# Integral equations, numerical methods

Methods for finding approximate solutions of integral equations.

It is required to find the solution $\phi ( x)$ of a one-dimensional Fredholm equation of the second kind,

$$\tag{1 } \phi ( x) = \lambda \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) \ d s + f ( x) ,$$

where $f ( x)$ is continuous on $[ a , b ]$, $\lambda$ is a numerical parameter and $K ( x , s )$ is continuous on $a \leq x , s \leq b$.

Suppose that $\lambda$ is not an eigen value of $K ( x , s )$. Then equation (1) has a unique solution $\phi ( x)$, which is continuous on $[ a , b ]$. Under these conditions one can give the following methods for obtaining an approximate solution.

## First method.

Let $a$ and $b$ be finite numbers. The integral in (1) is replaced by an integral sum over a grid $\{ s _ {j} \}$, $j = 0 \dots n$, while the variable $x$ takes the values $x _ {1}, \dots, x _ {n}$. One then obtains a system of linear algebraic equations with respect to the $\phi _ {j}$,

$$\tag{2 } \sum _ { j= 1} ^ { n } A _ {ij} \phi _ {j} = \ f _ {i} ,\ i = 1 \dots n ,$$

where $A _ {ij} = 1 - \lambda C _ {j} K ( x _ {i} , x _ {j} )$, and $C _ {j}$ are the coefficients of the quadrature formula by means of which the integral in (1) is replaced by the integral sum. For sufficiently large $n$, the system (2) has a unique solution $\{ \overline \phi _ {j} \}$. As an approximate solution to (1) one can take the function

$$\overline \phi _ {n} ( x) = \ f ( x) + \lambda \sum _ { j= 1} ^ { n } C _ {j} K ( x , x _ {j} ) ,$$

since as $n \rightarrow \infty$ and $\Delta _ {n} = \max _ {i} \{ | x _ {i+ 1} - x _ {i} | \} \rightarrow 0$, the sequence of functions $\overline \phi _ {n} ( x)$ converges uniformly on $[ a , b ]$ 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 $f ( x)$) needs to be.

In the case when the range of integration $( a , b )$ is infinite, it is replaced by a finite interval $( a _ {1} , b _ {1} )$ by using a priori information concerning the behaviour of the required solution $\overline \phi ( x)$ for large values of $| x |$. 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 $K ( x , s )$ is replaced by a degenerate kernel approximating it:

$$K _ {1} ( x , s ) = \ \sum _ { i= 1} ^ { n } a _ {i} ( x) b _ {i} ( s) ,$$

in which the functions $\{ a _ {i} ( x) \}$ are linearly independent. The equation

$$\tag{3 } \phi ( x) = \lambda \int\limits _ { a } ^ { b } \ \sum _ { i= 1} ^ { n } a _ {i} ( x) b _ {i} ( s) \phi ( s) d s + f ( x)$$

obtained in this way has a solution $\widehat \phi ( x)$ of the form

$$\tag{4 } \widehat \phi ( x) = \lambda \sum _ { i= 1} ^ { n } C _ {i} a _ {i} ( x) + f ( x) ,$$

in which the constants

$$C _ {i} = \ \int\limits _ { a } ^ { b } \widehat \phi ( s) b _ {i} ( s) d s$$

have to be determined. On substituting the function $\widehat \phi ( x)$ into equation (3) and comparing the coefficients at the functions $a _ {i} ( x)$ one obtains a system of linear algebraic equations for the $C _ {i}$:

$$C _ {i} - \lambda \sum _ { i= 1} ^ { n } \alpha _ {ij} C _ {j} = \ \beta _ {i} ,\ i = 1, \dots, n ,$$

where

$$\alpha _ {ij} = \ \int\limits _ { a } ^ { b } a _ {j} ( s) b _ {i} ( s) \ d s ,\ \beta _ {i} = \ \int\limits _ { a } ^ { b } f ( s) b _ {i} ( s) d s .$$

Having determined the $C _ {i}$ from the above system and by substituting them in (4) one obtains the function $\widehat \phi ( x)$, which is taken as the approximate solution of (1), since for a sufficiently good approximation of the kernel $K ( x , s )$ by a degenerate kernel the solution of equation (3) differs by an arbitrarily small amount from the required solution $\phi ( x)$ on any interval $[ a _ {1} , b _ {1} ] \subset ( a , b )$, as well as on $[ a , b ]$ in the case when $( a , b )$ is a finite interval (see [1], [4]).

## Third method.

As approximate solutions one takes functions $\phi _ {n} ( x)$ obtained by iteration based on the formula

$$\phi _ {n+ 1} ( x) = \lambda \int\limits _ { a } ^ { b } K ( x , s ) \phi _ {n} ( s) d s + f ( x) ,\ \ n = 0 , 1 \dots$$

$\phi _ {0} ( x) = f ( x)$; the sequence $\{ \phi _ {n} ( x) \}$ converges uniformly to the required solution as $n \rightarrow \infty$, provided that $| \lambda | < 1 / M ( b - a )$, where $M = \sup _ {x,s} | K ( x , s ) |$. Convergence of $\{ \phi _ {n} ( x) \}$ 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) $\lambda$ is one of the eigen values of the kernel $K ( x , s )$, 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)

The Fredholm equation (1) is said to be of the first kind if $f$ 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:

$$\phi ( x) = \int\limits _ {a ( x) } ^ { {b } ( x ) } K ( x , s ) \phi ( s) d s + f ( x) .$$

This equation is often referred to as Andreoli's integral equation. If $a ( x) = a$ and $b ( x) = x$, this equation reduces to a Volterra integral equation (cf. Volterra equation) of the second $( f \neq 0 )$ or first kind $( f = 0 )$.

In the numerical analysis of integral equations (including Fredholm and Voltera equations as well), one uses the terminology degenerate kernel (of rank $n$) or Pincherle–Goursat kernel for indicating kernels of the form $K _ {1} ( x , s )$. In the non-linear case, where the integrand in (1) is of the form $k ( x , s , \phi )$, one may approximate $k$ by a finite sum of terms $a _ {i} ( x) b _ {i} ( s , \phi )$. 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 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_equations,_numerical_methods&oldid=51876
This article was adapted from an original article by V.Ya. Arsenin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article