Computational Mathematics and Modeling, Vol. 15, No. 2, 2004

NUMERICAL METHODS

SOLVING AN INTEGRAL EQUATION OF THE FIRST KINDBY SPLINE APPROXIMATION

V. I. Dmitriev and Zh. Ingtem UDC 519.6

A method based on a special minimum-derivative spline is proposed for solving a Fredholm in-tegral equation of the first kind.

Introduction

Consider the integral equation

a

b

K x u d f x∫ =( , ) ( ) ( )ξ ξ ξ . (1)

Its solution is known to be unstable, and this equation is solved by special regularization methods [1]. Thebasic regularization approach ensures that the solution is contained in a compact set, e.g., boundedness of the de-rivative produces a compactum in C.

If the algorithm for Eq. (1) stipulates a bounded derivative, it produces a stable solution of an integralequation of the first kind. One of the appropriate techniques uses a spline approximation of the solution with aminimum-norm derivative of the spline function.

1. Constructing the Spline-Interpolation Function

We need to construct a quadratic spline that passes through the values of the function u x( ) at the points

x a n hn = + −( )1 , n N∈[ ]1, , where hb a

N= −

− 1, x a1 = , x bN = . Represent the spline function in the form

S x ux x

hu

x x

h

x x x x

hnn

nn

nn n( )

( )( )= − + − + − −+

+ +1

1 1α ,

(2)x x xn n∈[ ]+, 1 , n N∈ −[ ]1 1, ,

′ = − + + −( )+ +S xh

u u x x xn n n n n( ) ( )1

21 1α , (3)

where u u xn n= ( ).We seek the spline S x C( ) ∈ 1. Then the function and its derivative are continuous at the grid nodes

x xn= , n N∈ −[ ]2 1, , and their continuity gives conditions for the determination of the coefficient αn,

Translated from Prikladnaya Matematika i Informatika, No. 14, pp. 5–10, 2003.

1046–283X/04/1502–0099 © 2004 Plenum Publishing Corporation 99

100 V. I. DMITRIEV AND ZH. INGTEM

n N∈ −[ ]1 1, :

u u

h

u u

hn n

nn n

n+ + +

+− − = − +1 2 1

1α α , n N∈ −[ ]1 2, .

Hence we find that each element αn is expressible in terms of its predecessor:

α αn nn n nu u u

h++ += − − − +

11 22

, n N∈ −[ ]1 2, . (4)

Expression (4) enables us to express all αn in terms of α1:

α αnn k K K K

k

n u u u

h= − + − − +− + +

=

−

∑( ) ( )1 121

11 2

1

1

, n N∈ −[ ]2 1, . (5)

The parameter α1 is chosen so that the norm ′S x L( )2

2 is minimized. Using (3), we obtain a problem

for α1:

minα

αn

n

n u u

h

x x x

hdxn n

nn n

x

x

n

N+ +

=

− − + + −

+

∫∑ 1 12

1

1 21

. (6)

From (5) we have d

dn nα

α1

11= −( ) − , and we thus obtain the condition for the minimum (6) in the form

( )− − + + −

+ − =− + + +

=

− +

∫∑ 12 2

01 1 1 1

1

1 1n n n

nn n n n

x

x

n

N u u

h

x x x

h

x x x

hdx

n

n

α . (7)

Substituting in (7) the expression for αn in terms of α1 (5), we obtain an equation for α1:

u u

h

x x x

h

x x x

hdx

x

x2 1

12 1 2 12 2

1

2 − + + −

+ −∫ α

+ ( ) ( )− − + − + −

−

=

−+ − +∑ ∫

+

1 121

2

11 1

11

1n

n

Nn n n n n

x

xu u

h

x x x

hn

n

α

+ ( )− − +

+ −

+ − =+ +

=

−+ +∑ 1

2 2 201 2

21

11 1k k k k

k

nn n n nu h u h u h

h

x x x

h

x x x

hdx .

Hence we finally obtain

α1 1 21

1

2

111 1 2=

−− − − ++ +

=

−

=

−

∑∑b au u un k

k k kk

n

n

N

( ) ( ) ( ). (8)

SOLVING AN INTEGRAL EQUATION OF THE FIRST KIND BY SPLINE APPROXIMATION 101

Substituting (8) in (5), we obtain an expression for αn for n N∈ −[ ]2 1, :

αn b au u un n k

k k kk

n

n

N

= −−

− − − +

−

+ +=

−

=

−

∑∑( ) ( ) ( ) ( )11

1 1 211 2

1

1

2

1

+ ( )− − ++ +

=

−

∑ 12 1 2

1

1k K K K

k

n u u u

h. (9)

We have thus completely specified the spline-interpolation function S ( x ) defined by expressions (2), (8),and (9). The spline depends linearly on the values of the grid function u u xn n= ( ) and it is the smoothestamong all possible splines. This property enables us to interpolate the grid function using a spline with aminimum-norm derivative. This spline efficiently computes the derivative of a tabularly defined function. Itcan be applied to solve integral equations of the first kind.

