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)

Comments

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)