Formal Solutions of Differential Equations

We give a survey of some methods for finding formal solutions of differential equations. These include methods for finding power series solutions, elementary and liouvillian solutions, first integrals, Lie theoretic methods, transform methods, asymptotic methods. A brief discussion of difference equations is also included.

[1]  Bengt Malm A Program in REDUCE for Finding Explicit Solutions to Certain Ordinary Differential Equations , 1982, EUROCAM.

[2]  F. Olver Asymptotic Approximations and Error Bounds , 1980 .

[3]  Shunro Watanabe,et al.  A technique for solving ordinary differential equations using Riemann's P-functions , 1981, SYMSAC '81.

[4]  A. C. Norman Integration in Finite Terms , 1983 .

[5]  Jean Della Dora,et al.  Formal solutions of linear difference equations: method of Pincherle-Ramis , 1986, SYMSAC '86.

[6]  Fritz Schwarz,et al.  Automatically determining symmetries of ordinary differential equations , 1983, EUROCAL.

[7]  E. Hilb,et al.  Lineare Differentialgleichungen im komplexen Gebiet , 1921 .

[8]  Implicitly elementary integrals , 1976 .

[9]  J. Lafon Summation in Finite Terms , 1983 .

[10]  A. C. Norman,et al.  Computing with Formal Power Series , 1975, TOMS.

[11]  Michael Rothstein Aspects of symbolic integration and simplification of exponential and primitive functions. , 1976 .

[12]  Carol Tretkoff,et al.  Solution of the Inverse Problem of Differential Galois Theory in the Classical Case , 1979 .

[13]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[14]  Nicholas M. Katz,et al.  A SIMPLE ALGORITHM FOR CYCLIC VECTORS , 1987 .

[15]  Evelyne Tournier An algebraic form of a solution of a system of linear differential equations with constant coefficients , 1979, EUROSAM.

[16]  Bruce W. Char,et al.  A Tutorial Introduction to Maple , 1986, J. Symb. Comput..

[17]  R. Risch The problem of integration in finite terms , 1969 .

[18]  F. Neuman Ordinary linear differential equations — A survey of the global theory , 1986 .

[19]  M. Rosenlicht,et al.  The rank of a Hardy field , 1983 .

[20]  Michael F. Singer,et al.  On elementary, generalized elementary, and liouvillian extension fields , 1977 .

[21]  Allan Reiman Computer-aided closure of the lie algebra associated with a nonlinear partial differential equation , 1981 .

[22]  J. Jouanolou,et al.  Equations de Pfaff algébriques , 1979 .

[23]  D. Armbruster,et al.  "Perturbation Methods, Bifurcation Theory and Computer Algebra" , 1987 .

[24]  B. F. Caviness,et al.  An Extension of Liouville's Theorem on Integration in Finite Terms , 1985, SIAM J. Comput..

[25]  K. S. Kölbig Explicit Evaluation of Certain Definite Integrals Involving Powers of Logarithms , 1985, J. Symb. Comput..

[26]  Moulay A. Barkatou,et al.  On the reduction of linear systems of difference equations , 1989, ISSAC '89.

[27]  Jean Della Dora,et al.  An Algorithm to Obtain Formal Solutions of a Linear Homogeneous Differential Equation at an Irregular Singular Point , 1982, EUROCAM.

[28]  Barry M. Trager,et al.  Integration of algebraic functions , 1984 .

[29]  Fritz Schwarz,et al.  A factorization algorithm for linear ordinary differential equations , 1989, ISSAC '89.

[30]  D. V. Chudnovsky,et al.  On expansion of algebraic functions in power and Puiseux series, I , 1986, J. Complex..

[31]  A counterexample to elimination in systems of algebraic differential equations , 1983 .

[32]  Keith O. Geddes Convergence behavior of the Newton iteration for first order differential equations , 1979, EUROSAM.

[33]  L. Lipshitz,et al.  The diagonal of a D-finite power series is D-finite , 1988 .

[34]  Robert L. Grossman,et al.  Labeled trees and the efficient computation of derivations , 1989, ISSAC '89.

