backward equations”, International Workshop on Numerical Analysis and Computational Methods for Functional Differential and Integral Equations, December 2007.
• (with P.M Lima, P.M Lumb & F Teodoro), “Finite element solution of a linear mixed-type functional differential equation”, 2nd Dolomites Workshop on Constructive Approximation and Applications (DWCAA09), September 2009.
• (with M.L Morgado), “Structural Stability for Fractional Differential Equations as Boundary Value Problems”, International Conference on Numerical Analysis and Applied Mathematics, September 2010.