Andrei D. Polyanin and Alexander V. Manzhirov HANDBOOK OF INTEGRAL EQUATIONS

 Andrei D. Polyanin and Alexander V. Manzhirov HANDBOOK  OF INTEGRAL EQUATIONS

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. Lots of equations of physics, chemistry and biology contain functions or parameters which are obtained from experiments and hence are not strictly fixed. Therefore, it is expedient to choose the structure of these functions so that it would be easier to analyze and solve 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. 

      More than 2100 integral equations and their solutions are given in the first part of the book (Chapters 1–6). A lot of new exact solutions to linear and nonlinear equations are included. Special attention is paid to equations of general form, which depend on arbitrary functions. The other equations contain one or more free parameters (the book actually deals with families of integral equations); it is the reader’s option to fix these parameters. Totally, the number of equations described in this handbook is an order of magnitude greater than in any other book currently available. 

    The second part of the book (Chapters 7–14) presents exact, approximate analytical, and numerical methods for solving linear and nonlinear integral equations. Apart from the classical methods, some new methods are also described. When selecting the material, the authors have given a pronounced preference to practical aspects of the matter; that is, to methods that allow effectively “constructing” the solution. For the reader’s better understanding of the methods, each section is supplied with examples of specific equations. Some sections may be used by lecturers of colleges and universities as a basis for courses on integral equations and mathematical physics equations for graduate and postgraduate students.

Topics are discussed in this book are

1.  Linear  Equations  of  the  First  Kind  With  Variable  Limit  of  Integration

2.  Linear  Equations  of  the  Second  Kind  With  Variable  Limit  of  Integration

3.  Linear  Equation  of  the  First  Kind  With  Constant  Limits  of  Integration

4.  Linear  Equations  of  the  Second  Kind  With  Constant  Limits  of  Integration

5.  Nonlinear  Equations  With  Variable  Limit  of  Integration

6.  Nonlinear  Equations  With  Constant  Limits  of  Integration

7  Main Definitions  and  Formulas.  Integral  Transforms

8.  Methods  for  Solving  Linear  Equations  of  the  Form xa K(x, t)y(t) dt  =  f(x)

9.Methods for Solving Linear Equations of the Form y(x) –xaK(x, t)y(t) dt = f(x) 

10.  Methods  for  Solving  Linear  Equations  of  the  Form baK(x, t)y(t) dt  =  f(x)1

11.  Methods  for  Solving  Linear  Equations  of  the  Form  y(x)  –baK(x, t)y(t) dt  =  f(x) 

12.  Methods  for  Solving  Singular  Integral  Equations  of  the  First  Kind

13.  Methods  for  Solving  Complete  Singular  Integral  Equations

14.  Methods  for  Solving  Nonlinear  Integral  Equations

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