2. Integral equations
• Any equation where an unknown function, to be
determined appears under one or more integral
signs it can be called an integral equation. (if the
derivatives of the function are involved it is
called an integro -differential equation)
• The actual development of the theory of integral
equations began only at the end of the Integral
functions are such where unknown is only
outside the integral sign
2

3. Classification of integral
equations
Fredholm equation of the first type
b
f x K x, t t dt
a
Fredholm equation of the second type
b
x f x K x, t t dt
a
3

4. Classification of integral
equations
Volterra equation of the first type
x
f x K x, t t dt
a
Volterra equation of the second type
x
x f x K x, t t dt
a
4

5. Classification of integral equations
Limits of the integration fixed: Fredholm
One limit not fixed: Volterra
Unkonwn function only inside integral: first kind
Also outside integral: second kind
Known function f(x)=0 for all x: homogeneous
f(x) 0 for some x:
inhomogeneous
Not all integral equations can be classified with the
above.
5

6. Integral functions
• When differential equations are solved,
integration is the final step. Sometimes
there is no analytical solution to the
integral. In those cases the solution is given
in terms of integral functions
x
2
Error function erf x exp t 2 dt
0
• Solution of diffusion equation
• Probability theory (cumulative normal distribution)
6

7. Some integral functions
Gamma function
• Generalization of factorial n! x t x 1e t dt
0
Incomplete gamma
function x, t x 1e t dt
• Some bubble breakage models 0
/2
d
Complete elliptic integral Kk
• Pendulum movement 0 1 k 2 sin 2
There are solution methods (series solution
etc.) for these integrals. They are also widely
tabulated 7

8. Distributions
y
fs y s, L g L dL
0
This is a Fredholm equation of the first kind:
L
b
f x K x, t t dt
a
Notation does not matter, important is to identify what is the unknown or
known function, distributed property etc.
s x, y K, g ,L t
8