# Integral equations, numerical methods

Methods for finding approximate solutions of integral equations.

It is required to find the solution of a one-dimensional Fredholm equation of the second kind,

(1) |

where is continuous on , is a numerical parameter and is continuous on .

Suppose that is not an eigen value of . Then equation (1) has a unique solution , which is continuous on . Under these conditions one can give the following methods for obtaining an approximate solution.

## Contents

## First method.

Let and be finite numbers. The integral in (1) is replaced by an integral sum over a grid , , while the variable takes the values . One then obtains a system of linear algebraic equations with respect to the ,

(2) |

where , and are the coefficients of the quadrature formula by means of which the integral in (1) is replaced by the integral sum. For sufficiently large , the system (2) has a unique solution . As an approximate solution to (1) one can take the function

since as and , the sequence of functions converges uniformly on to the required solution of equation (1). See [1]–[4].

When replacing the integral by a quadrature formula, one has to bear in mind that the greater the precision of the quadrature formula used, the greater the smoothness of the kernel and solution (and hence also ) needs to be.

In the case when the range of integration is infinite, it is replaced by a finite interval by using a priori information concerning the behaviour of the required solution for large values of . The equation so obtained is then approximately solved by the above method. Alternatively, by a change of the integration variable the range of integration is reduced to a finite range. As another alternative, quadrature formulas for an infinite range can be applied.

## Second method.

In equation (1) the kernel is replaced by a degenerate kernel approximating it:

in which the functions are linearly independent. The equation

(3) |

obtained in this way has a solution of the form

(4) |

in which the constants

have to be determined. On substituting the function into equation (3) and comparing the coefficients at the functions one obtains a system of linear algebraic equations for the :

where

Having determined the from the above system and by substituting them in (4) one obtains the function , which is taken as the approximate solution of (1), since for a sufficiently good approximation of the kernel by a degenerate kernel the solution of equation (3) differs by an arbitrarily small amount from the required solution on any interval , as well as on in the case when is a finite interval (see [1], [4]).

## Third method.

As approximate solutions one takes functions obtained by iteration based on the formula

; the sequence converges uniformly to the required solution as , provided that , where . Convergence of to the exact solution also holds for kernels with an integrable singularity (see [1]). Estimates of the errors of these methods are given in [1]–[7]. In [8] the question is considered of the minimum number of arithmetical operations required in order to obtain an approximate value of the integral with a prescribed precision. The solution of this problem is equivalent to finding the size of the minimal error of an approximate solution of the problem under a prescribed number of arithmetical operations.

For the solution of Fredholm integral equations of the first kind special methods need to be applied, since these problems are ill-posed. If in equation (1) is one of the eigen values of the kernel , then the problem of finding a solution of (1) is ill-posed and requires special methods (see Ill-posed problems).

Non-linear equations of the second kind are usually solved approximately by an iterative method (see [3]).

The Galerkin method for obtaining approximate solutions of linear and non-linear equations is also used.

Similar methods can also be applied for obtaining approximate solutions of multi-dimensional Fredholm integral equations of the second kind. However, their numerical implementation is more complicated. See [5]–[10] for cubature formulas for the approximate computation of multiple integrals and their error estimates. A Monte-Carlo method of approximate numerical computation of multiple integrals is discussed in [10].

#### References

[1] | I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian) |

[2] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |

[3] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian) |

[4] | L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) |

[5] | I.P. Mysovskikh, "Cubature formulae for evaluating integrals on the surface of a sphere" Sibirsk. Mat. Zh. , 5 : 3 (1964) pp. 721–723 (In Russian) |

[6] | I.P. Mysovskikh, "The application of orthogonal polynomials to cubature formulae" USSR Comp. Math. Math. Phys. , 12 : 2 (1972) pp. 228–239 Zh. Vychisl. Mat. i Mat. Fiz. , 12 : 2 (1972) pp. 467–475 |

[7] | I.P. Mysovskikh, "On Chakalov's theorem" USSR Comp. Math. Math. Phys. , 15 : 6 (1976) pp. 221–227 Zh. Vychisl. Mat. i Mat. Fiz. , 15 : 6 (1975) pp. 1589–1593 |

[8] | K.B. Emel'yanov, A.M. Il'in, "Number of arithmetical operations necessary for the approximate solution of Fredholm integral equations of the second kind" USSR Comp. Math. Math. Phys. , 7 : 4 (1970) pp. 259–266 Zh. Vychisl. Mat. i Mat. Fiz. , 7 : 4 (1967) pp. 905–910 |

[9] | S.L. Sobolev, "Introduction to the theory of cubature formulas" , Moscow (1974) (In Russian) |

[10] | I.M. Sobol', "Multi-dimensional cubature formulas and Haar functions" , Moscow (1969) (In Russian) |

#### Comments

The Fredholm equation (1) is said to be of the first kind if vanishes and of the second kind otherwise (cf. also Fredholm equation, numerical methods). It is a special case of the more general integral equation with variable limits of integration:

This equation is often referred to as Andreoli's integral equation. If and , this equation reduces to a Volterra integral equation (cf. Volterra equation) of the second or first kind .

In the numerical analysis of integral equations (including Fredholm and Voltera equations as well), one uses the terminology degenerate kernel (of rank ) or Pincherle–Goursat kernel for indicating kernels of the form . In the non-linear case, where the integrand in (1) is of the form , one may approximate by a finite sum of terms . Kernels of this type are called separable or finitely decomposable.

A thorough discussion of numerical methods for linear integral equations of the second kind including Fortran programs can be found in [a2]; see also [a3]. The "first method" of the main article is usually called the Nyström method. A functional-analytic basis for numerical methods for both linear and non-linear integral equations is the theory of collectively-compact operators ([a1]). Numerical methods for integral equations of the first kind are the so-called "regularization methodregularization methods" (cf. Regularization method, [a4]).

#### References

[a1] | P.M. Anselone, "Collectively compact operator approximation theory and applications to integral equations" , Prentice-Hall (1971) |

[a2] | K.E. Atkinson, "A survey of numerical methods for the solution of Fredholm integral equations of the second kind" , SIAM (1976) |

[a3] | C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) |

[a4] | C.W. Groetsch, "The theory of Tikhonov regularization for Fredholm equations of the first kind" , Pitman (1984) |

[a5] | J.L. Walsh, L.M. Delver, "Numerical solution of integral equations" , Oxford Univ. Press (1974) |

**How to Cite This Entry:**

Integral equations, numerical methods. V.Ya. Arsenin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Integral_equations,_numerical_methods&oldid=14561