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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: 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.
  • 5: 18.38 Mathematical Applications
    Differential Equations: Spectral Methods
    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. …
    Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
    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. …
    6: Bibliography C
  • L. D. Carr, C. W. Clark, and W. P. Reinhardt (2000) Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.
  • P. A. Clarkson and E. L. Mansfield (2003) The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16 (3), pp. R1–R26.
  • P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.
  • P. A. Clarkson (2006) Painlevé EquationsNonlinear Special Functions: Computation and Application. In Orthogonal Polynomials and Special Functions, F. Marcellàn and W. van Assche (Eds.), Lecture Notes in Math., Vol. 1883, pp. 331–411.
  • E. A. Coddington and N. Levinson (1955) Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London.
  • 7: 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.