Bibliography for Series Solutions and Frobenius Method

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  1. The Frobenius power series solution for cylindrically anisotropic radially inhomogeneous elastic materials
    Shuvalov, A. L.
    Quarterly Journal of Mechanics and Applied Mathematics, 2003, vol. 56, no. 3, pp. 327-346, Ingenta.
  2. Series solutions of coupled differential equations with one regular singular point
    Tomantschger, K.W.   
    Journal of Computational and Applied Mathematics, v 140, n 1-2, Mar 1, 2002, p 773-783, Compendex.
  3. Formal Solutions of Linear PDEs and Convex Polyhedra
    Aroca F.; Cano J.
    Journal of Symbolic Computation, December 2001, vol. 32, no. 6, pp. 717-737    , Ingenta.
  4. Solution of a linear differential equation in the form of power series and its application.
    Kitamoto, T.
    Computer mathematics (Matsuyama, 2001), 46--55, Lecture Notes Ser. Comput., 9, World Sci. Publishing, River Edge, NJ, 2001, MathSciNet.  
  5. Exact Integration of Reduced Fishers Equation, Reduced Blasius Equation, and the Lorenz Model
    Roman-Miller L.; Broadbridge P.
    Journal of Mathematical Analysis and Applications, November 2000, vol. 251, no. 1, pp. 65-83    , Ingenta.
  6. Approximated solutions in rational form for systems of differential equations
    Gonzalez-Concepcion C.; Pestano-Gabino C.
    Numerical Algorithms, 1999, vol. 21, no. 1/4, pp. 185-203    , Ingenta.
  7. Automatic programming of recurrent power series
    Lara M.; Elipe A.; Palacios M.
    Mathematics and Computers in Simulation, September 1999, vol. 49, no. 4, pp. 351-362    , Ingenta.
  8. Leibniz's Formula, Cauchy Majorants, and Linear Differential Equations  
    Michael Mezzino; Mark Pinsky  
    Mathematics Magazine, Vol. 71, No. 5. (Dec., 1998), pp. 360-368, Jstor.  
  9. New method to count the index numbers of the singular points of ordinary differential equations
    Gong, Peishan; Wei, Bin  
    Qingdao Daxue Xuebao/Journal of Qingdao University, v 10, n 3, Sep, 1997, p 97, Compendex.
  10. On a class of weakly regular singular two point boundary value problems--I
    Pandey R.K.
    Nonlinear Analysis, July 1996, vol. 27, no. 1, pp. 1-12, Ingenta.
  11. Computer Implementation of a Series Solution for Constant Coefficient Ordinary Differential Equations
    Wang C.L.-Y.
    Computers and Mathematics with Applications, February 1996, vol. 31, no. 3, pp. 43-60, Ingenta.
  12. A Power Series Approach For The Study Of Periodic Motion
    Qaisi M.I.
    Journal of Sound and Vibration, October 1996, vol. 196, no. 4, pp. 401-406    , Ingenta.
  13. Nonlinear Ordinary Differential Equations Resolvable with Respect to an Irregular Singular Point.
    Tovbis, A.
    Journal of differential equations, 1994, vol. 109, no. 1, pp. 201, Ingenta.
  14. Convergent power series solutions of a nonlinear partial differential equation.
    Gérard, Raymond; Sibuya, Yasutaka
    Analysis 13 (1993), no. 4, 395--401, MathSciNet.  
  15. Multisummability of formal power series solutions of nonlinear meromorphic differential equations.
    Braaksma, Boele L. J.
    Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 517--540, MathSciNet.  
  16. Multisummability of formal power series solutions of linear ordinary differential equations.
    Balser, W.; Braaksma, B. L. J.; Ramis, J.-P.; Sibuya, Y.
    Asymptotic Anal. 5 (1991), no. 1, 27--45, MathSciNet.  
  17. Frobenius Analysis of Higher Order Equations: Incipient Buoyant Thermal Convection  
    David L. Littlefield; Prateen V. Desai  
    SIAM Journal on Applied Mathematics, Vol. 50, No. 6. (Dec., 1990), pp. 1752-1763, Jstor.  
  18. Algorithm for computing the formal solutons of differential systems in the neighborhood of an irregular singular point
    Guoting, Chen   
    ISSAC '90 Proceedings of International Symposium on Symbolic and Algebraic Computation, 1990, p 231-235, Compendex.