[35]  Claudine Chaffy How to compute multivariate Pade approximants , 1986, SYMSAC '86.

[36]  A. Duval,et al.  Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes généralisées. , 1989 .

[37]  W. Wu,et al.  A MECHANIZATION METHOD OF GEOMETRY AND ITS APPLICATIONS——II. CURVE PAIRS OF BERTRAND TYPE , 1987 .

[38]  M. Clarkson MACSYMS's inverse Laplace transform , 1989, SIGS.

[39]  P. Hartman Ordinary Differential Equations , 1965 .

[40]  Michael F. Singer,et al.  Elementary first integrals of differential equations , 1983 .

[41]  Dieter Schramm,et al.  Application of NEWEUL in Robot Dynamics , 1989, J. Symb. Comput..

[42]  M. Rothstein,et al.  A new algorithm for the integration of exponential and logarithmic functions , 1977 .

[43]  Maxwell Rosenlicht Liouville's theorem on functions with elementary integrals. , 1968 .

[44]  Lawrence Goldman Integrals of ν-graded systems of ordinary differential equations , 1987 .

[45]  Erich Kaltofen A Note on the Risch Differential Equation , 1984, EUROSAM.

[46]  A. Seidenberg An elimination theory for differential algebra , 1959 .

[47]  Manuel Bronstein Simplification of real elementary functions , 1989, ISSAC '89.

[48]  R. Fateman Some comments on series solutions , 1977 .

[49]  "ORDERS OF INFINITY" GENERATED BY DIFFERENCE EQUATIONS , 1984 .

[50]  M. Rosenlicht,et al.  On Liouville’s theory of elementary functions , 1976 .

[51]  B. Char Algorithms using lie transformation groups to solve first order ordinary differential equations algebraically , 1980 .

[52]  D. Grigor'ev,et al.  Solving ordinary differential equations in terms of series with real exponents , 1991 .

[53]  Michael F. Singer,et al.  Solving homogeneous linear differential equations in terms of second order linear differential equations , 1985 .

[54]  Keith O. Geddes,et al.  Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms , 1989, Computers and Mathematics.

[55]  B. Dwork,et al.  On Second Order Linear Differential Equations with Algebraic Solutions , 1979 .

[56]  Robert H. Risch ALGEBRAIC PROPERTIES OF THE ELEMENTARY FUNCTIONS OF ANALYSIS. , 1979 .

[57]  Fritz Schwarz,et al.  Symmetries of Differential Equations: From Sophus Lie to Computer Algebra , 1988 .

[58]  J. Golden MACSYMA's symbolic ordinary differential equation solver , 1977 .

[59]  Manuel Bronstein,et al.  Integration of Elementary Functions , 1990, J. Symb. Comput..

[60]  James H. Davenport,et al.  Elementary and Liouvillian Solutions of Linear Differential Equations , 1986, J. Symb. Comput..

[61]  H. Q. Tan Symbolic Derivation of Equations for Mixed Formulation in Finite Element Analysis , 1989, Computers and Mathematics.

[62]  A. Hilali,et al.  Un algorithme de calcul de l'invariant de Katz d'un système différentiel linéaire , 1986 .

[63]  Guy W. Cherry Integration in Finite Terms with Special Functions: the Error Function , 1985, J. Symb. Comput..

[64]  James H. Davenport,et al.  Elementary and Liouvillian Solutions of Linear Differential Equations , 1985, J. Symb. Comput..

[65]  Stanly Steinberg,et al.  Lie series, Lie transformations, and their applications , 1986 .

[66]  William Y. Sit On Goldman's Algorithm for Solving First-Order Multinomial Autonomous Systems , 1988, AAECC.

[67]  N. Kamran,et al.  Classification of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries , 1989 .

[68]  Jeremy J. Gray,et al.  Linear Differential Equations and Group Theory from Riemann to Poincare , 1986 .

[69]  George Labahn,et al.  A fast, reliable algorithm for calculating Pade´-Hermite forms , 1989, ISSAC '89.

[70]  Paul S. Wang EVALUATION OF DEFINITE INTEGRALS BY SYMBOLIC MANIPULATION , 1971 .

[71]  Lou van den Dries,et al.  Exponential rings, exponential polynomials and exponential functions , 1984 .

[72]  Willard Miller,et al.  Symmetry and Separation of Variables , 1977 .

[73]  M. Singer Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations , 1981 .

[74]  Aziz Hilali,et al.  Characterization of a linear differential system with a regular singularity , 1983, EUROCAL.

[75]  Thomas Wolf Analytical Decoupling, Decision of Compatibility and Partial Integration of Systems of Non-Linear Ordinary and Partial Differential Equations , 1985, European Conference on Computer Algebra.

[76]  Michael F. Singer Functions satisfying elementary relations , 1977 .

[77]  Jean C. Piquette,et al.  Special function integration , 1989, SIGS.

[78]  Jacques Cohen,et al.  Symbolic Solution of Finite-Difference Equations , 1977, TOMS.

[79]  Shigeru Maeda,et al.  The Similarity Method for Difference Equations , 1987 .

[80]  Jerald J. Kovacic,et al.  An Algorithm for Solving Second Order Linear Homogeneous Differential Equations , 1986, J. Symb. Comput..

[81]  Michael Karr,et al.  Summation in Finite Terms , 1981, JACM.

[82]  Michael F. Singer ELEMENTARY SOLUTIONS OF DIFFERENTIAL EQUATIONS , 1975 .

[83]  Paul S. Wang,et al.  FINGER: A Symbolic System for Automatic Generation of Numerical Programs in Finite Element Analysis , 1986, J. Symb. Comput..

[84]  Doron Zeilberger,et al.  The Method of Creative Telescoping , 1991, J. Symb. Comput..

[85]  Aziz Hilali,et al.  On the Algebraic and Differential Newton-Puiseux Polygons , 1987, J. Symb. Comput..

[86]  Fritz Schwarz,et al.  A reduce package for determining lie symmetries of ordinary and partial differential equations , 1984 .

[87]  Volker Weispfenning,et al.  Non-Commutative Gröbner Bases in Algebras of Solvable Type , 1990, J. Symb. Comput..

[88]  J. Baddoura Integration in Finite Terms and Simplification with Dilogarithms: A Progress Report , 1989, Computers and Mathematics.

[89]  B. David Saunders An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations , 1981, SYMSAC '81.

[90]  M. H. Protter,et al.  THE SOLUTION OF THE PROBLEM OF INTEGRATION IN FINITE TERMS , 1970 .

[91]  N. M. Katz On the calculation of some differential galois groups , 1987 .

[92]  Françoise Lamnabhi-Lagarrigue,et al.  Algebraic computation of the statistics of the solution of some nonlinear stochastic differential equations , 1983, EUROCAL.

[93]  Guy W. Cherry Integration in Finite Terms with Special Functions: The Logarithmic Integral , 1986, SIAM J. Comput..

[94]  D. Grigor'ev,et al.  Complexity of factoring and calculating the GCD of linear ordinary differential operators , 1990 .

[95]  M. Rosenlicht,et al.  Integration in Finite Terms , 1972 .

[96]  A. H. M. Levelt,et al.  Invariants mesurant l'irrégularité en un point singulier des systèmes d'équations différentielles linéaires , 1972 .

[97]  Abdelaziz Hilali,et al.  Algorithm for computing formal invariants of linear differential systems , 1986, SYMSAC '86.

[98]  Willard Miller,et al.  Lie Theory and Special Functions , 1969 .

[99]  Francesco Baldassarri,et al.  ON SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ALGEBRAIC SOLUTIONS ON ALGEBRAIC CURVES , 1980 .

[100]  N. A. Kostov,et al.  Computer Algebra in the Theory of Ordinary Differential Equations of Halphen Type , 1989, Computers and Mathematics.

[101]  P. Winternitz LIE GROUPS AND SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS , 1983 .

[102]  Barry M. Trager Integration of simple radical extensions , 1979, EUROSAM.

[103]  Manuel Bronstein The Trascendental Risch Differential Equation , 1990, J. Symb. Comput..

