Strip method (integral equations)

A method for the approximate solution of one-dimensional Fredholm integral equations of the second kind (cf. also Fredholm equation; Fredholm equation, numerical methods), based on replacing the kernel in a special way by a degenerate kernel, evaluating the resolvent of the degenerate equation and then improving the approximate solution through the use of a rapidly-convergent iterative algorithm.

Let the original integral equation be written as

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

To construct the degenerate kernel, divide the square

$$ \{ a \leq x \leq b , a \leq s \leq b \} $$

into $ N $ strips

$$ \left \{ b- \frac{a}{N} i \leq x \leq b- \frac{a}{N} ( i + 1 ) , a \leq s \leq b \right \} ,\ \ i = 0 \dots N - 1 . $$

In each strip, say the $ i $- th, the function $ K ( x , s ) $ is approximated in the mean square, or uniformly, by functions

$$ K _ {i} ( x , s ) = C _ {i} ( x) + P _ {i} ( x) Q _ {i} ( s) . $$

In the simplest case,

$$ K _ {i} ( x , s ) = K ( \xi _ {i} , s ) ,\ \ \xi _ {i} \in \left [ b- \frac{a}{N} i , b- \frac{a}{N} ( i + 1 ) \right ] . $$

The function $ K _ {i} ( x , s ) $ is now used to construct a degenerate kernel:

$$ \tag{2 } K _ {N} ( x , s ) = \sum _ { i= } 0 ^ { N- } 1 [ \widehat{C} _ {i} ( x) + \widehat{P} _ {i} ( x) Q _ {i} ( s)] , $$

$$ \widehat{P} _ {i} ( x) = \left \{ \begin{array}{ll} P _ {i} ( x) , &\ x \in \left [ b- \frac{a}{N} i , b- \frac{a}{N} ( i + 1 ) \right ] , \\ 0, & x \notin \left [ b- \frac{a}{N} i , b- \frac{a}{N} ( i + 1 ) \right ] , \\ \end{array} $$

$$ \widehat{C} _ {i} ( x) = \left \{ \begin{array}{ll} C _ {i} ( x) , &\ x \in \left [ b- \frac{a}{N} i , b- \frac{a}{N} ( i + 1 ) \right ] , \\ 0, & x \notin \left [ b- \frac{a}{N} i , b- \frac{a}{N} ( i + 1 ) \right ] . \\ \end{array} $$

The solution of the equation with the degenerate kernel (2) approximates the solution of equation (1), generally, the better the larger the number $ N $ of strips and the better the approximation of $ K ( x , s ) $ in each strip is. The approximate solution $ \phi _ {0} ( x) $ can be further improved by using the iterative algorithm

$$ \tag{3 } \phi _ {k} ( x) - \lambda \int\limits _ { a } ^ { b } K _ {N} ( x , s ) \phi _ {k} ( s) d s = $$

$$ = \ f ( x) + \lambda \int\limits _ { a } ^ { b } [ K ( x , s ) - K _ {N} ( x , s ) ] \phi _ {k-} 1 ( s) d s . $$

The iterations (3) converge in the mean square, or uniformly, to the solution of equation (1), provided the kernels $ K _ {N} ( x , s ) $ approach $ K ( x , s ) $.

References

[1]	G.N. Polozhii, P.I. Chalenko, "The strip method for solving integral equations" Dop. Akad. Nauk UkrSSR : 4 (1962) pp. 427–431 (In Ukrainian) ((English abstract.))

Comments

Excellent surveys on the numerical solution of Fredholm equations of the second kind can be found in [a1] and [a2]; the strip method is not mentioned there, although other degenerate kernel methods are discussed, cf. also Degenerate kernels, method of.

References

[a1]	K.E. Atkinson, "A survey of numerical methods for the solution of Fredholm integral equations of the second kind" , SIAM (1976)
[a2]	C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4