  19. The Radius of Convergence of Power Series Solutions to Linear Differential Equations (in The Teaching of Mathematics)  
    Isom H. Herron  
    The American Mathematical Monthly, Vol. 96, No. 9. (Nov., 1989), pp. 824-827, Jstor.  
  20. Power series approximation to solutions of nonlinear systems of differential equations.
    Fairén, Victor; López, Vicente; Conde, Luis
    Amer. J. Phys. 56 (1988), no. 1, 57--61, MathSciNet.  
  21. Power series solutions of linear differential equations with polynomial coefficients.
    Kongsakorn, Kannika; Laohakosol, Vichian
    Southeast Asian Bull. Math. 11 (1987), no. 1, 13--17, MathSciNet.  
  22. A Gap Theorem for Power Series Solutions of Algebraic Differential Equations  
    Leonard Lipshitz; Lee A. Rubel  
    American Journal of Mathematics, Vol. 108, No. 5. (Oct., 1986), pp. 1193-1213, Jstor.  
  23. Twisted differential equations with regular singular points. (Spanish)  
    Plá, Héctor
    Cienc. Mat. (Havana)  7  (1986),  no. 2, 23--32, MathSciNet.  
  24. Power series solutions of algebraic differential equations.
    Denef, J.; Lipshitz, L.
    Math. Ann. 267 (1984), no. 2, 213--238, MathSciNet.  
  25. Algorithm To Obtain Formal Solutions Of A Linear Homogeneous Differential Equation At An Irregular Singular Point.
    Della Dora, J.; Di Crescenzo, Cl.; Tournier, E.  
    Lecture Notes in Computer Science, 1982, p 273-280, Compendex.
  26. Arithmetic Properties of Power Series Solutions of Algebraic Differential Equations  
    Yasutaka Sibuya; Steven Sperber  
    The Annals of Mathematics, 2nd Ser., Vol. 113, No. 1. (Jan., 1981), pp. 111-157, Jstor.
  27. Recurrence Relations for the Coefficients in Chebyshev Series Solutions of Ordinary Differential Equations  
    T. S. Horner  
    Mathematics of Computation, Vol. 35, No. 151. (Jul., 1980), pp. 893-905, Jstor.
  28. Formal and Convergent Power Series Solutions of Singular Partial Differential Equations  
    Stanley Kaplan  
    Transactions of the American Mathematical Society, Vol. 256. (Dec., 1979), pp. 163-183, Jstor.
  29. Difference Equations: Disconjugacy, Principal Solutions, Green's Functions, Complete Monotonicity  
    Philip Hartman  
    Transactions of the American Mathematical Society, Vol. 246. (Dec., 1978), pp. 1-30, Jstor.  
  30. The Frobenius method for complex roots of the indicial equation.  
    Neuringer, Joseph L.
    Internat. J. Math. Ed. Sci. Tech.  9  (1978), no. 1, 71--77, MathSciNet.  
  31. Taylor Series Methods for the Solution of Volterra Integral and Integro-Differential Equations  
    Alan Goldfine  
    Mathematics of Computation, Vol. 31, No. 139. (Jul., 1977), pp. 691-707, Jstor.  
  32. Expanding the solutions of implicit sets of ordinary differential equations in power series.
    Norman, A. C.
    Comput. J. 19 (1976), no. 1, 63--68, MathSciNet.  
  33. A necessary condition for a power series to be a formal solution of a singular linear differential equation of order k.
    Gingold, H.
    J. Math. Anal. Appl. 52 (1975), no. 3, 546--552, MathSciNet.  
  34. Power series solution of the matrix linear differential equation.
    Lupas, L.
    Rev. Roumaine Sci. Tech. Sér. Électrotech. Énergét. 19 (1974), 137--152, MathSciNet.  
  35. Analytic Solutions of a Neutral Differential Equation Near a Singular Point  
    L. J. Grimm
    Proceedings of the American Mathematical Society, Vol. 36, No. 1. (Nov., 1972), pp. 187-190, Jstor.  
  36. Hadamard's Elementary Solution and Frobenius's Method  
    E. T. Copson  
    SIAM Review, Vol. 13, No. 2. (Apr., 1971), pp. 222-230, Jstor.  
  37. Asymptotic Behavior of Solutions of Linear Systems of Ordinary Differential Equations Near an Irregular Singular Point  
    Donald A. Lutz  
    American Journal of Mathematics, Vol. 91, No. 1. (Jan., 1969), pp. 95-105, Jstor.  
