Handbook of Integral Equations
Integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine.
Exact (closed-form) solutions of integral equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Equations of physics, chemistry, and biology contain functions or parameters obtained from experiments - hence, they are not strictly fixed. Therefore, it is expedient to choose the structure of these functions for more easily analyzing and solving the equation. As a possible selection criterion, one may adopt the requirement that the model integral equation admit a solution in a closed form. Exact solutions can be used to verify the consistency and estimate errors of various numerical, asymptotic, and approximate methods.
The first part of Handbook of Integral Equations:
Other equations contain one or more free parameters (the book actually deals with families of integral equations); the reader has the option to fix these parameters.
The second part of the book - chapters 7 through 14 - presents exact, approximate analytical, and numerical methods for solving linear and nonlinear integral equations. Apart from the classical methods, the text also describes some new methods. When selecting the material, the authors emphasize practical aspects of the matter, specifically for methods that allow an effective "constructing" of the solution. Each section provides examples of applicatio
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Linear Equations of the First Kind With Variable Limit of Integration
Equations Whose Kernels Contain Trigonometric Functions
Linear Equations of the Second Kind With Variable Limit of Integration
Linear Equation of the First Kind With Constant Limits of Integration
Linear Equations of the Second Kind With Constant Limits of Integration
Nonlinear Equations With Variable Limit of Integration
Nonlinear Equations With Constant Limits of Integration
Main Definitions and Formulas Integral Transforms
Methods for Solving Singular Integral Equations of the First Kind
Methods for Solving Complete Singular Integral Equations
Methods for Solving Nonlinear Integral Equations
Elementary Functions and Their Properties
Tables of Indefinite Integrals
Tables of Definite Integrals
Supplements Tables of Inverse Laplace Transforms
Tables of Fourier Cosine Transforms
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algebraic equations algebraic or transcendental analytic function approximate arbitrary constant arctan Assume boundary condition boundary value problem Cauchy kernel Cauchy principal value characteristic value constants Bk constants determined contour cos(Ax cosh difference kernel domain eigenfunction equation corresponding Equations Whose Kernels expression form y(x formulas presented Fredholm equation Fredholm integral equation given half-plane Holder condition homogeneous equation infinity initial conditions integrand Inverse transform J-oo kernel K(x Kernels Containing Laplace transform Logarithmic Mellin transform method of undetermined nonlinear obtain ordinary differential equation original equation original integral equation parameter polynomial real axis References for Section regularization relation Riemann problem right-hand side roots satisfies the Holder satisfy the conditions second kind second-order linear sin(Ax singular integral equation sinh sinh[A(x solution of Eq solving Subsection substitution w(x theorem undetermined coefficients variable Volterra equation Volterra integral equation yi(x zero