Hybrid methods for direct integration of special third order ordinary differential equations

Abstract In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y ′ ′ ′ = f ( x , y ) ; namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper.

[1]  Tim G. Myers,et al.  Thin Films with High Surface Tension , 1998, SIAM Rev..

[2]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[3]  J. Dormand,et al.  Numerical Methods for Differential Equations: A Computational Approach , 2017 .

[4]  A New Block Method for Special Third Order Ordinary Differential Equations , 2009 .

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

[6]  D. O. Awoyemi,et al.  A class of hybrid collocation methods for third-order ordinary differential equations , 2005, Int. J. Comput. Math..

[7]  V. Varlamov.  The Third‐Order Nonlinear Evolution Equation Governing Wave Propagation in Relaxing Media , 1997 .

[8]  Solving directly general third order ordinary differential equations using two-point four step block method , 2012 .

[9]  Juan Li,et al.  Two-derivative Runge-Kutta-Nyström methods for second-order ordinary differential equations , 2015, Numerical Algorithms.

[10]  Fudziah Bt. Ismail,et al.  An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations , 2014, J. Appl. Math..

[11]  D. O. Awoyemi.  A P-Stable Linear Multistep Method For Solving General Third Order Ordinary Differential Equations , 2003, Int. J. Comput. Math..

[12]  John P. Coleman,et al.  Order conditions for a class of two‐step methods for y″ = f (x, y) , 2003 .

[13]  Siamak Mehrkanoon,et al.  A direct variable step block multistep method for solving general third-order ODEs , 2011, Numerical Algorithms.

[14]  Kadanoff,et al.  Traveling-wave solutions to thin-film equations. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  Fudziah Ismail,et al.  A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem , 2013 .

[16]  Xiong You,et al.  Direct integrators of Runge-Kutta type for special third-order ordinary differential equations , 2013 .

[17]  W. Troy.  Solutions of third-order differential equations relevant to draining and coating flows , 1993 .

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

[19]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .