Numerical-Analytic Methods in the Theory of Boundary-Value Problems

The following sections are included:

• PREFACE

• CONTENTS

The development of the theory of boundary-value problems for ordinary differential equations was initiated by the works of G. Floquet [Flo], A. M. Lyapunov [Lyal], [Lya2], S. N. Bernstein [Ber], G. D. Birkhoff [Bir], E. L. Bunitsky [Bunl], [Bun2], D.Jackson [Jacl], [Jac2], G. A. Bliss [Blil], [Bli2], G. D. Birkhoff and R. E. Langer [BiL], and by E. Kamke [Kam], which appeared at the end of the 19th century and in the first decades of the 20th century. The analysis of bibliography shows that boundary-value problems have always attracted particular attention.

The following sections are included:

• Abstract Scheme of the Method

• Choice of the Form of Successive Approximations and Their Uniform Convergence

• Sufficient Conditions for the Existence of Solutions

• Necessary Conditions for the Solvability of the Boundary-Value Problem

• Error of Calculation of the Initial Value of a Solution

• Special Types of Successive Approximations and Estimates

• Numerical-Analytic Method in the Case of Nonlinear Two-Point Boundary Conditions

• Boundary-Value Problems with Small Parameter

The following sections are included:

• Periodic Boundary-Value Problem

• Theorems on the Properties of Determining Functions of a Periodic Boundary-Value Problem

• Solvability of the Approximate Determining Equation and the Error of the Initial Value of a Periodic Solution

• Modification of the Method for Two-Point Problems

• Relationship between Exact and Approximate Determining Equations

• Determination of Initial Values of Solutions of Two-Point Boundary-Value Problems

• Realization of the Method for Systems of Two Equations

The following sections are included:

• Successive Approximations for Problems with One Parameter in Linear Boundary Conditions

• Sufficient Solvability Conditions and Determination of the Initial Value of a Solution of the Boundary-Value Problem with Parameter

• Boundary-Value Problems with Nonfixed Right Boundary

• Solvability Theorems for Problems with Nonfixed Right Boundary

• The Case of Several Parameters in Boundary Conditions

• Linear Dependence of Boundary Conditions on Two Parameters

The following sections are included:

• Green Function of a Homogeneous Two-Point Boundary-Value Problem

• Inhomogeneous Linear Impulsive Boundary-Value Problems

• Convergence of the Algebraic Collocation Method for Nonlinear Systems

• Method of Trigonometric Collocation for Periodic Systems

• Practical Solution of Impulsive Problems

• Green Function for a Three-Point Boundary-Value Problem with Single Impulse Influence

• Semihomogeneous Linear Two-Point Boundary-Value Problem with m-Impulse Influence

• Inhomogeneous Linear m-Impulse Two-Point Boundary-Value Problem

• Multipoint m-Impulse Boundary-Value Problem

• Equations with Piecewise-Continuous Right-Hand Side

• Construction of Solutions of Two-Impulse Systems

The following sections are included:

• APPENDIX. THE THEORY OF THE NUMERICAL-ANALYTIC METHOD: ACHIEVEMENTS AND NEW TRENDS OF DEVELOPMENT
  • History of the Method of Periodic Successive Approximations
  • Relation to Other Investigations
  • Application of Modifications of the Numerical-Analytic Method to the Investigation of Various Boundary-Value Problems
    • Second-Order Equations
    • Applications of the Method to the Investigation of Differential-Functional Equations
    • Differential-Operator Equations
    • Abstract Scheme of the Method
    • Autonomous Systems of Differential Equations
    • Autonomous Systems with Periodic External Influence
    • Impulsive Equations

• REFERENCES