Integral Equation

If the limits are fixed, an integral equation is called a Fredholm integral equation. If one limit is variable, it is called a Volterra integral equation. If the unknown function is only under the integral sign, the equation is said to be of the ``first kind.'' If the function is both inside and outside, the equation is called of the ``second kind.'' A Fredholm equation of the first kind is of the form

$$f(x) = \int^b_a K(x,t)\phi(t)\,dt.$$ (1)

A Fredholm equation of the second kind is of the form

$$\phi (x) = f(x) + \int^b_a K(x,t)\phi(t)\,dt.$$ (2)

A Volterra equation of the first kind is of the form

$$f(x) = \int^x_a K(x,t)\phi(t)\,dt.$$ (3)

A Volterra equation of the second kind is of the form

$$\phi (x) = f(x)+ \int^x_a K(x,t)\phi(t)\,dt,$$ (4)

where the functions $K(x,t)$ are known as Kernels. Integral equations may be solved directly if they are Separable. Otherwise, a Neumann Series must be used.

A Kernel is separable if

$$K(x,t) = \lambda \sum_{j=1}^n M_j(x)N_j(t).$$ (5)

This condition is satisfied by all Polynomials and many Transcendental Functions. A Fredholm Integral Equation of the Second Kind with separable Kernel may be solved as follows:

$$\phi(x) = f(x)+\int^b_a K(x,t)\phi (t)\,dt$$

$$= f(x)+\lambda \sum_{j=1}^n M_j(x) \int^b_a N_j(t)\phi (t)\,dt$$

$$= f(x)+\lambda \sum_{j=1}^n c_jM_j(x),$$ (6)

where

$$c_j \equiv \int^b_a N_j(t)\phi (t)\,dt.$$ (7)

Now multiply both sides of (7) by $N_i(x)$ and integrate over $dx$.

$$\int^b_a \phi(x)N_i(x)\,dx= \int^b_a f(x)N_i(x)\,dx +\lambda\sum_{j=1}^n c_j\int^b_a M_j(x)N_i(x)\,dx.$$ (8)

By (7), the first term is just $c_i$. Now define

$$b_i \equiv \int^b_a N_i(x)f(x)\,dx$$ (9)

$$a_{ij} = \int^b_a N_i(x)M_j(x)\,dx,$$ (10)

so (8) becomes

$$c_i = b_i + \lambda \sum_{j=1}^n a_{ij}c_j.$$ (11)

Writing this in matrix form,

$${\bf C}={\bf B}+\lambda {\hbox{\sf A}}{\bf C},$$ (12)

so

$$({\hbox{\sf I}}-\lambda{\hbox{\sf A}}){\bf C} = {\bf B}$$ (13)

$${\bf C} = ({\hbox{\sf I}}-\lambda{\hbox{\sf A}})^{-1}{\bf B}.$$ (14)

See also Fredholm Integral Equation of the First Kind, Fredholm Integral Equation of the Second Kind, Volterra Integral Equation of the First Kind, Volterra Integral Equation of the Second Kind

References

Integral Equations

Corduneanu, C. Integral Equations and Applications. Cambridge, England: Cambridge University Press, 1991.

Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. New York: Dover, 1962.

Kondo, J. Integral Equations. Oxford, England: Clarendon Press, 1992.

Lovitt, W. V. Linear Integral Equations. New York: Dover, 1950.

Mikhlin, S. G. Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd rev. ed. New York: Macmillan, 1964.

Mikhlin, S. G. Linear Integral Equations. New York: Gordon & Breach, 1961.

Pipkin, A. C. A Course on Integral Equations. New York: Springer-Verlag, 1991.

Porter, D. and Stirling, D. S. G. Integral Equations: A Practical Treatment, from Spectral Theory to Applications. Cambridge, England: Cambridge University Press, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Integral Equations and Inverse Theory.'' Ch. 18 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 779-817, 1992.

Tricomi, F. G. Integral Equations. New York: Dover, 1957.