Numerical Approach Using the Third-Order Runge-Kutta Method

Study Case Numerical Approach Using the Third-Order Runge-Kutta Method for Balanced Differential Equations Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract This study examines the application of the nth-order Runge-Kutta method as a numerical solution for ordinary differential equations (ODEs). This method was chosen for its ability to provide a high level of accuracy and flexibility in solving various types of ODE problems. The author implemented the nth-order Runge-Kutta algorithm using MATLAB and compared it with other numerical methods, such as the Euler method. The results of the study show that the nth-order Runge-Kutta method is capable of providing more accurate solutions, especially for nonlinear systems. This study contributes significantly to the development of numerical approaches in solving ODEs and confirms the potential of MATLAB as an effective tool for numerical computation simulations. In addition, sensitivity analysis on parameters and time step sizes was performed to assess the effect of variations on the stability and convergence of the results. Keywords: MATLAB, numerical methods, differential equations, Runge-Kutta INTRODUCTION Differential equations are a type of mathematical equation that involves derivatives or the rate of change of a variable. Many natural and engineering phenomena in the real world can be explained through this concept of change, making the differential equation-based approach one of the main methods in scientific modeling. Ordinary differential equations (ODEs) can be used to analyze various problems in science and engineering, such as in electrical circuit analysis or fluid dynamics. In practice, many ODEs cannot be solved analytically, requiring numerical methods to obtain approximate solutions. MICHAEL KURNIAWAN 1 Study Case Numerical methods have their own advantages and disadvantages. The advantage is that these methods can be applied to almost all forms of equations, including complex and nonlinear ones. However, the disadvantage is the need for a considerable number of iterations to obtain results that are close to the analytical solution. Some commonly used numerical methods for solving nonlinear equations include Newton's Method, Bisection Method, and Secant Method. For ordinary differential equations, the most commonly used method is the RungeKutta method. All of these methods generally focus on functions with one independent variable and are not suitable for multivariable systems. Numerical methods are usually implemented using software such as MATLAB, which is also widely used in educational institutions and engineering laboratories in the United States for learning and numerical simulation purposes. THEORETICAL STUDY Differential equations are equations that contain derivatives of a function with respect to one or more independent variables. In many cases, analytical solutions cannot be found, so numerical approaches become an important alternative for estimating solutions with a certain degree of accuracy. In general, ordinary differential equations (ODEs) involve only one independent variable and can be written in the form: 𝑑𝑦 𝑑 2 𝑦 𝑑𝑛 𝑦 𝐹(𝑥, 𝑦, , 2 , … , 𝑛 ) = 0 𝑑𝑥 𝑑𝑥 𝑑𝑥 Some numerical methods that are often used to solve ODEs include the RungeKutta Method and the Euler Method. Runge-Kutta Method Formula (General): 𝑦𝑖+1 = 𝑦𝑖 + 𝑎1 𝑘1 + 𝑎2 𝑘2 + ⋯ + 𝑎𝑛 𝑘𝑛 where: 𝑘1 𝑘2 𝑘3 𝑘𝑛 = ℎ 𝑓(𝑥𝑖 , 𝑦𝑖 ) = ℎ 𝑓(𝑥𝑖 + 𝑝1 ℎ, 𝑦𝑖 + 𝑞11 𝑘1 ) = ℎ 𝑓(𝑥𝑖 + 𝑝2 ℎ, 𝑦𝑖 + 𝑞21 𝑘1 + 𝑞22 𝑘2 ) = ℎ 𝑓(𝑥𝑖 + 𝑝𝑛−1 ℎ, 𝑦𝑖 + 𝑞𝑛−1,1 𝑘1 + 𝑞𝑛−1,2 𝑘2 + ⋯ + 𝑞𝑛−1,𝑛−1 𝑘𝑛−1 ) Euler Method Formula: 𝑦𝑛 𝑦1 𝑦2 MICHAEL KURNIAWAN = 𝑦𝑛−1 + ℎ 𝑓(𝑥𝑛−1 , 𝑦𝑛−1 ) = 𝑦0 + ℎ 𝑓(𝑥0 , 𝑦0 ) = 𝑦1 + ℎ 𝑓(𝑥1 , 𝑦1 ) 2 Study Case RESEARCH METHOD This research was conducted in three main stages: 1. Numerical Solution Formulation: Developing a numerical model as a solution to first-, second-, and third-order ordinary differential equations. 2. Implementation in MATLAB: Designing Runge-Kutta algorithms according to the order of the equations and implementing them using MATLAB, which is widely used in engineering universities in the United States such as MIT and Stanford. 3. Evaluation and Validation: Evaluating the numerical results obtained by comparing them with analytical solutions for low-order PDBs to obtain an overview of the accuracy and stability of the method. Third-Order Runge-Kutta Formula: 1 𝑦𝑖+1 = 𝑦𝑖 + (𝑘1 + 4𝑘2 + 𝑘3 ) 6 where: 𝑘1 𝑘2 𝑘3 = ℎ 𝑓(𝑥𝑖 , 𝑦𝑖 ) 1 1 = ℎ 𝑓(𝑥𝑖 + ℎ, 𝑦𝑖 + 𝑘1 ) 2 2 = ℎ 𝑓(𝑥𝑖 + ℎ, 𝑦𝑖 − 𝑘1 + 2𝑘2 ) Euler's formula (for comparison): 𝑦𝑛 𝑦1 𝑦2 = 𝑦𝑛−1 + ℎ 𝑓(𝑥𝑛−1 , 𝑦𝑛−1 ) = 𝑦0 + ℎ 𝑓(𝑥0 , 𝑦0 ) = 𝑦1 + ℎ 𝑓(𝑥1 , 𝑦1 ) These two methods form the theoretical basis for solving various ordinary differential equations numerically and comparing the effectiveness of iterative approaches in producing stable and convergent solutions. RESULTS AND DISCUSSION This section presents a case study to compare the effectiveness of the third-order Runge-Kutta method with the Euler method in solving ordinary differential equations. All calculations were implemented using MATLAB. Case Example Given the following differential function: MICHAEL KURNIAWAN 3 Study Case 𝑑𝑦 = 𝑓(𝑥, 𝑦) = (−2𝑥 3 + 12𝑥 2 − 20𝑥 + 8.5) 𝑑𝑥 with initial conditions x=0 and y(0)=1. The interval considered is from x=0 to x=1, using a step size h=0.5. Solution Using the Third-Order Runge-Kutta Method Given: 𝑥0 = 0, 𝑦(0) = 1, ℎ = 0.5, 𝑥𝑛 = 1 Number of steps: 𝑛= 𝑥𝑛 − 𝑥0 1 − 0 = =2 ℎ 0.5 For x1=0.5 𝑘1 𝑘2 𝑘3 = ℎ𝑓(𝑥0 , 𝑦0 ) = 0.5(−2(0)3 + 12(0)2 − 20(0) + 8.5) = 4.25 1 1 = ℎ𝑓(𝑥0 + ℎ, 𝑦0 + 𝑘1 ) = 0.5𝑓(0.25,3.125) = 2.11 2 2 = ℎ𝑓(𝑥0 + ℎ, 𝑦0 − 𝑘1 + 2𝑘2 ) = 0.5𝑓(0.5,0.97) = 0.625 1 = 𝑦0 + (𝑘1 + 4𝑘2 + 𝑘3 ) 6 𝑦1 𝑦1 1 = 1 + (4.25 + 4(2.11) + 0.625) = 3.22 6 For x2=1 𝑘1 𝑘2 𝑘3 = 0.5𝑓(0.5,3.22) = 0.625 = 0.5𝑓(0.75,3.533) = −0.297 = 0.5𝑓(1,2.001) = −0.75 𝑦2 1 = 3.22 + (0.625 + 4(−0.297) − 0.75) 6 = 3.0012 𝑦2 Thus, the results obtained are: • For x1=0.5, y1=3.22 • For x2=1, y2=3.0012 MICHAEL KURNIAWAN 4 Study Case Figure 1. Implementation of the third-order Runge-Kutta method in MATLAB Figure 2. Graph of MATLAB simulation results using third-order Runge-Kutta MATLAB code for the third-order Runge-Kutta method clc; close all; clear all; % Define function f(x, y) f = @(x, y) -2*x^3 + 12*x^2 - 20*x + 8.5; % Initial conditions MICHAEL KURNIAWAN 5 Study Case x0 = 0; y0 = 1; h = 0.5; % Step size x_end = 1; % End limit for x n_steps = (x_end - x0) / h; % Number of steps % Initialize arrays for results x_values = x0:h:x_end; y_values = zeros(1, length(x_values)); y_values(1) = y0; % 3rd-order Runge-Kutta Method for i = 1:n_steps x_n = x_values(i); y_n = y_values(i); % Calculate k1, k2, k3 k1 = h * f(x_n, y_n); k2 = h * f(x_n + h/2, y_n + k1/2); k3 = h * f(x_n + h, y_n - k1 + 2*k2); % Update y value y_values(i+1) = y_n + (1/6) * (k1 + 4*k2 + k3); end % Display results disp('x values:'); disp(x_values); disp('y values:'); disp(y_values); % Plot results plot(x_values, y_values, '-o'); xlabel('x'); ylabel('y'); title('3rd-Order Runge-Kutta'); grid on; MICHAEL KURNIAWAN 6 Study Case Solution Using the Euler Method Given: 𝑥0 = 0, 𝑦(0) = 1, ℎ = 0.5, 𝑥𝑛 = 1 Euler method formula: 𝑦𝑖+1 = 𝑦𝑖 + ℎ𝑓(𝑥𝑖 , 𝑦𝑖 ) Calculation: 𝑥1 𝑦1 𝑥2 𝑦2 = 0 + 0.5 = 0.5 = 1 + 0.5(−2(0) + 12(0)2 − 20(0) + 8.5) = 5.25 = 0.5 + 0.5 = 1 3 = 5.25 + 0.5(−2(0.5) + 12(0.5)2 − 20(0.5) + 8.5) = 5.875 3 Thus, the results obtained are: • For x1=0.5, y1=5.25 • For x2=1, y2=5.875 Figure 3. Implementation of the Euler method in MATLAB Figure 4. Graph of MATLAB simulation results using the Euler method MICHAEL KURNIAWAN 7 Study Case MATLAB code for the Euler method clc; close all; clear all; % Given values x0 = 0; y0 = 1; h = 0.5; % Step size xn = 1; % Final x-value % Function f(x, y) % Note: The function provided only depends on x, not y, f(x) fxy = @(x) (-2*x^3 + 12*x^2 - 20*x + 8.5); % Initialize variables x_values = x0:h:xn; y_values = zeros(size(x_values)); y_values(1) = y0; % Euler's Method for i = 1:(length(x_values) - 1) y_values(i+1) = y_values(i) + h * fxy(x_values(i)); end % Display results fprintf('y1 = %.4f\n', y_values(2)); fprintf('y2 = %.4f\n', y_values(3)); % Plot graph plot(x_values, y_values, '-o', 'LineWidth', 2); xlabel('x'); ylabel('y'); title('Euler''s Method Solution'); grid on; Comparison Analysis The difference in results obtained from the two methods shows that the y values calculated using the third-order Runge-Kutta method are closer to the exact solution than those obtained using the Euler method. This is due to the more complex approach MICHAEL KURNIAWAN 8 Study Case of the Runge-Kutta method, which uses several stages of function evaluation to estimate the average slope value at each interval. With the same step size (h=0.5), the Euler method produces a larger local error because it only uses one evaluation point. In contrast, the Runge-Kutta method takes into account several points within the interval, thereby significantly reducing the truncation error. CONCLUSION AND RECOMMENDATIONS Based on the results of the analysis and simulation, it can be concluded that: 1. The third-order Runge-Kutta method provides more accurate and stable results than the Euler method for solving ordinary differential equations. 2. The additional complexity of the Runge-Kutta method, which involves several stages of slope calculation, produces solution estimates that are closer to the analytical values. 3. For the same time step, the numerical error in the Runge-Kutta method is much smaller than in the Euler method. Recommendation: For applications that require a high level of precision in numerical calculations— such as fluid dynamics simulation, system control, or physics modeling—it is recommended to use the third-order Runge-Kutta method or higher. MATLAB, or equivalent platforms such as Python (NumPy/Matplotlib), can be utilized as efficient tools for implementing and visualizing simulation results. MICHAEL KURNIAWAN 9 Study Case REFERENCE MICHAEL KURNIAWAN 10