EIGEN MATHEMATICS JOURNAL

Numerical Solution of the Korteweg-De Vries Equation Using Finite Difference Method

Maulana Rifky Haizar — 2024-02-15

The Korteweg-de Vries (KdV) equation is a nonlinear partial differential equation that has a key role in wave physics and many other disciplines. In this article, we develop numerical solutions of the KdV equation using the finite difference method with the Crank-Nicolson scheme. We explain the basic theory behind the KdV equation and the finite difference method, and outline the implementation of the Crank-Nicolson scheme in this context. We also give an overview of the space and time discretization and initial conditions used in the simulation. The results of these simulations are presented through graphical visualizations, which allow us to understand how the KdV solution evolves over time. Through analysis of the results, we explore the behavior of the solutions and perform comparisons with exact solutions in certain cases. Our conclusion summarizes our findings and discusses the advantages and limitations of the method used. We also provide suggestions for future research in this area.

Forecasting Non-Metal and Rock Mineral (MBLB) Tax Revenue Using the Fuzzy Time Series Markov Chain Method in East Lombok Regency

Baiq Siti Patimah Zohrah — 2024-02-15

Indonesia is one of the countries that is included in a developing countries. Therefore, the Indonesian Goverment is trying to carry out various developments in various regions. Regional development is one of the Indonesian government’s ways of achieving national goals. In carrying out regional development, of course funds are needed as the main source to support the achievement of national development. The main source of funds obtained by the Government comes from Regional Oroginal Income. One source of Regional Oroginal Income is tax. There are various types of taxes managed by the government in East Lombok Regency. One of them is the Non-Metal Minerals and Rocks, which is a tax on the extraction of non-metallic minerals and rock Tax, which is a tax on the extraction of of non-metallic minerals and rocks from natural sources within or on the surface of the earth for use. This Non-Metal and Rock Mineral tax provides quite large revenues for East Lombok district regional taxes. Non-Metal and Rock Mineral tax income is often not constant, meaning that there is an increases and there is a decreases in the amount of income. For this reason, it is necessary to forecast Non-Metal and Rock Mineral tax revenue to predict income in the future. The method used in this study is the FTS Markov Chain order 1 and order 2. Based on the MAPE indicator, the results of forecasting using the FTS Markov Chain method of order 1 amounted to Rp. 1.117.069.497 with an accuracy of 48,55% with a just good forecasting classification. While the results of forecasting using the FTS Markov Chain method of order 2 amounted to Rp.1.761.652.173 with an accuracy of 39,12% with a just good forecasting classification. If seen from the MAPE value obtained, the forecasting results using the 2nd order FTS Markov Chain are more accurate than using the 1st order Markov Chain FTS method.

Solution of The Duffing Equation Using Exponential Time Differencing Method

Ramadian Ridho Illahi — 2024-02-15

To describe the spring stiffening effect that occurs in physics and engineering problems, Georg Duffing added the cubic stiffness term to the linear harmonic oscillator equation and is now known as the Duffing oscillator. Despite its simplicity, its dynamic behavior is very diverse. In this research, the Exponential Time Difference method is introduced to solve the Duffing oscillator numerically. To formulate the ETD method, we were using the integration factors. It is a function which, when multiplied by an ordinary differential equation, produces a differential equation that can be integrated. This method is an effective numerical method for solving complex differential equations, especially equations that have strong non-linearity The ETD method delivers highly accurate numerical solutions for the Duffing oscillator, with minimal discrepancy from the analytical results. Through parameter variation, the ETD method's applicability extends to diverse Duffing oscillator configurations.

Modeling of the Spread of Malaria in the Bangka Belitung Islands Province Using the SEIR Method

Nikken Halim — 2024-04-05

Malaria is an infectious disease caused by plasmodium through the bite of the Anopheles sp. female mosquito. (Roach, 2012). Malaria disease which hit the Bangka Belitung Islands Province in 2005 experienced a spike, reaching 36,901 people out of 981,573 residents and claimed the lives of 12 local residents. In 2011, the Bangka Belitung Islands Province was declared an endemic area for malaria. This research aims to model and interpret the spread of malaria using the SEIR model and predict the spread of malaria using parameter estimates. The steps in analyzing the SEIR model on the spread of malaria are making assumptions, forming a SEIR model, determining the equilibrium point and analyzing the stability of the equilibrium point, determining the basic reproduction number, and carrying out a simulation of the SEIR model that has been obtained. The SEIR model is classified into 4 classes, namely Susceptible (susceptible individuals), Exposed (individuals who have symptoms), Infected (infected individuals), and Recovered (recovered individuals). The data used in this research is data on the number of Susceptible, Exposed, Infected, and Recovered malaria cases in 2022 obtained from the Bangka Belitung Islands Provincial Health Service. The SEIR mathematical model is used to calculate the equilibrium point and basic reproduction number. Based on the SEIR model simulation results, it was found that the susceptible population decreased from the 0^th month to the 48^th month. As for the exposed population, there were 9,623 people in month 0, but in this condition the population decreased drastically per month. Furthermore, for the infected population there were 129 people in month 0, but in this condition the number of infected decreased drastically per month along with the decrease in the exposed population. For individuals who recovered, there was a increase from the 0^th month to the 48^th month.

The ARIMA-GARCH Method in Case Study Forecasting the Daily Stock Price Index of PT. Jasa Marga (Persero)

Ihsan Fathoni Amri — 2024-04-07

PT Jasa Marga is a large company in Indonesia that develop and operation the toll roads and is known as one of the blue chip companies with LQ45 shares. However, share prices have high volatility or rise and fall quickly so their value is always changing. Therefore, forecasting is needed to predict the share price of PT Jasa Marga in the future in order to know the movement of its share price. The Autoregressive Integrated Moving Average (ARIMA) method is a method that can predict data with high volatility, but has the disadvantage of residuals containing heteroscedasticity. So, the addition of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model was carried out to overcome the heteroscedasticity problem that was initially caused by the ARIMA model so it could predict data with high volatility more optimally. Therefore, this research applies the ARIMA-GARCH method to find the best model for forecasting the daily share price index of PT Jasa Marga. The data used comes from the daily closing stock price index of PT Jasa Marga (Persero) for the period January 2015 to May 2023. The measurement of forecasting accuracy uses the Mean Absolute Percentage Error (MAPE). The forecasting results show that the best model uses ARIMA (2,1,1) - GARCH (1,3) with a MAPE value of 6.825728%, which indicates very good forecasting results because the MAPE value is <10%.