Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations

Leibnitz in 1663 introduced the derivative notation for the order of natural numbers, and then the idea of fractional derivatives appeared. Only a century later, this idea began to be realized with the discovery of the concepts of fractional derivatives by several mathematicians, including Riemann (1832), Grundwal, Fourier, and Caputo in 1969. The concepts in the definitions of fractional derivatives by Riemann-Liouville and Caputo are more frequently used than other definitions, this paper will discuss the Grunwald-Letnikov (GL) operator, which has been discovered in 1867. This concept is less popular when compared to the Riemann-Liouville and Caputo concepts, however, this concept is quite interesting because the concept of derivation is developed from the definition of ordinary derivatives. In this paper will be shown that the formulas for the fractional derivative using the GL concept are the same as the results obtained using the Riemann-Liouville and Caputo concepts. As a complement, we will give an example of solving a fractional differential equation using Modified Homotopy Perturbation Methods

Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras

In this paper, we study low-dimensional Filiform Lie algebras. Specifically, three-dimensional standard Filiform Lie algebras and five-dimensional non-standard Filiform Lie algebras. The classification method was given in the following stage. For given a low-dimensional Filiform Lie algebra, we compute its second centre. We showed that three-dimensional Filiform Lie algebra-called Heisenberg Lie algebra-is type and + as well. On the other hand, for ≥3, the standard Filiform Lie algebras are type but not type +. In this case, we give a concrete example of case five-dimensional Heisenberg Lie algebra. Moreover, we proved that five-dimensional non-standard Filiform Lie algebra is type but not type +. It is still an open problem to classify types and + for the general case of non-standard Filiform Lie algebra of dimension ≥6

Factors Affecting the Number of Infant Morality Cases in West Java for the 2019-2020 Period using Generalized Poisson Regression (GPR)

The number of infant mortality cases is data in the form of counts which is modeled by Poisson regression. There is an assumption that needs to be met, namely equidispersion. Equidispersion is a condition in which the mean and variance of the variables are the same, but in practice this assumption is often not met. There are two possible events, namely overdispersion and underdispersion. The Generalized Poisson Regression (GPR) model is one solution to solve this problem. In estimating the GPR parameter, the Maximum Likelihood Estimation (MLE) method is used, but the derivation of the log-likelihood function does not always produce explicit results, so the Newton-Raphson iteration method is used. Poisson regression analysis conducted on the number of infant mortality cases in West Java showed that the model had overdispersion as seen from the value of the dispersion parameter which was more than zero, so the GPR model was used. Parameter significance test was carried out on three factors, namely the poverty gap index , the percentage of low birth weight infants , and the percentage of exclusive breastfeeding for infants  the results obtained that all factors affected the number of infant mortality cases in West Java

Application of Single Index Model to Determine Optimal Stock Portfolio (A Case Study on IDX30 in 2022)

Stock represent proof of ownership or participation of an individual or entity in a company. Investors gain profits from shares through capital gains and dividends. The difficulty in selecting an optimal composition of a stock portfolio is a major concern for investors. This study aims to determine the optimal composition of a stock portfolio, calculate the expected returns in the future, and assess the potential risks that investors may encounter later on. The data for this research consists of stocks listed on the IDX30 Index throughout the year 2022, which consistently appear in every six-month evaluation. The analysis is conducted using a single-index model. Based on the findings of this study, the following ten stocks are identified as the optimal portfolio constituents: KLBF with a weight of 17.20%, BBRI with a weight of 17.18%, BBCA with a weight of 17.08%, PTBA with a weight of 12.46%, BBNI with a weight of 9.89%, UNVR with a weight of 8.33%, INKP with a weight of 8.66%, ICBP with a weight of 5.56%, BMRI with a weight of 3.25%, and UNTR with a weight of 0,39%. The expected return from the formed portfolio is 0,1% per day, with a corresponding risk of 0,004%

Application of Threshold Generalized Autoregressive Conditional Heteroscedastic (TGARCH) Model in Forecasting the LQ45 Stock Price Return

Economics is one of the most important fields for a country. One of the activities that illustrate the importance of the economy in a country is an investment. Investment activities, especially stock investment, are included in the capital market activities that various age groups currently carry out. Stocks are generally known to have high-risk, high-return characteristics. Therefore we need a way to minimize losses in investing. This study uses time series analysis theory to analyze LQ45 stock data.The data used is the closing price of PT. Bank Central Asia, Tbk. obtained from finance.Yahoo.com. The results of this study indicate that the return of daily closing price data of PT. Bank Central Asia, Tbk. during the period 2017-2021, there are heteroscedasticity and asymmetric shocks, so variations of the ARCH/GARCH model are needed to obtain accurate forecasting results. One suitable model is Threshold GARCH (TGARCH). The results of this study indicate that the suitable forecasting model for the data is the MA(3)-TGARCH(1,1) model. The model produces forecasts with an accuracy rate based on MAPE of 0.895% for the next seven day

