Systems of Fredholm Integral Equations

Systems of Volterra and Fredholm integral equations have attracted much concern in applied sciences. The systems of Fredholm integral equations appear in two kinds. The system of Fredholm integral equations of the first kind [1–5] reads 11.1 $$\begin{gathered} {f_1}\left( x \right) = \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ {f_2}\left( x \right) = \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ \end{gathered} $$ where the unknown functions u(x) and v(x) appear only under the integral sign, and a and b are constants. However, for systems of Fredholm integral equations of the second kind, the unknown functions u(x) and v(x) appear inside and outside the integral sign. The second kind is represented by the form 11.2 $$\begin{gathered} u\left( x \right) = {f_1}\left( x \right) + \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ v\left( x \right) = {f_2}\left( x \right) + \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt. \hfill \\ \end{gathered} $$ The systems of Fredholm integro-differential equations have also attracted a considerable size of interest. These systems are given by 11.3 $$\begin{gathered} {u^{\left( i \right)}}\left( x \right) = {f_1}\left( x \right) + \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ {v^{\left( i \right)}}\left( x \right) = {f_2}\left( x \right) + \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ \end{gathered} $$ where the initial conditions for the last system should be prescribed.