Simulation of Linear Systems: Analysis and Application in Urban Transportation

چکیده

Linear systems form the foundation of linear algebra and showcase its extensive applications in various fields, including engineering, physics, chemistry, computer science, and others. Mathematics often connects with other sciences indirectly, particularly through simulation. The connection of mathematics, especially linear algebra, with other sciences is established in this way. Network systems, such as electric currents in circuits, the number of services passing through a network system in a unit of time, systems involving the combination and interaction of chemical substances, and more, can be simulated as linear systems. The simulation of dynamic systems (systems with flow) is based on an important hypothesis known as the "Fundamental Hypothesis of System Analysis." This hypothesis states that the total inflow to a system equals the total outflow from the system. Every system contains several points called nodes, which are connected by direct branches or arc-like edges. The fundamental hypothesis of systems is applied to every node within the system. By applying this hypothesis to each node, a linear equation is obtained. From multiple nodes, a system of equations can be constructed. By solving the resulting system of equations, the general flow of the network system can be determined. Furthermore, based on the existing criteria for each system, maximum and minimum values for each flow within the system can also be determined.