It is proposed to develop numerical methods for solving partial differential equations with nonlinearity in the differentiation operator and with the effect of functional heredity. Among the studied equations are one-dimensional and multidimensional transport equations, equations of parabolic and hyperbolic types, as well as equations with fractional spatial and temporal derivatives with heredity effect. The basic approach is related to the separation of the history of the object and its current state. This approach, combined with suitable methods of multidimensional interpolation and extrapolation, allows to build the analogues of numerical algorithms used for objects without delay. Systems of non-linear difference equations are supposed to be reduced to sequences of linear systems; to prove the convergence a special technique is developed. Another essential point is the use of the common property of time delay and fractional derivatives with respect to time – memory. It is supposed to create open source software packages and apply them for solving a number of evolutionary problems of mathematical modeling in population theory, immunology and in the theory of control and stability.