Design and Implementation of MATLAB GUI

Study Case Design and Implementation of MATLAB GUI as a Medium for Analyzing Ordinary Differential Equations Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Mathematical equations are one way to represent various natural phenomena and everyday life. Many of these phenomena can be approached using Ordinary Differential Equations (ODE), both first-order and second-order, to aid the analysis process. ODE has been proven capable of describing physical phenomena such as falling objects, organism growth, and even socio-economic phenomena such as population dynamics and stock price movements. Numerical solutions to ODEs, such as the second-order Runge–Kutta (RK2) method, fourth-order Runge–Kutta (RK4) method, Euler method, and Modified Euler method, are important alternatives because many equations are difficult to solve analytically. The implementation of these numerical solutions through a MATLAB-based Graphical User Interface (GUI) is expected to become an interactive learning medium that facilitates the understanding of concepts and the visualization of natural phenomena and everyday life. Keywords: First- and second-order ODEs; Numerical solutions; GUI; MATLAB. INTRODUCTION Differential equations are a type of equation closely related to the concept of change (derivative). Many natural phenomena exhibit change over time, making differential equations a highly relevant mathematical tool for analyzing them. The use of differential equations is not a new approach. Various studies have utilized them, for example, to analyze bridge cable tension, heat transfer on a conductor plate, population growth over time, population estimates, and river flow simulations. All of these cases can be modeled using differential equations. MICHAEL KURNIAWAN 1 Study Case One of the most commonly used types is the Ordinary Differential Equation (ODE), both first and second order. These equations can range from simple to complex, depending on the phenomenon being modeled. ODEs are often used in various fields such as electrical circuit analysis, solute transport in rivers, and mechanical and thermodynamic systems. The methods for solving ODEs vary greatly depending on the approach used and the complexity of the equation. For example, the characteristic method is used for linear equations, the Adams–Bashforth–Moulton method is used for homogeneous equations with constant coefficients, and the boundary condition analysis approach is used for more complex cases. However, even though ODEs play an important role in various fields of science and engineering, this topic remains a challenge for many students. The complexity of the equation form and the number of solution methods are the main causes of this difficulty. To overcome these obstacles, numerical methods can be used as an alternative solution. The main advantage of the numerical approach is that it can provide solutions for almost all forms of ODEs without the need for explicit analytical solutions. However, this method requires a fairly long iterative process to achieve high accuracy, depending on the form of the equation being solved. Thus, numerical methods are considered a powerful approach to solving ODEs. Based on these challenges, this study proposes the development of a user-friendly MATLAB-based application to analyze first- and second-order ODEs using several numerical methods, such as second- and fourth-order Runge–Kutta, Euler, and Modified Euler. Although these methods