  38. The Solution of a Second Order Linear Differential Equation Near a Regular Singular Point  
    John W. Dettman  
    The American Mathematical Monthly, Vol. 71, No. 4. (Apr., 1964), pp. 378-385, Jstor.  
  39. Generalized Laurent Series for Singular Solutions of Elliptic Partial Differential Equations  
    Murray Wachman  
    Proceedings of the American Mathematical Society, Vol. 15, No. 1. (Feb., 1964), pp. 101-108, Jstor.  
  40. Expansions of solutions of differential equations with retardation in power series of small retardation.
    Rodionov, A. M.
    Prikl. Mat. Meh. 26 947--949 (Russian); translated as J. Appl. Math. Mech. 26 1962 1430--1435, MathSciNet.  
  41. A Note on the  [Frobenius] Indical Equation (in Miscellaneous Notes)   
    William Squire  
    Mathematics Magazine, Vol. 34, No. 4. (Mar. - Apr., 1961), pp. 226-229, Jstor.  
  42. Motivating the Method of Frobenius (in Classroom Notes)  
    Robert H. Owens  
    The American Mathematical Monthly, Vol. 67, No. 3. (Mar., 1960), pp. 278-279, Jstor.  
  43. A note on the indicial equation.    
    Squire, William
    Math. Mag.  34  1960/1961 226--229, MathSciNet.  
  44. Solution About a Singular Point of a Linear Differential Equation Involving A Large Parameter  
    Robert McKelvey  
    Transactions of the American Mathematical Society, Vol. 91, No. 3. (Jun., 1959), pp. 410-424, Jstor.  
  45. An Eigenfunction Series Solution of a Certain Hyperbolic Partial Differential Equation  
    E. J. Scott  
    SIAM Review, Vol. 1, No. 2. (Jul., 1959), pp. 160-166, Jstor.  
  46. Solution of Nonlinear Differential Equations with a Parameter by Asymptotic Series  
    Wolfgang Wasow  
    The Annals of Mathematics, 2nd Ser., Vol. 69, No. 2. (Mar., 1959), pp. 486-509, Jstor.  
  47. Relations between properties of solutions of partial differential equations and the coefficients of their power series development.
    Kreyszig, Erwin
    J. Math. Mech. 6 (1957), 361--381, MathSciNet.  
  48. Asymptotic Solution with Respect to a Parameter of a Differential Equation Having an Irregular Singular Point  
    Nicholas D. Kazarinoff  
    Proceedings of the American Mathematical Society, Vol. 7, No. 1. (Feb., 1956), pp. 62-69, Jstor.  
  49. Erratum to Asymptotic and Convergent Factorial Series in the Solution of Linear Ordinary Differential Equations  
    Robert L. Evans  
    Proceedings of the American Mathematical Society, Vol. 5, No. 6. (Dec., 1954), p. 1000, Jstor.  
  50. Asymptotic and Convergent Factorial Series in the Solution of Linear Ordinary Differential Equations  
    Robert L. Evans  
    Proceedings of the American Mathematical Society, Vol. 5, No. 1. (Feb., 1954), pp. 89-92, Jstor.  
  51. Notes on Numerical Analysis--3: Solution of Differential Equations by Recurrence Relations (in Automatic Computing Machinery; Discussions)  
    C. W. Clenshaw; F. W. J. Olver  
    Mathematical Tables and Other Aids to Computation, Vol. 5, No. 33. (Jan., 1951), pp. 34-39, Jstor.  
  52. Solution of Differential Equations by Recurrence Relations (in Automatic Computing Machinery; Discussions)  
    John Todd  
    Mathematical Tables and Other Aids to Computation, Vol. 4, No. 29. (Jan., 1950), pp. 39-44, Jstor.  
  53. A New Method for Determining a Series Solution of Linear Differential Equations with Constant or Variable Coefficients  
    W. O. Pennell  
    The American Mathematical Monthly, Vol. 33, No. 6. (Jun. - Jul., 1926), pp. 293-307, Jstor.  
  54. On a Simple Type of Irregular Singular Point    
    George D. Birkhoff  
    Transactions of the American Mathematical Society, Vol. 14, No. 4. (Oct., 1913), pp. 462-476, Jstor.  
  55. A Simplified Treatment of the Regular Singular Point  
    George D. Birkhoff  
    Transactions of the American Mathematical Society, Vol. 11, No. 2. (Apr., 1910), pp. 199-202, Jstor.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004