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nonlinear ordinary differential equations

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1: 22.19 Physical Applications
§22.19(iii) Nonlinear ODEs and PDEs
Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. …
2: 3.7 Ordinary Differential Equations
§3.7 Ordinary Differential Equations
Consideration will be limited to ordinary linear second-order differential equations
§3.7(v) Runge–Kutta Method
The Runge–Kutta method applies to linear or nonlinear differential equations. … An extensive literature exists on the numerical solution of ordinary differential equations by Runge–Kutta, multistep, or other methods. …
3: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
  • A. B. Olde Daalhuis and F. W. J. Olver (1998) On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.
  • A. B. Olde Daalhuis (2004b) On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.
  • F. W. J. Olver (1993a) Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.
  • F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.
  • 4: 18.38 Mathematical Applications
    Differential Equations
    Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). This process has been generalized to spectral methods for solving partial differential equations. … While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. …with H n ( x ) as in §18.3, satisfies the Toda equation
    5: Bibliography
  • M. J. Ablowitz and P. A. Clarkson (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.
  • V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.
  • D. W. Albrecht, E. L. Mansfield, and A. E. Milne (1996) Algorithms for special integrals of ordinary differential equations. J. Phys. A 29 (5), pp. 973–991.
  • U. M. Ascher, R. M. M. Mattheij, and R. D. Russell (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • U. M. Ascher and L. R. Petzold (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 6: Bibliography W
  • W. Wasow (1965) Asymptotic Expansions for Ordinary Differential Equations. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney.
  • E. J. Weniger (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.
  • G. B. Whitham (1974) Linear and Nonlinear Waves. John Wiley & Sons, New York.
  • H. S. Wilf and D. Zeilberger (1992a) An algorithmic proof theory for hypergeometric (ordinary and “ q ”) multisum/integral identities. Invent. Math. 108, pp. 575–633.
  • R. Wong and H. Y. Zhang (2007) Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.
  • 7: Bibliography D
  • K. Dekker and J. G. Verwer (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.
  • T. M. Dunster (1996c) Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.
  • T. M. Dunster (2001a) Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.
  • T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.
  • T. M. Dunster (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.