have been widely used, most applications are still limited to specific cases or phenomena. With this Graphical User Interface (GUI)-based MATLAB application, users—both students and researchers—can solve various forms of ODEs without the need for complex manual programming. The equations to be analyzed can be entered directly through the application interface. METHOD The solution of first- and second-order ordinary differential equations (ODEs) in this study was performed using a numerical approach, namely the second-order Runge– Kutta (RK2) method, fourth-order Runge–Kutta (RK4) method, Euler method, and Modified Euler method. All of these methods were applied with relatively simple numerical equations and could be implemented directly in MATLAB (Sihombing & MICHAEL KURNIAWAN 2 Study Case Dahlia, 2018). The main medium used was the MATLAB Graphical User Interface (GUI), which was designed to be easy to use. This GUI allows users to solve ODEs numerically by entering several initial parameters such as initial conditions, final time values, and the desired number of iterations. This study was divided into three main stages: 1. First stage: Developing numerical solution formulations for first- and second-order ODEs. 2. Second stage: Designing the MATLAB GUI display and logic to match the form of the equations to be analyzed. 3. Stage three: Evaluating the numerical results by comparing them to the analytical solutions for simple ODEs. The developed MATLAB GUI is capable of solving ODEs in the form of x(t), its derivative v(t), and its second derivative. Numerical Formulation of Second-Order ODEs 1. Second-Order Runge–Kutta Method (RK2): 𝑡(𝑖 + 1) = 𝑡(𝑖) + ℎ; 𝑘1𝑣 = ℎ ⋅ 𝑑𝑓1(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); 𝑘1𝑥 = ℎ ⋅ 𝑑𝑓2(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); 𝑘2𝑣 = ℎ ⋅ 𝑑𝑓1(𝑡(𝑖) + ℎ/2, 𝑣(𝑖) + 𝑘1𝑣/2, 𝑥(𝑖) + 𝑘1𝑥/2); 𝑘2𝑥 = ℎ ⋅ 𝑑𝑓2(𝑡(𝑖) + ℎ/2, 𝑣(𝑖) + 𝑘1𝑣/2, 𝑥(𝑖) + 𝑘1𝑥/2); 𝑣(𝑖 + 1) = 𝑣(𝑖) + 𝑘2𝑣; 𝑥(𝑖 + 1) = 𝑥(𝑖) + 𝑘2𝑥; 2. Fourth-Order Runge–Kutta Method (RK4): 𝑡(𝑖 + 1) 𝑘1𝑣 𝑘1𝑥 𝑘2𝑣 𝑘2𝑥 𝑘3𝑣 𝑘3𝑥 𝑘4𝑣 𝑘4𝑥 𝑣(𝑖 + 1) 𝑥(𝑖 + 1) MICHAEL KURNIAWAN = 𝑡(𝑖) + ℎ; = ℎ ⋅ 𝑑𝑓1(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); = ℎ ⋅ 𝑑𝑓2(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); = ℎ ⋅ 𝑑𝑓1(𝑡(𝑖) + ℎ/2, 𝑣(𝑖) + 𝑘1𝑣/2, 𝑥(𝑖) + 𝑘1𝑥/2); = ℎ ⋅ 𝑑𝑓2(𝑡(𝑖) + ℎ/2, 𝑣(𝑖) + 𝑘1𝑣/2, 𝑥(𝑖) + 𝑘1𝑥/2); = ℎ ⋅ 𝑑𝑓1(𝑡(𝑖) + ℎ/2, 𝑣(𝑖) + 𝑘2𝑣/2, 𝑥(𝑖) + 𝑘2𝑥/2); = ℎ ⋅ 𝑑𝑓2(𝑡(𝑖) + ℎ/2, 𝑣(𝑖) + 𝑘2𝑣/2, 𝑥(𝑖) + 𝑘2𝑥/2); = ℎ ⋅ 𝑑𝑓1(𝑡(𝑖) + ℎ, 𝑣(𝑖) + 𝑘3𝑣, 𝑥(𝑖) + 𝑘3𝑥); = ℎ ⋅ 𝑑𝑓2(𝑡(𝑖) + ℎ, 𝑣(𝑖) + 𝑘3𝑣, 𝑥(𝑖) + 𝑘3𝑥); = 𝑣(𝑖) + (𝑘1𝑣 + 2𝑘2𝑣 + 2𝑘3𝑣 + 𝑘4𝑣)/6; = 𝑥(𝑖) + (𝑘1𝑥 + 2𝑘2𝑥 + 2𝑘3𝑥 + 𝑘4𝑥)/6; 3 Study Case 3. Euler Method: 𝑡(𝑖 + 1) = 𝑡(𝑖) + ℎ; 𝑣(𝑖 + 1) = 𝑣(𝑖) + ℎ ⋅ 𝑑𝑓1(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); 𝑥(𝑖 + 1) = 𝑥(𝑖) + ℎ ⋅ 𝑑𝑓2(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); 4. Modified Euler Method: 𝑡(𝑖 + 1) 𝑣new 𝑣(𝑖 + 1) 𝑥new 𝑥(𝑖 + 1) = 𝑡(𝑖) + ℎ; = 𝑣(𝑖) + ℎ ⋅ 𝑑𝑓1(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); = 𝑣(𝑖) + ℎ ⋅ 𝑑𝑓1(𝑡(𝑖), 𝑣new , 𝑥(𝑖)); = 𝑥(𝑖) + ℎ ⋅ 𝑑𝑓2(𝑡(𝑖), 𝑣(𝑖), 𝑥(𝑖)); = 𝑥(𝑖) + ℎ ⋅ 𝑑𝑓2(𝑡(𝑖), 𝑣(𝑖), 𝑥new ); Numerical Formulation of First-Order ODEs 1. Second-Order Runge–Kutta Method (RK2): 𝑥(𝑖 + 1) = 𝑥(𝑖) + ℎ; 𝑘1𝑦 = ℎ ⋅ 𝑑𝑓1(𝑥(𝑖), 𝑦(𝑖)); 𝑘2𝑦 = ℎ ⋅ 𝑑𝑓1(𝑥(𝑖) + ℎ/2, 𝑦(𝑖) + 𝑘1𝑦/2); 𝑦(𝑖 + 1) = 𝑦(𝑖) + 𝑘2𝑦; 2. Fourth-Order Runge–Kutta Method (RK4): 𝑥(𝑖 + 1) = 𝑥(𝑖) + ℎ; 𝑘1𝑦 = ℎ ⋅ 𝑑𝑓1(𝑥(𝑖), 𝑦(𝑖)); 𝑘2𝑦 = ℎ ⋅ 𝑑𝑓1(𝑥(𝑖) + ℎ/2, 𝑦(𝑖) + 𝑘1𝑦/2); 𝑘3𝑦 = ℎ ⋅ 𝑑𝑓1(𝑥(𝑖) + ℎ/2, 𝑦(𝑖) + 𝑘2𝑦/2); 𝑘4𝑦 = ℎ ⋅ 𝑑𝑓1(𝑥(𝑖) + ℎ, 𝑦(𝑖) + 𝑘3𝑦); 𝑦(𝑖 + 1) = 𝑦(𝑖) + (𝑘1𝑦 + 2𝑘2𝑦 + 2𝑘3𝑦 + 𝑘4𝑦)/6; 3. Euler Method: 𝑥(𝑖 + 1) = 𝑥(𝑖) + ℎ; 𝑦(𝑖 + 1) = 𝑦(𝑖) + ℎ ⋅ 𝑑𝑓1(𝑥(𝑖), 𝑦(𝑖)); 4. Modified Euler Method: 𝑥(𝑖 + 1) = 𝑥(𝑖) + ℎ; 𝑦new = 𝑦(𝑖) + ℎ ⋅ 𝑑𝑓1(𝑥(𝑖), 𝑦(𝑖)); 𝑦(𝑖 + 1) = 𝑦(𝑖) + ℎ ⋅ 𝑑𝑓1(𝑥(𝑖), 𝑦new ); In all of the above formulations, the df1 and df2 functions represent the form of the ODE to be analyzed. Additional inputs required include the initial conditions of the variable x, its first derivative v, the end time t_“end”, and the number of iterations. The difference between the four numerical methods lies in the level of discretization. Because differential equations are derivative in nature, the more iterations used, the closer the numerical results will be to the analytical solution. With a large enough number of iterations, all methods will produce relatively similar solutions. MICHAEL KURNIAWAN 4 Study Case RESULTS The MATLAB-based Graphical User Interface (GUI) application that was developed is equipped with a popup menu feature to make it easier for users to select the desired Ordinary Differential Equation (ODE) solution method. The main GUI display is shown in Figure 1, where users can freely enter ODE functions, while the output is displayed in the form of a graph of the numerical solution of the equation. To test the application's precision level, a simulation of the free-falling object phenomenon is used, as shown in Figure 2. In addition, Figure 3 displays a question dialogue that helps users—especially students—operate the application more easily and purposefully. Figure 1. MATLAB GUI interface display Figure 2. Evaluation of MATLAB GUI on the phenomenon of free-falling objects MICHAEL KURNIAWAN 5 Study Case Figure 3. Question Dialogue display on MATLAB GUI DISCUSSION This MATLAB-based Graphical User Interface (GUI) application was developed as an interactive alternative learning medium. The application is designed to analyze Ordinary Differential Equations (ODEs) using four numerical methods, namely secondorder Runge–Kutta (RK2), fourth-order Runge–Kutta (RK4), Euler, and Modified Euler. These four methods differ in how they perform discretization of differential equations. As shown in Figure 1, the GUI interface requires input in the form of the first and second derivatives of x(t) for second-order ODEs, and the first derivative of x(t) for first-order ODEs. In MATLAB, these functions are defined as df1 and df2. The program will then calculate the numerical solution based on the method selected via the popup menu. In addition to the ODE function itself, the application also requires several input parameters such as the initial time value (t_0), initial position value (x_0), initial velocity value (v_0), and number of iterations. As an example of application, Figure 2 shows the use of the MATLAB GUI to analyze the phenomenon of free fall. This phenomenon describes the motion of an object affected by the acceleration due to gravity, with the initial condition that the object is at rest. In the simulation, the initial value x=100 meters is given—indicating that the object is dropped from a height of 100 meters—and the initial velocity v=0, which means that the object is not moving at the start of the experiment. The input function for the second derivative x is -10, representing an acceleration due to gravity of -10 m/s^2. The negative sign indicates the MICHAEL KURNIAWAN 6 Study Case direction of acceleration towards the center of the earth. Thus, this model represents the conditions of a free-falling object