H

where k(u,t) is the kernel of the equation. The limits of the integral may be either constants or functions of time. If u, and u2 are constants, Eq. (F-34) is called a Fredholm equation, whereas if u, is a constant and u2=t, then Eq. (F-34) is called a Volterra equation of the second kind [Korn and Korn, 1968]. If the functional form of the kernel may be expressed as k(u,t) = k(u-t) (F-35)

then the Volterra equation

may be solved by Laplace transform methods. Taking the Laplace transform of Eq. (F-36) and rearranging, we obtain y(5)=F(5)/(i-K(i))

where Y(s)= £ (y(t)), F(s)= £ (f(t)), and K(s)= e (*(/)), which may be solved for y(t) by taking the inverse Laplace transform.

References

1. Abramowitz, Milton, and Irene A. Stegun, Handbook of Mathematical Functions. Washington, D.C., National Bureau of Standards, 1970.

2. Churchill, R. V., Operational Mathematics, Second Edition. New York: McGraw-Hill, Inc., 1958.

3. DiStefano, Joseph J., Ill, Allen R. Stubberud, and Ivan J. Williams, Feedback and Control Systems, Schaum's Outline Series. New York: McGraw Hill, Inc., 1967.

4. Erdelyi, A., et al., Tables of Integral Transforms. New York: McGraw Hill, Inc., 1954.

5. Korn, Granino A., and Theresa M. Kora, Mathematical Handbook for Scientists and Engineers. New York: McGraw Hill, Inc., 1968.

6. Spiegel, Murray R., Laplace Transform, Schaum's Outline Series. New York: Schaum Publishing Co., 1965.

0 0

Post a comment