Question

(2) (Volterra Integral Theoretical) Consider the equation (1.3) o(t) + k(t – $)() dě = f(t), in which f and k are known functions, and o is to be determined. Since the unknown function o appears under an integral sign, the given equation is called an integral equation; in particular, it belongs to a class of integral equations knowns as Volterra integral equations. Take the Laplace transform of the given integral equation and obtain an expression for L(o(t)) in terms of the trans- forms L(f(t)) and L(k(t)) of the given functions f and k. The inverse transform of L(o(t)) is the solution of the original integral equation.

Answer 1

Soln. Here, the volterra integnal aquation is given by- sue +6) + Sk (4-5) ¢CE) df = 4(1) $(t) = f(A) - Jkle-&) & (&) dę 1 heces) & (8) dę - 0 and om Now, given that there we have to obtain expression for 2 ($(t)) in terms of the transform 2 ( f(t)) and 2 (K(b)) of the giren function of andk. let the Laplace transforms of the function f (t) and f(t) are given by - 2 Cf, (ts) and L (f (t)). The Laplace convolution product of these two functions is defined by it (f, * })(+) 1 t, (A - {) f (CE) d is given by and the laptace transformation of the convolution {{{4*430}=-{$8cept($34944 (40), 4 (4 (9) Thus, taking taplace transformation on the equation ® e get- 2 (&(t)) = 2 (f(t)) - 2 (k (t)) . 2 (6 (t)) L(& (t)) LCH (t)) 1+ Lek(t))