without air resistance. After all inputs are entered and the numerical method is selected, the application will display the simulation results in graphical form: position x(t) versus time, velocity v(t) versus time, and a combined graph of both. The results in Figure 2 show that the object's velocity increases negatively (towards the bottom) over time, while the object's position decreases rapidly to near zero. This visualization is consistent with the physics theory of free fall motion. In addition to the falling object phenomenon, this application can also be used for various other cases in science and engineering, because the ODE function used can be customized by the user. Thus, this application is not limited to just one phenomenon. To help users understand the modeling process, the MATLAB GUI is equipped with a question dialogue as shown in Figure 3. This feature provides guidance and recommendations on the steps to take when users want to solve first- and second-order ODEs. The system will display different messages depending on the type of ODE being analyzed. If the user ignores these suggestions, the application will refuse to run the calculation process until the input is adjusted to the correct format. The user-friendly GUI design aims to prevent input errors so that the simulation results remain accurate and consistent with the phenomenon being analyzed. However, this application still has limitations. For example, the GUI does not yet support the solution of high-dimensional ODEs (multi-dimensional systems) and does not yet include various other numerical methods such as Adams–Bashforth or Predictor– Corrector. In addition, testing by users such as students or researchers has not been carried out, so the effectiveness and efficiency of the application as a learning medium cannot yet be measured quantitatively or qualitatively. This kind of evaluation is important for developing the next version to be more comprehensive and applicable in the context of education and research. CONCLUSION The application of numerical methods for solving Ordinary Differential Equations (ODEs) through a MATLAB-based GUI has proven to be an effective learning medium for understanding various natural phenomena that are modeled mathematically. Numerical methods such as second-order Runge–Kutta (RK2), fourth-order Runge– MICHAEL KURNIAWAN 7 Study Case Kutta (RK4), Euler, and Modified Euler are capable of providing accurate results for both simple and complex ODE forms. The main difference between the methods lies in the discretization method, but the results will approach the analytical solution when the number of iterations is increased. ODEs are one of the important mathematical formulas in explaining various physical and biological phenomena, but they are often considered difficult by students. Therefore, interactive learning media such as MATLAB GUI can be an interesting tool to improve understanding of ODE concepts. Unfortunately, this study has not included direct testing of its effectiveness on users. Further research is needed to assess quantitatively and qualitatively the extent to which this application is effective in improving learning. Further development could also include adding support for highdimensional systems, other numerical methods, and integration with digital learning platforms commonly used in the United States, such as MATLAB Online or Simulink. MICHAEL KURNIAWAN 8 Study Case REFERENCE MICHAEL KURNIAWAN 9 Study Case MICHAEL KURNIAWAN 10