Solving a Sequence of Recurrence Relations for First-Order Differential Equations

Abstract

A whole category of engineering and economic problems can be reduced to solving a set of differential equations. Downsides of known approaches for their solutions include limited accuracy numerical methods with stringent requirements for computational power. A direct analytical solution should be derived to eliminate such flaws. This research intends to derive such a solution for an n-dimensional set of recurrence relations for first-order differential equations, linearly dependent on the right side. The research methodology relies on successive integration of the considered set in view of the initial conditions. The overall solution was derived as a sum of products of exponential multipliers with constant coefficients that are defined through weights of a tree graph, which is a descriptor of successive integration. An analytical solution for an n-dimensional set of recurrent differential equations in view of the initial conditions has been derived for the first time in this research.

Development and experimental verification of the mathematical model of thermal inertia for a branched heat supply system

“…The values of the components Ψ can be obtained by recurrent substitution with sequential integration according to the methodology of [38].…”

“…Value of t y nTN can be found from the solution of the system of auxiliary balances: (22) System (20) is transformed with regard to the transition to a dimensionless complex: (23) System (21) is transformed into the recurrence relation: (24) Solution to relation (22) was obtained as follows:…”

“…The exact solution to the problem of modeling the unsteady temperature condition of heating networks, taking into account their heat-accumulating properties and the multivariate confi guration (it is necessary to consider the temperature condition of a complex of consecutive sections from the heat supply source to a specifi c consumer) can be obtained by solving a recurrently related system n TN of differential equations: (28) where In addition to system (22), the system of auxiliary heat balances is used.…”

“…can be reduced to a sequence of recurrent relations of fi rst order differential equations with a linear dependence in the right-hand side. Such tasks include thermodynamic analysis models [4][5][6][7][8][9][10][11], heat and power system models [12][13][14][15][16], combustion models [17][18][19][20][21][22], models of local heat exchange systems (for example, solar collectors [23][24][25][26][27][28]), and others. Nowadays, systems of homogeneous and inhomogeneous differential equations are one of the main tools for mathematical modeling of physical processes [14].…”

“…Most of the modeling problems are reduced to the numerical solution of systems of differential equations in this system [34] and similar ones. Such studies can be exemplifi ed by modeling projects (based on numerical solutions) of hybrid solar photovoltaic-thermal systems [35][36][37][38], as well as by the projects on modeling the optimal strategy of heating control planning using green technologies based on wind energy [39]. Advanced intelligent tools for modeling daily distribution schedules include neural networks based on various control signal generation algorithms, such as described in [40,41].…”