Existence and uniqueness of the solution to the problem of increase in the main factors of production

This study considers the solution to the problem of finding the growth of production factors at known rates of investment development, given initial and boundary conditions of economic development. The solution process begins with the analysis of a homogeneous parabolic problem, for which the analytical solution is known in advance. In the process of studying, we find the eigenfunctions and eigenvalues of the Sturm – Liouville problem, which are necessary to calculate the growth of production factors and describe the dynamics of the economic system. To solve the problem under consideration, taking into account the influence of investment development rates, we formulate a new problem using the values of the growth of production factors specified in two extreme time intervals. In the case of continuity of the investment development rate function, it is possible to reduce the resulting problem to a known one-dimensional spatial problem of heat conductivity, for which the existence and uniqueness theorem has been proven. We apply the method of decomposing the solution into homogeneous and inhomogeneous parts. We write the general solution as the sum of the solution of the homogeneous problem and an additional component that considers the investment development rate. To find this additional component, we use the parameter variation method and approaches related to functions that satisfy the corresponding boundary conditions. In the final theoretical part of the work, we obtain a general solution, which consists of the sum of solutions of a homogeneous problem and additional functions that consider the influence of the speed of investment deve lopment. This approach allows us to obtain a complete picture of the behavior of production factors when investment conditions change and provides a basis for further research in the field of modeling economic processes. Numerical modeling of the growth of fixed assets in the United States in the period 2011–2022 was carried out, which allowed a detailed study of the dynamics of changes in fixed assets depending on time and volume. The Frankel – Dufort scheme was used, which showed good results, which indicates its effectiveness in predicting changes in the structure of fixed assets. The proposed model allowed us to identify key trends and dependencies, as well as determine possible development scenarios for making informed management decisions.

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