The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t={\left(k\left(x\right)u_x(x,t)\right)}_x+F(x,t)$ from the measured data at the final time $u(x,T)=\mu \left(x\right)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J\left(F\right)=\parallel {\mu }_T{\left(x\right)-u(x,T)\parallel }_0^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved....