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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 ...

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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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...Chester research online in mathematics and its applications series editors cth baker nj ford the numerical solution of forward backward differential equations decomposition related issues neville j patricia m lumb pedro lima filomena teodoro rma issn please reference this article as follows n p f journal computational applied doi cam author biographies prof is dean at university where he has been employed since holds degrees from universities oxford ma manchester msc liverpool phd founded group became professor interests theory analysis modelling using functional including delay mixed fractional dr pat with york ba senior lecturer she a member special interest so called small solutions which relate to problems degeneracy associate instituto superior tecnico ist part universidade tecnica de lisboa utl portugal involved collaborative projects currently student studying dierential b c adepartment ch bj bcemat t ecnico cdepartamento matem atica est polit setub al abstract paper focuses on ...

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