Difference between revisions of "Integral equations, numerical methods"
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Methods for finding approximate solutions of integral equations. | Methods for finding approximate solutions of integral equations. | ||
− | It is required to find the solution | + | It is required to find the solution $ \phi ( x) $ |
+ | of a one-dimensional [[Fredholm equation|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 | + | 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 | + | 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.== | ==First method.== | ||
− | Let | + | 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 | + | 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 | + | 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 [[#References|[1]]]–[[#References|[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 | + | 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 | + | 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.== | ==Second method.== | ||
− | In equation (1) the kernel | + | 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 | + | 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 | + | 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 | 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 | + | 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 | 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 | + | 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 [[#References|[1]]], [[#References|[4]]]). | ||
==Third method.== | ==Third method.== | ||
− | As approximate solutions one takes functions | + | 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 [[#References|[1]]]). Estimates of the errors of these methods are given in [[#References|[1]]]–[[#References|[7]]]. In [[#References|[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) | + | 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|Ill-posed problems]]). | ||
Non-linear equations of the second kind are usually solved approximately by an iterative method (see [[#References|[3]]]). | Non-linear equations of the second kind are usually solved approximately by an iterative method (see [[#References|[3]]]). | ||
Line 68: | Line 176: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S.L. Sobolev, "Introduction to the theory of cubature formulas" , Moscow (1974) (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I.M. Sobol', "Multi-dimensional cubature formulas and Haar functions" , Moscow (1969) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S.L. Sobolev, "Introduction to the theory of cubature formulas" , Moscow (1974) (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I.M. Sobol', "Multi-dimensional cubature formulas and Haar functions" , Moscow (1969) (In Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | The Fredholm equation (1) is said to be of the first kind if | + | 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|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 | + | 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|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|degenerate kernel]] (of rank | + | In the numerical analysis of integral equations (including Fredholm and Voltera equations as well), one uses the terminology [[Degenerate kernel|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 [[#References|[a2]]]; see also [[#References|[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 ([[#References|[a1]]]). Numerical methods for integral equations of the first kind are the so-called "regularization methodregularization methods" (cf. [[Regularization method|Regularization method]], [[#References|[a4]]]). | A thorough discussion of numerical methods for linear integral equations of the second kind including Fortran programs can be found in [[#References|[a2]]]; see also [[#References|[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 ([[#References|[a1]]]). Numerical methods for integral equations of the first kind are the so-called "regularization methodregularization methods" (cf. [[Regularization method|Regularization method]], [[#References|[a4]]]). |
Revision as of 22:12, 5 June 2020
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 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) |
Integral equations, numerical methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_equations,_numerical_methods&oldid=14561