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CHESTER RESEARCH ONLINE
Research in Mathematics and its Applications
Series Editors: CTH Baker, NJ Ford
The numerical solution of forward-backward
differential equations: Decomposition and
related issues
Neville J. Ford, Patricia M. Lumb, Pedro M. Lima, M. Filomena Teodoro
2012:RMA:6
© 2010 Neville J. Ford, Patricia M. Lumb, Pedro M. Lima, M. Filomena Teodoro
ISSN 2050-0661
Chester Research Online Research in Mathematics and its Applications ISSN 2050-0661
Please reference this article as follows:
Ford, N. J., Lumb, P. M., Lima, P. M. & Teodoro, M. F. (2010). The numerical
solution of forward–backward differential equations: Decomposition and related
issues. Journal of Computational and Applied Mathematics, 234(9), 2745-2756.
doi: 10.1016/j.cam.2010.01.039
2012:RMA:6
Chester Research Online Research in Mathematics and its Applications ISSN 2050-0661
Author Biographies
Prof. Neville J. Ford
Neville Ford is Dean of Research at the University of Chester, where he has been
employed since 1986. He holds degrees from Universities of Oxford (MA),
Manchester (MSc) and Liverpool (PhD). He founded the Applied Mathematics
Research Group in 1991 and became Professor of Computational Applied
Mathematics in 2000. He has research interests in theory, numerical analysis and
modelling using functional differential equations, including delay, mixed and
fractional differential equations.
Dr. Patricia M. Lumb
Pat Lumb (with degrees from the Universities of York (BA) and Liverpool (MSc, PhD)
has been Senior Lecturer in Mathematics at the University of Chester since 1999.
She is a member of the Applied Mathematics Research Group and has research
interests in functional and delay differential equations with special interest in so-
called small solutions which relate to problems of degeneracy.
Prof. Pedro M. Lima
Pedro Lima is Associate Professor in Mathematics at Instituto Superior Técnico
(IST), part of the Universidade Técnica de Lisboa (UTL) in Portugal. He has been
involved in collaborative research projects with the University of Chester since 2000.
M. Filomena Teodoro
Filomena Teodoro is currently a PhD student studying Mathematics at IST UTL in
Portugal.
2012:RMA:6
Chester Research Online Research in Mathematics and its Applications ISSN 2050-0661
The numerical solution of forward-backward differential equations:
Decomposition and related issues
a a,∗ b b,c
Neville J. Ford , Patricia M. Lumb , Pedro M. Lima , M. Filomena Teodoro
aDepartment of Mathematics, University of Chester, CH1 4BJ
bCEMAT, Instituto Superior T´ecnico, UTL, 1049-001 Lisboa, Portugal
cDepartamento de Matem´atica, EST, Instituto Polit´ecnico de Setub´ al, 2910-761 Setub´ al, Portugal
Abstract
This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional
differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider
′
equations of the form x (t) = ax(t) + bx(t − 1) + cx(t + 1) and develop a numerical approach, using a
central difference approximation, which leads to the desired decomposition and propagation of the solution.
Weinclude illustrative examples to demonstrate the success of our method, along with an indication of its
current limitations.
Key words:
Mixed-type functional differential equations, decomposition of solutions, central differences
AMSSubject Classification:, 34K06, 34K10, 34K28, 65Q05
1. Introduction
Interest in the study of mixed-type functional equations (MFDEs), or forward-backward equations,
developed following the pioneering work of Rustichini in 1989 [19, 20]. The analysis of such equations,
with both advanced and delayed arguments, presents a significant challenge to both analysts and numerical
analysts alike. We are reminded in the opening section of [12] that “the dichotomy of insight and numbers is
specific to numerical analysis”, that “computation should not wait until analysis has run out of steam” but
that we should “employ computational algorithms that reflect known qualitative features of the underlying
system”. The analytical decomposition of solutions of mixed-type equations as sums of “forward” solutions
and“backward”solutions has been studied by J. Mallet-Paret and S. M. Verduyn Lunel in [18]. It is our aim
in this paper to present an algorithm to decompose the solution of a particular class of MFDE into growing
and decaying components and to provide further insight into issues related to the success or otherwise of
this approach. We choose not to provide a more detailed review of current literature here. Instead we refer
the reader to [1, 17, 19, 20] and for further examples of applications of MFDEs to [2, 3].
Wefocus our attention on the linear autonomous functional equation given by
x′(t) = ax(t) + bx(t − 1) + cx(t + 1), (1)
a particular case of the nonautonomous equation
′
x(t) = a(t)x(t)+b(t)x(t−1)+c(t)x(t+1). (2)
∗Corresponding author [copyright reserved]
Email addresses: njford@chester.ac.uk (Neville J. Ford), p.lumb@chester.ac.uk (Patricia M. Lumb),
plima@math.ist.pt (Pedro M. Lima), mteodoro@est.ips.pt (M. Filomena Teodoro)
1
2012:RMA:6
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