[104]  E. Tournier Computer algebra and differential equations , 1989 .

[105]  Thomas Wolf An analytic algorithm for decoupling and integrating systems of nonlinear partial differential equations , 1985 .

[106]  L. Lipshitz,et al.  Power series solutions of algebraic differential equations , 1984 .

[107]  W. A. Beyer Lie-group theory for symbolic integration of first-order differential equations: a modern treatment. [In MACSYMA system] , 1979 .

[108]  Françoise Lamnabhi-Lagarrigue,et al.  Algebraic Computation of the Solution of Some NonLinear Differential Equations , 1982, EUROCAM.

[109]  M. A. Ostrowski Sur l'intégrabilité élémentaire de quelques classes d'expressions , 1945 .

[110]  M. Singer Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem , 1988 .

[111]  Jean-Pierre Ramis,et al.  Théorèmes d’indices Gevrey pour les équations différentielles ordinaires , 1984 .

[112]  Michael F. Singer,et al.  Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients , 1989, J. Symb. Comput..

[113]  Fritz Schwarz An Algorithm for Determining Polynomial First Integrals of Autonomous Systems of Ordinary Differential Equations , 1985, J. Symb. Comput..

[114]  John Fitch The automatic derivation of periodic solutions to a class of weakly nonlinear differential equations , 1981, SYMSAC '81.

[115]  Nikolaos Glinos,et al.  Operational Calculus Technique for Solving Differential Equations , 1984, EUROSAM.

[116]  E. Horowitz Algorithms for partial fraction decomposition and rational function integration , 1971, SYMSAC '71.

[117]  David Y. Y. Yun Fast Algorithm for Rational Function Integration , 1977, IFIP Congress.

[118]  Michael B. Hayden,et al.  Summation of binomial coefficients using hypergeometric functions , 1986, SYMSAC '86.

[119]  James H. Davenport,et al.  On the Integration of Algebraic Functions , 1979, Lecture Notes in Computer Science.

[120]  A. H. M. Levelt,et al.  Jordan decomposition for a class of singular differential operators , 1975 .

[121]  Bruce W. Char,et al.  Using Lie transformation groups to find closed form solutions to first order ordinary differential equations , 1981, SYMSAC '81.

[122]  James H. Davenport,et al.  Symbolic integration — the dust settles? , 1979 .

[123]  Michael Karr Theory of Summation in Finite Terms , 1985, J. Symb. Comput..

[124]  Vladimir V. Kornyak,et al.  Computer Algebra Application for Determining Lie and Lie-Bäcklund Symmetries of Differential Equations , 1989, J. Symb. Comput..

[125]  Peter Schmidt Substitution methods for the automatic symbolic solution of differential equations of first order and first degree , 1979, EUROSAM.

[126]  R. W. Gosper Indefinite hypergeometric sums in MACSYMA , 1977 .

[127]  A. Hilali Calcul des invariants de Malgrange et de Gérard-Levelt d'un système différentiel linéaire en un point singulier irrégulier , 1987 .

[128]  Patrick J. Roache,et al.  Symbolic manipulation and computational fluid dynamics , 1985 .

[129]  S. Watanabe Formula Manipulations Solving Linear Ordinary Differential Equations (II) , 1970 .

[130]  G. Rw Decision procedure for indefinite hypergeometric summation , 1978 .

[131]  M. Rosenlicht Rank change on adjoining real powers to Hardy fields , 1984 .

[132]  L. Lipshitz,et al.  D-finite power series , 1989 .

[133]  Maxwell Rosenlicht On the Value Group of a Differential Valuation II , 1979 .

[134]  F. Marotte Les équations différentielles linéaires et la théorie des groupes , 1898 .

[135]  Barry M. Trager,et al.  Algebraic factoring and rational function integration , 1976, SYMSAC '76.

[136]  John ffitch,et al.  Alkahest III: automatic analysis of periodic weakly nonlinear ODEs , 1986, SYMSAC '86.

[137]  James H. Davenport The Risch Differential Equation Problem , 1986, SIAM J. Comput..