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Integral Equations and their Applications by M. Rahman



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Integral Equations and their Applications written by M. Rahman, Dalhousie University, Canada.

Integral Equations and their Applications written by M. Rahman cover the following topics.

  • 1. Introduction
    1.1 Preliminary concept of the integral equation
    1.2 Historical background of the integral equation
    1.3 An illustration from mechanics
    1.4 Classification of integral equations
    1.4.1 Volterra integral equations
    1.4.2 Fredholm integral equations
    1.4.3 Singular integral equations
    1.4.4 Integro-differential equations
    1.5 Converting Volterra equation to ODE
    1.6 Converting IVP to Volterra equations
    1.7 Converting BVP to Fredholm integral equations
    1.8 Types of solution techniques
    1.9 Exercises
    References

  • 2. Volterra integral equations
    2.1 Introduction
    2.2 The method of successive approximations
    2.3 The method of Laplace transform
    2.4 The method of successive substitutions
    2.5 The Adomian decomposition method
    2.6 The series solution method
    2.7 Volterra equation of the first kind
    2.8 Integral equations of the Faltung type
    2.9 Volterra integral equation and linear differential equations
    2.10 Exercises
    References

  • 3. Fredholm integral equations
    3.1 Introduction
    3.2 Various types of Fredholm integral equations
    3.3 The method of successive approximations: Neumann’s series
    3.4 The method of successive substitutions
    3.5 The Adomian decomposition method
    3.6 The direct computational method
    3.7 Homogeneous Fredholm equations
    3.8 Exercises
    References

  • 4. Nonlinear integral equations
    4.1 Introduction
    4.2 The method of successive approximations
    4.3 Picard’s method of successive approximations
    4.4 Existence theorem of Picard’s method
    4.5 The Adomian decomposition method
    4.6 Exercises
    References

  • 5. The singular integral equation
    5.1 Introduction
    5.2 Abel’s problem
    5.3 The generalized Abel’s integral equation of the first kind
    5.4 Abel’s problem of the second kind integral equation
    5.5 The weakly-singular Volterra equation
    5.6 Equations with Cauchy’s principal value of an integral and Hilbert’s transformation
    5.7 Use of Hilbert transforms in signal processing
    5.8 The Fourier transform
    5.9 The Hilbert transform via Fourier transform
    5.10 The Hilbert transform via the ±π/2 phase shift
    5.11 Properties of the Hilbert transform
    5.11.1 Linearity
    5.11.2 Multiple Hilbert transforms and their inverses
    5.11.3 Derivatives of the Hilbert transform
    5.11.4 Orthogonality properties
    5.11.5 Energy aspects of the Hilbert transform
    5.12 Analytic signal in time domain
    5.13 Hermitian polynomials
    5.14 The finite Hilbert transform
    5.14.1 Inversion formula for the finite Hilbert transform
    5.14.2 Trigonometric series form
    5.14.3 An important formula
    5.15 Sturm–Liouville problems
    5.16 Principles of variations
    5.17 Hamilton’s principles
    5.18 Hamilton’s equations
    5.19 Some practical problems
    5.20 Exercises
    References

  • 6. Integro-differential equations
    6.1 Introduction
    6.2 Volterra integro-differential equations
    6.2.1 The series solution method
    6.2.2 The decomposition method
    6.2.3 Converting to Volterra integral equations
    6.2.4 Converting to initial value problems
    6.3 Fredholm integro-differential equations
    6.3.1 The direct computation method
    6.3.2 The decomposition method
    6.3.3 Converting to Fredholm integral equations
    6.4 The Laplace transform method
    6.5 Exercises
    References

  • 7. Symmetric kernels and orthogonal systems of functions
    7.1 Development of Green’s function in one-dimension
    7.1.1 A distributed load of the string
    7.1.2 A concentrated load of the strings
    7.1.3 Properties of Green’s function
    7.2 Green’s function using the variation of parameters
    7.3 Green’s function in two-dimensions
    7.3.1 Two-dimensional Green’s function
    7.3.2 Method of Green’s function
    7.3.3 The Laplace operator
    7.3.4 The Helmholtz operator
    7.3.5 To obtain Green’s function by the method of images
    7.3.6 Method of eigenfunctions
    7.4 Green’s function in three-dimensions
    7.4.1 Green’s function in 3D for physical problems
    7.4.2 Application: hydrodynamic pressure forces
    7.4.3 Derivation of Green’s function
    7.5 Numerical formulation
    7.6 Remarks on symmetric kernel and a process of orthogonalization
    7.7 Process of orthogonalization
    7.8 The problem of vibrating string: wave equation
    7.9 Vibrations of a heavy hanging cable
    7.10 The motion of a rotating cable
    7.11 Exercises
    References

  • 8. Applications
    8.1 Introduction
    8.2 Ocean waves
    8.2.1 Introduction
    8.2.2 Mathematical formulation
    8.3 Nonlinear wave–wave interactions
    8.4 Picard’s method of successive approximations
    8.4.1 First approximation
    8.4.2 Second approximation
    8.4.3 Third approximation
    8.5 Adomian decomposition method
    8.6 Fourth-order Runge−Kutta method
    8.7 Results and discussion
    8.8 Green’s function method for waves
    8.8.1 Introduction
    8.8.2 Mathematical formulation
    8.8.3 Integral equations
    8.8.4 Results and discussion
    8.9 Seismic response of dams
    8.9.1 Introduction
    8.9.2 Mathematical formulation
    8.9.3 Solution
    8.10 Transverse oscillations of a bar
    8.11 Flow of heat in a metal bar
    8.12 Exercises
    References

  • Appendix A Miscellaneous results

  • Appendix B Table of Laplace transforms

  • Appendix C Specialized Laplace inverses

  • Answers to some selected exercises

  • Subject index

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