Abstract 
In this study, we consider the initial value problem
u_t(t,x)+a(x)u_x(t,x)+δu(t,x)=f(t,x),00 and u_0(x) (x∈(0,l)) and f(t,x) (t,x)∈(0,T)×(0,l) ) are given smooth functions and they satisfy every compatibility conditions which guarantees problem (1) has a smooth solution u(t,x). The positivity of the space operator A generated by problem (1) in the space C with maximum norm is established. The structure interpolation spaces of the space operator A are investigated. The positivity of this space operator in Hölder spaces is established.
The finite difference method for the approximate solution of problem (1) is presented. The positivity of the difference analogy of the space operator generated by this problem in the space C with maximum norm is established. The structure interpolation spaces generated by this difference operator are studied. The positivity of this difference operator in Hölder spaces is established.
Following some mathematical models [13], in application the mathematical modeling of lake pollution is described and an example is given.
