Finite element solution of a linear mixed-type functional differential equation

This paper is devoted to the approximate solution of a linear first-order functional differential equation which involves delayed and advanced arguments. We seek a solution x, defined for t ∈ (0, k − 1],(k ∈ IN ), which takes given values on the intervals [ − 1, 0] and (k − 1, k]. Continuing the work started in previous articles on this subject, we introduce and analyse a computational algorithm based on the finite element method for the solution of this problem which is applicable both in the case of constant and variable coefficients. Numerical results are presented and compared with the results obtained by other methods.

[1]  David de la Croix,et al.  MODELLING VINTAGE STRUCTURES WITH DDEs: PRINCIPLES AND APPLICATIONS , 2004 .

[2]  Björn Sandstede,et al.  Exponential dichotomies for linear non-autonomous functional differential equations of mixed type , 2002 .

[3]  Neville J. Ford,et al.  Mixed-type functional differential equations: A numerical approach , 2009 .

[4]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[5]  Fabrice Collard,et al.  The Short-Run Dynamics of Optimal Growth Models with Delays , 2005 .

[6]  J. Bell,et al.  Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory , 1986, Journal of mathematical biology.

[7]  M. Filomena Teodoro,et al.  The numerical solution of forward-backward differential equations: Decomposition and related issues , 2010, J. Comput. Appl. Math..

[8]  N. J. Ford,et al.  Numerical Approximation of Forward-Backward Differential Equations by a Finite Element Method , 2009 .

[9]  Aido Rust,et al.  Hopf Bifurcation for Functional Differential Equations of Mixed Type , 2022 .

[10]  Neville J. Ford,et al.  Mixed-type functional dierential equations: A numerical approach (Extended version) , 2007 .

[11]  Aldo Rustichini,et al.  Functional differential equations of mixed type: The linear autonomous case , 1989 .

[12]  A. R. Humphries,et al.  Computation of Mixed Type Functional Differential Boundary Value Problems , 2005, SIAM J. Appl. Dyn. Syst..

[13]  John Mallet-Paret,et al.  The Fredholm Alternative for Functional Differential Equations of Mixed Type , 1999 .

[14]  Hippolyte d'Albis,et al.  Competitive Growth in a Life-Cycle Model: Existence and Dynamics , 2009 .

[15]  John Mallet-Paret,et al.  MIXED-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS, HOLOMORPHIC FACTORIZATION, AND APPLICATIONS , 2005 .

[16]  M. Filomena Teodoro,et al.  Analytical and numerical investigation of mixed-type functional differential equations , 2010, J. Comput. Appl. Math..

[17]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[18]  Neville J. Ford,et al.  Numerical Modelling of a Functional Differential Equation with Deviating Arguments Using a Collocation Method , 2008 .

[19]  Neville J. Ford,et al.  New approach to the numerical solution of forward-backward equations , 2009 .