A LEGENDRE-GAUSS COLLOCATION METHOD FOR NONLINEAR DELAY DIFFERENTIAL EQUATIONS

In this paper, we introduce an efficient Legendre-Gauss collocation method for solving nonlinear delay differential equations with variable delay. We analyze the convergence of the single-step and multi-domain versions of the proposed method, and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. We also make numerical comparison with other methods.

[1]  John C. Butcher,et al.  Integration processes based on Radau quadrature formulas , 1964 .

[2]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[3]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[4]  Alfredo Bellen,et al.  One-step collocation for delay differential equations , 1984 .

[5]  Marino Zennaro,et al.  Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method , 1985 .

[6]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[7]  J. Boyd.  Chebyshev and Fourier Spectral Methods , 1989 .

[8]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[9]  Bradley K. Alpert,et al.  A Fast Algorithm for the Evaluation of Legendre Expansions , 1991, SIAM J. Sci. Comput..

[10]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[11]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

[12]  A. Manitius,et al.  A fully-discrete spectral method for delay-differential equations , 1991 .

[13]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

[14]  Christopher T. H. Baker,et al.  Issues in the numerical solution of evolutionary delay differential equations , 1995, Adv. Comput. Math..

[15]  J. Turi,et al.  On numerical solutions for a class of nonlinear delay equations with time- and state-dependent delays , 1996 .

[16]  B. Guo,et al.  Spectral Methods and Their Applications , 1998 .

[17]  Stefano Maset,et al.  Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems , 2000, Numerische Mathematik.

[18]  M. S. Salim,et al.  Convergence Of The Spline Function For Delay Dynamic System , 2003, Int. J. Comput. Math..

[19]  Haitao Ma,et al.  Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials , 2004 .

[20]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[21]  David J. Evans,et al.  The Adomian decomposition method for solving delay differential equation , 2005, Int. J. Comput. Math..

[22]  Guo Ben-yu,et al.  Numerical integration based on Laguerre-Gauss interpolation , 2007 .

[23]  Ed Bueler,et al.  Error Bounds for Approximate Eigenvalues of Periodic-Coefficient Linear Delay Differential Equations , 2007, SIAM J. Numer. Anal..

[24]  Ben-yu Guo,et al.  Integration processes of ordinary differential equations based on Laguerre-Radau interpolations , 2008, Math. Comput..

[25]  Ben-Yu Guo,et al.  Legendre--Gauss collocation method for initial value problems of second order ordinary differential equations , 2009 .

[26]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[27]  Ben-yu Guo,et al.  Legendre–Gauss collocation methods for ordinary differential equations , 2009, Adv. Comput. Math..

[28]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .