Mathematical methods with applications
著者
書誌事項
Mathematical methods with applications
WIT, c2000
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内容説明・目次
内容説明
In this textbook the author applies differential equations to physical problems, and highlights solution techniques with practical examples. Emphasis is placed on: operator methods; the convolution integral; periodic signals; the energy and power spectra; Frobenius method; Laplace, Fourier, Hankel and Z-transforms; Green's Function method; the similarity technique; the method of characteristics; the separation of variable method; and Bessel functions and Legendre polynomials. Four tables of integral transforms are also included, while exercises and answers are given on an accompanying CD-ROM.
目次
- Ordinary differential equations - Classification of first order differential equations
- First order nonlinear differential equations
- Singular solutions of differential equations
- Orthogonal trajectories
- Higher order linear differential equations
- The solution of the nonhomogeneous equations
- The method of variation of parameters
- The method of differential operator
- Euler-Cauchy differential equations
- Applications to practical problems. Fourier series and Fourier transform - Introduction
- Definition of a periodic function
- Fourier series and Fourier coefficients
- Complex form of Fourier series
- Half-range Fourier sine and cosine series
- Parseval's theorem
- Gibbs' phenomenon
- Development of Fourier integral and transform
- Relationship of Fourier and Laplace transforms
- Applications of Fourier transforms
- Parseval's theorem for energy signals
- Heaviside unit step function and Dirac delta function
- Some Fourier transforms involving impulse functions
- Properties of the Fourier transform
- The frequency transfer function. Laplace transforms - Introduction
- Definition of Laplace transform
- Laplace transform properties
- Laplace transforms of special functions
- Some important theorems
- The unit step function and the Dirac delta function
- The Heaviside expansion theorems to find inverses
- The method of residues to find inverses
- The Laplace transform of a periodic function
- Convolution. Series solution: method of Frobenius - Introduction
- Definition of ordinary and singular points
- Series expansion about an ordinary point
- Series expansion about a regular singular point. Partial differential equations - Introduction
- Mathematical formulation of equations
- Classification of PDE: Method of characteristics
- The D'Alembert solution of the wave equation
- The method of separation of variables
- Laplace and Fourier transform methods
- Similarity technique
- Applications to miscellaneous problems
- Sturm-Liouville problems. Bessel functions and Legendre polynomials - Introduction
- Series solution of Bessel's equation
- Modified Bessel functions
- Ber, Bei, Ker and Kei functions
- Equations solvable in terms of Bessel functions
- Recurrence relations of Bessel functions
- Orthogonality of Bessel functions
- Legendre polynomials
- Applications. Applications - Applications of Fourier series
- Applications of Fourier integrals
- Applications of Laplace transforms
- Applications with PDE
- Transmission lines
- The heat conduction problem
- The chemical diffusion problem
- Vibration of beams
- The hydrodynamics of waves and tides. Green's function - One-dimensional Green's function
- Green's function using variation of parameters
- Developments of Green's function in 2D
- Development of Green's function in 3D
- Numerical formulation. Integral transforms - Introduction
- The Hankel transform
- The Mellin transform
- The Z-transform.
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