Details

Subject(s)

• Differential equations [Browse]

Related name

• Rota, Gian-Carlo, 1932-1999 [Browse]

Notes

Includes index.

Bibliographic references

Bibliography: p. 392-394.

Contents

• First-order of differential equations : Fundamental theorem of the calculus ; First-order linear equations ; Separable equations ; Quasilinear equations; implicit solutions ; Exact differentials; integrating factors ; Linear fractional equations ; Graphical and numerical integration ; The initial value problem ; Uniqueness and continuity ; A comparison theorem ; Regular and normal curve families
• Second-order linear equations : Bases of solutions ; Initial value problems ; Qualitative behavior; stability ; Uniqueness theorem ; The Wronskian ; Separation and comparison theorems ; The phase plane ; Adjoint operators; Lagrange identity ; Green’s functions ; Two-endpoint problems ; Green’s functions, II
• Linear equations with constant coefficients : The characteristic polynomial ; Complex exponential functions ; The operational calculus ; Solution bases ; Inhomogeneous equations ; Stability ; The transfer function ; The Nyquist diagram ; The Green’s function
• Power series solutions : Method of undetermined coefficients ; More examples ; Three first-order DEs ; Analytic functions ; Method of majorants ; Sine and cosine functions ; Bessel functions ; First-order nonlinear DEs ; Radius of convergence ; Method of majorants, II ; Complex solutions
• Plane autonomous systems : Autonomous systems ; Plane autonomous systems ; The phase plane, II ; Linear autonomous systems ; Linear equivalence ; Equivalence under diffeomorphisms ; Stability ; Method of Liapunov ; Undamped nonlinear oscillations ; Soft and hard springs ; Damped nonlinear oscillations ; Limit cycles
• Existence and uniqueness theorems : Lipschitz conditions ; Well-posed problems ; Continuity ; Normal systems ; Equivalent integral equation ; Successive approximation ; Linear systems ; Local existence theorem ; The Peano existence theorem ; Analytic equations ; Continuation of solutions ; The perturbation equation
• Approximate solutions : Error bounds ; Deviation and error ; Mesh-halving: Richardson extrapolation ; Midpoint quadrature ; Trapezoidal quadrature ; Trapezoidal integration ; The improved Euler method ; The modified Euler method ; Cumulative error bound
• Efficient numerical integration : Difference operators ; Polynomial interpolation ; Stability ; Numerical differentiation; roundoff ; Higher order quadrature ; Gaussian quadrature ; Fourth-order Runge-Kutta ; Milne’s method ; Multistep methods
• Regular singular points : Movable singular points ; First-order linear equations ; Continuation principle; circuit matrix ; Canonical bases ; Regular singular points ; Bessel equation ; The fundamental theorem ; Alternative proof of the fundamental theorem ; Hypergeometric functions ; The Jacobi polynomials ; Singular points at infinity ; Fuchsian equations
• Sturm-Liouville systems : Sturm-Liouville systems ; Sturm-Liouville series ; Physical interpretations ; Singular systems ; Prüfer substitution ; Sturm comparison theorem ; Sturm oscillation theorem ; The sequence of eigenfunctions ; The Liouville normal form ; Modified Prüfer substitution ; The asymptotic behavior of Bessel functions ; Distribution of eigenvalues ; Normalized eigenfunctions ; Inhomogeneous equations ; Green’s functions ; The Schroedinger equation ; The square-well potential ; Mixed spectrum
• Expansions in eigenfunctions : Fourier series ; Orthogonal expansions ; Mean-square approximation ; Completeness ; Orthogonal polynomials ; Properties of orthogonal polynomials ; Chebyshev polynomials ; Euclidean vector spaces ; Completeness of eigenfunctions ; Hilbert space ; Proof of completeness.

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ISBN

• 0471860034
• 9780471860037
• 0471500208
• 9780471500209

LCCN

88014231

OCLC

18379836

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