2. Solution of Fredholm Integral Equation of the First Kind

Consider the solution of Eq. (1) by the spline function constructed in the previous section. Substitutingspline (2) in Eq. (1), we obtain

K x y ux x

hu

x x

h

x x x x

hdx f y

x

x

m

N

mm

mm

mm m

m

m

( , )( )( )

( )+

∫∑=

−

++ +− + − + − −

=1

1

1

11 1α . (10)

Denote

1 1

hK x y x x dxn m

x

x

mn

m

m

( , )( )− =+

∫ β ,

11

1

hK x y x x dxn m

x

x

mn

m

m

( , )( )+ − =+

∫ θ , (11)

11

1

hK x y x x x x dxn m m

x

x

mn

m

m

( , )( )( )− − =+

+

∫ γ .

Satisfying Eq. (10) at the grid points y yn= , y a n hn = + −( )1 , n N∈[ ]1, , hb a

N= −

− 1, we obtain a linear

algebraic system

( )u u fm mn m mn m mnm

N

n+=

−+ + =∑ 1

1

1

β θ α γ , n N∈[ ]1, , (12)

where αn are linearly expressible by (8), (9) in terms of uk , k N∈[ ]1, . The coefficients (11) are expressiblein terms of the local moments of the kernel of the integral equation:

102 V. I. DMITRIEV AND ZH. INGTEM

δ ξ ξmn m n

h

K x y d= +∫ ( , )0

,

β ξ ξ ξmn m n

h

hK x y d= +∫1

0

( , ) , (13)

ϕ ξ ξ ξmn m n

h

hK x y d= +∫1 2

0

( , ) .

Then we have

θ δ βmn mn mn= − , γ β ϕmn mn mnh= − , (14)

and Eq. (12) takes the form

u u h fm mn m mn mn m mn mnm

N

n+=

−+ − +( ) =∑ 1

1

1

β δ β α β ϕ( ) ( - ) , n N∈[ ]1, . (15)

Rewrite (8), (9) in a compact form

α αm mk kk

N

u==

∑1

. (16)

From Eq. (15) after some transformations we finally obtain a system for u u yk k= ( ):

u h u hn n m mn mnm

N

N N n mN mn mnm

N

1 1 1 11

1

11

1

( ) ( ) ( )δ β α β ϕ β α β ϕ− + −

+ + −

=

−

−=

−

∑ ∑

+ u u h fm m n mn mn kk

N

mk mn mnm

N

m

N

nβ δ β α β ϕ−=

−

=

−

=

−+ −( ) + −

=∑ ∑∑ 1

2

1

1

1

2

1

( ) ( ) ,

n N∈[ ]1, .

This linear system produces a stable solution of the Fredholm equation of the first kind. The solution isregularized because we use a bounded-derivative spline.

The method has been tested for the equation

ln ( )y x l u x dx f y−( ) +( )⋅ = ( )∫ 2 2

0

1

, y ∈[ ]0 1, .

The tests have shown that the solution is sufficiently accurate when the number of points does not exceed afew tens. With N = 50 the solution deviates from the exact solution (Figs. 1 and 2). In these figures the exact

SOLVING AN INTEGRAL EQUATION OF THE FIRST KIND BY SPLINE APPROXIMATION 103

solution is plotted by the continuous curve, and the approximate solution slightly oscillates about the exact solu-tion. This effect is attributable to errors in the spline function that arise when a large number of points are used.If we have to take a large number of grid points N, then an exact solution can be obtained by partitioning theintegration interval into several sections. A separate spline function is constructed on each section and we obtaina system of equations. In our tests the interval was divided into two parts and the spline solution was very closeto the exact solution.

Fig. 1

Fig. 2

104 V. I. DMITRIEV AND ZH. INGTEM

The study has been supported by the Russian Foundation for Basic Research grants 02-01-00300 and 03-05-64167.

REFERENCES

1. A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems [in Russian], Nauka, Moscow (1986).