• A. Silaev State University of Management
  • G. Parshikova State University of Management
  • A. Perfilyev State University of Management



distributed lag, exogenous and endogenous variables, degenerate kernel, Fredholm integral equation, resolvent, eigenvalues of the integral operator.


In this paper, the authors propose to model the dynamic processes of economics and ecology not by differential, but by integral Fredholm (or Volterra) equations. Since with this approach, the overwhelming majority of simultaneous values of indicators representing exogenous and endogenous variables of the corresponding economic and mathematical models are considered, the authors attempt to synthesize distributed lag models (continuous and discrete) with Fredholm (Volterra) integral equations. The integral core, which is part of the transforming operator, transfers its accumulating action, transforming, in essence, an exogenous indicator into an endogenous one. Three schemes are proposed, according to which the resulting integral indicator is realized in relation to a stimulus with a lagging argument (lag). The synthesis of the distributed lag model and the model, the apparatus of which is the Fredholm and Volterra integral equations, will increase the reliability (probability of reliability) of the forecast estimates of the economic indicators of the models.

Author Biographies

A. Silaev , State University of Management

PhD in Economics, associate Professor

G. Parshikova , State University of Management

Senior Lecturer

A. Perfilyev , State University of Management

PhD of Physico-mathematical Sciences,  associate Professor


Amel'kin V. V. Differencial'nye uravnenija v prilozhenijah. M.: URSS; 2021.

[Amelkin VV Differential equations in applications. Moscow: URSS; 2021. (In Russ).]

Arnol'd V. I. Teorija katastrof. M.: Lenand; 2020. [Arnold VI Theory of catastrophes. Moscow: Lenand; 2020. (In Russ).]

Volkov E. A. Chislennye metody. M.: Lan'; 2021. [Volkov EA Tchislennii metodi. Moscow: Lan; 2020. (In Russ).]

Kolmogorov A.N., Fomin S.V. Jelementy teorii funkcij i funkcional'nogo analiza. M.: MGU im. Lomonosova; 2019. [Kolmogorov AN, Fomin SV Elementy teorii funktcii i funktcionalnogo analiza. Moscow: Lomonosov Moscow State

University; 2019. (In Russ).]

Parshikova G. Ju., Perfil'ev A. A., Silaev A. A. Linejnye integral'nye uravnenija Fredgol'ma vtorogo roda v prilozhenijah jekonomiki // Innovacii i investicii. 2020. 9: 162-169. [Parshikova GYu, Perfilyev AA, Silaev AA (2020). Lineinie integralnie uravneniya Fredholma vtorogo roda v prilozhenyii k ekonomike. Inovatsii i investitsii. 2020. 9: 162 -169. (In Russ).]

Poljanin A. D., Manzhirov A. V. Integral'nye uravnenija. Chast' 1: spravochnik dlja vuzov. M.: Jurajt; 2017. [Polyanin AD, Manzhirov AV Integral Equations. Tchast 1: spravochnik dlya vuzov. Moscow: Yurait; 2017. (In Russ).]

Silaev A. A., Parshikova G. Ju.

Issledovanie struktury investicionnogo laga v modeli razvitija jekonomiki goroda M.: Sputnik+; 2009. [Parshikova GYu, Silaev AA The study of the structure of the investment lag in the model of the development of the city's economy. Moscow:

Sputnik+; 2009. (In Russ).]

Hoang Lan Vu, Kelvin Tsun WaiNG, Damien Bolingbroke Time-lagged effects of weekly climatic and socio-economic factors on ANN municipal yard waste prediction models // Waste Management. 2019. 84: 129-140.

Luo H, Liu J, Liu X, Yang Y Predicting temporal violations for parallel business cloud workflows // Software: Practice and Experience. 2018. 48(4): 775-795.

Ustaoglu E, Lavalle C Examining lag effects between industrial land development and regional economic changes // PLOS one. 2017. 12 (9). https: //