simple maths equations

Chapter 7 INTEGRAL EQUATIONS




Chapter 7 Integral Equations



7.1 Normed Vector Spaces
1. Euclidian vector space \ n
2. Vector space of continuous functions C G ( )
3. Vector Space L2 ( G )
4. Cauchy-Bunyakowski inequality
5. Minkowski inequality

7.2 Linear Operators
- continuous operators
- bounded operators
- Lipschitz condition
- contraction operator
- successive approximations
- Banach fixed point theorem

7.3 Integral Operator

7.4 Integral equations
- Fredholm integral equations
- Volterra integral equations
- integro-differential equations
- solution of integral equation

7.5 Solution Methods for Integral Equations
1. Method of successive approximations for Fredholm IE
(Neumann series)
2. Method of successive substitutions for Fredholm IE
(Resolvent method)
3. Method of successive approximations for Volterra IE

7.6 Connection between integral equations and initial and boundary value
problems

1. Reduction of IVP to the Volterra IE
2. Reduction of the Volterra IE to IVP
3. Reduction of BVP to the Fredholm IE

7.7 Exercises


Future Topics:

7.7 Fixed point theorem (see [Hochstadt “Integral equations”, p.25]) (added in 7.2)
Elementary existence theorems

7.8 Practical applications (see [Jerri “Introduction to Integral Equations with Applications”])

7.9 Inverse problems (see [ Jerri, p.17])

7.10 Fredholm’s alternatives

,Chapter 7 INTEGRAL EQUATIONS


7.1 Normed Vector Spaces
We will start with some definitions and results from the theory of normed vector
spaces which will be needed in this chapter (see more details in Chapter 10).

1. Euclidian vector space \ n The n-dimensional Euclidian vector space consists of all points
{
\ n = x = ( x1 ,x2 ,...,xn ) xk ∈ \ }
for which the following operations are defined:

Scalar product ( x, y ) = x1 y1 + x2 y2 + ... + xn yn x, y ∈ \ n

Norm x = ( x,x ) = x12 + x22 + ... + xn2
Distance ρ ( x, y ) = x − y
Convergence lim xk = x if lim x − xk = 0
k →∞ k →∞
n
\ is a complete vector space (Banach space) relative to defined norm x .


2. Vector space C G( ) ( )
Vector space C G consists of all real valued continuous functions defined on
the closed domain G ⊂ \ n :
( ) {
C G = f ( x ) : D ⊂ \ n → \ continuous }
Norm f C
= max f ( x )
x∈G

Convergence lim f k = f if lim f − f k C
=0
k →∞ k →∞

( )
C G is a complete vector space (Banach space) relative to defined norm f C
.


3. Vector space L2 ( G ) The space of functions integrable according to Lebesgue (see Section 3.1)
 
L2 ( G ) =  f ( x ) : G ⊂ \ n → ^ ∫ f ( x)
2
dx < ∞ 
 G 
Inner product ( f ,g ) = ∫ f ( x ) g ( x )dx
G

( f , f ) = ∫ f ( x)
2
Norm f 2
= dx
G
The following property follows from the definition of the Lebesgue integral


∫ f (x )dx ≤ ∫ f (x )dx
G G



L2 ( G ) is a complete normed vectors spaces (Banach spaces) relative to f 2
.

4. Cauchy-Bunyakovsky-Schwarz Inequality (see also Theorem 10.1, p.257)

( f ,g ) ≤ f 2
⋅ g 2
for all f ,g ∈ L2 ( G )

Proof:
If f ,g ∈ L2 ( G ) , then functions f , g and any combination α f + β g are
also integrable and therefore belong to L 2 (G ) .
Consider

f + λ g ∈ L 2 (G ) , λ ∈ R for which we have

, Chapter 7 INTEGRAL EQUATIONS


0 ≤ ∫ ( f + λ g ) dx = ∫ f dx + 2λ ∫ fg dx + λ 2 ∫ g dx
2 2 2

G G G G


The right hand side is a quadratic function of λ . Because this function is non-
negative, its discrimenant ( D = b 2 − 4ac ) is non-positive

2
   2  2 
4  ∫ fg dx  − 4  ∫ f dx   ∫ g dx  ≤ 0
  
G  G  G 
2
   2  2 
∫
 fg dx  ≤  f dx   g dx 
G     ∫ ∫
  G G 

and because ( f , g ) = ∫ fgdx ≤ ∫ fg dx ≤ ∫ f g dx ,
G G G


( f ,g)
2 2 2
≤ f 2
⋅ g 2


from which the claimed inequality yields

( f ,g) ≤ f 2
⋅ g 2

■
5. Minkowski Inequality (3rd property of the norm “Triangle Inequality”), (see Example 10.7 on p.257)



f +g 2
≤ f 2
+ g 2
for all f ,g ∈ L2 ( G )

Proof:

= ( f + g, f + g)
2
Consider f +g 2

= ( f , f ) + ( f , g ) + (g , f ) + (g , g )
+ ( f , g ) + (g , f ) + g
2 2
≤ f 2 2
2 2
≤ f 2
+2 f 2
g 2
+ g 2
from C-B inequality

= f ( 2
+ g 2
) 2




Then extraction of the square root yields the claimed result. ■

Note that the Minkowski inequality reduces to equality only if functions f and
g are equal up to the scalar multiple, f = αg , α ∈ R (why?).