Differential equations are mathematical equations that describe how a function changes over time. In the case of electricity, they are used to calculate the flow of current through a conductor, resistors, capacitors, and inductors, as well as the analysis of more complex circuits. The flow of electricity in a circuit can be modelled using differential equations that describe the behaviour of voltage and current over time. This study will explore the different differential equations commonly used in electrical circuit analysis, including ordinary and partial differential equations (Kobyzev et al., 2023). The flow of electricity is a very important part of our society. It is responsible for powering devices in our homes and businesses, and it helps us to stay connected to the outside world. The flow of electricity can be complicated, but it can also be modelled using ordinary differential equations (ODEs). This essay will discuss the use of ODEs in calculating the flow of electricity and show how they can be used to solve problems relating to this vital sector of our economy.
The challenges faced in accurately modelling and analyzing electrical circuits require a numerical solution in the calculation of the flow of electricity. Indeed, the use of electrical circuits in modern technology is applied to various electrical appliances. Therefore, using electrical computational equations in the accurate modelling and analysis of the flow of electricity in these circuits is essential for their design, optimization, and efficient operation (Constantin, 2023). However, the behaviour of electrical circuits can be complex and dynamic, and using simple equations makes it challenging to model and analyze using such traditional methods. This has led to the need for more advanced mathematical models and techniques, such as Ordinary Differential Equations (ODEs), to accurately describe and predict the behaviour of these circuits. ODE in mathematical computation is appropriate for the flow of electricity as it employs initial and boundary conditions. Therefore, proper numerical methods for engineering are needed to ensure accurate measurement and estimation of circuit parameters (Kobyzev et al., 2023). Moreover, the emphasis is on the significance of accurate modelling and analysis of electrical circuits, given the potential safety and economic implications of circuit failures or inefficiencies.
Ordinary Differential Equations are commonly used to compute the flow of electricity in various applications, such as electric circuits, power systems, and electromagnetic devices. The solutions that ODE provides are accurate in modelling the circuit and give a meaningful computation of identifying any source of instability. In most cases, the ODE cannot be solved analytically, and numerical methods must be used. This involves discretizing the ODE by dividing the independent variable (usually time) into discrete points. The ODE is then approximated by a set of algebraic equations that relate the values of the dependent variables at each time point. Many numerical methods are available for solving ODEs, such as Euler’s method. The numerical method is implemented to compute the values of the dependent variables at the next time point.
This research will use a literature review of case studies in scholarly journals, books and articles to review the use of the ODE in solving the flow of electricity in the circuits. This study will investigate using Euler’s method to provide numerical solutions when employing ODE. The first step is to formulate the ODEs describing the behaviour of the studied electrical system. This involves identifying the relevant variables and parameters, such as voltages, currents, resistances, and capacitances, and formulating the differential equations that relate them (Constantin, 2023). The ODEs can be derived from fundamental physical laws, such as Kirchhoff’s, Ohm’s, and Faraday’s laws. The ODEs are then approximated by a set of algebraic equations that relate the values of the dependent variables at each time point.
Further, the study will use Euler’s numerical methods for solving ODEs. The choice of method depends on the specific characteristics of the ODEs and the application requirements. Once the numerical method has been selected, computer software is in use for the computational analysis of the results. In addition, the computation involves handling boundary conditions, initial conditions, and constraints. Further, after the numerical computation, the results are analyzed to identify if there are any sources of error or instability in the numerical method through visualization by plotting graphs.