Mathematical Model of Paddy Production using Cobb Douglas Method Based On Weather Factors

This research was conducted to model paddy production based on weather factors. This needs to be done to predict crop yields and regulate paddy cropping patterns. In setting the cropping pattern, the weather is selected which consists of temperature, wind speed, and rainfall, as a variable factor of production. Meanwhile, other factors (such as fertilization, sunshine, air humidity, etc.) are assumed to be in catteries paribus conditions. The research method used is a mixed method between qualitative methods which are descriptive details and quantitative methods which are based on weather data and Paddy's harvest data. The aim of this research is to analyze the influence of weather on paddy production results. Analysis is done to get the production function. Parameters are estimated using the Ordinary Least Square (OLS) method by minimizing the sum of squared errors. Based on data analysis, a correlation of 0.899 was obtained with a standard error of .051665515. the results of model testing also show significant results with the F statistic obtained at 33.98 with a p-value of 0.028 which is less than 5%. So it can be concluded that there is a significant relationship between weather and paddy productivity. In such a way that the weather can be used as a reference in determining the prediction of loss risk and paddy production. This model can also be recommended for further research, namely to determine insurance losses that may arise when extreme weather events occur.

Estimasi Parameter Model Regresi Nonparametrik Birespon berdasarkan Penalized Spline Pada Data Tindak Kriminal di Indonesia (Studi Kasus Jumlah Kejadian Kejahatan terhadap Kesusilaan dan Jumlah Kejadian Kejahatan terhadap Fisik di Indonesia Tahun 2020)

Untuk mencapai terciptanya kehidupan bermasyarakat yang aman dan damai, tindak kriminal menjadi salah satu hal yang sangat diperhatikan. Pada tahun 2020, di Indonesia terjadi 6.872 kejadian kejahatan terhadap kesusilaan dan 36.672 kejadian kejahatan terhadap fisik. Salah satu upaya yang bisa dilakukan untuk menekan jumlah kejadian kejahatan terhadap kesusilaan dan jumlah kejadian kejahatan terhadap fisik di Indonesia adalah dengan memodelkan hal tersebut atas faktor-faktor yang memengaruhinya sehingga dapat diperoleh prediksinya.  Pada penelitian ini, dilakukan estimasi parameter model regresi nonparametrik birespon berdasarkan estimator penalized spline menggunakan pendekatan metode Weighted Least Square (WLS) untuk memprediksi jumlah kejadian kejahatan terhadap kesusilaan dan jumlah kejadian kejahatan terhadap fisik di Indonesia dengan variabel prediktor kepadatan penduduk (X1), rasio jenis kelamin (X2), persentase penduduk miskin (X3) dan rata-rata upah bersih buruh/karyawan/pegawai (X4). Estimator penalized spline digunakan untuk memperhitungkan titik knot dan parameter penghalus secara bersamaan sehingga menghasilkan ketepatan dan kehalusan bentuk kurva secara simultan. Model terbaik bergantung pada penentuan titik knot dan parameter pemulus optimal yaitu dengan nilai Generalized Cross Validation (GCV) minimum. Model terbaik diperoleh saat banyaknya titik knot untuk X1 adalah satu, X2 adalah tiga, X3 adalah tiga, dan X4 adalah satu serta lambda=0,000000171 dengan GCV sebesar 568359 dan nilai koefisien determinasi sebesar 0,652

Interest Rate in Pension Plan Premium Calculation

This research aims to analyze the relationship between the interest rate relationship is inversely proportional to the amount of the premium on the pension plan. The method used is to measure several variables, among others FSL (Future Service Liability), PVFSAL (Present Value Future Salary), PR (Pension Rate) and Premiums. Calculation, life annuity uses actuarial assumptions, one of which is the interest rate assumption, if the assumptions used are not in accordance with the actual conditions, then what happens is excessive payments or deficient payments. The interest rate has an influence in the process of calculating the defined benefit pension plan premium. Using the assumption of different interest rates (11%, 12 % and 13%), it is found that the interest rate relationship is inversely proportional to the amount of the premium. The results of this study are FSL, PVFSAL, PR and Premiums for the interest of 11%, 12% and 13% (participants aged 25 years) as follows 720,187.97; 554,000,24; 430,570.07 (FSL in Rupiah); 27,155,187.70; 24,922,770,59; 23,002,699.40 (PVFSAL in Rupiah); 2.6521; 2.2229; 1.8718 (PR in%) and 55,535.38; 46,546.85; 39,196.00 (Premiums in Rupiah)The higher the interest rate, the smaller the pension premium and vice versa

ANALISIS KORELASI KANONIK PERILAKU BELAJAR TERHADAP PRESTASI BELAJAR SISWA SMP (STUDI KASUS SISWA SMPN I SUKASARI PURWAKARTA)

Abstrak: Analisis korelasi kanonik merupakan salah satu teknik analisis multivariat yang dapat digunakan untuk mengidentifikasi dan mengukur hubungan linier yang melibatkan lebih dari satu variabel independen dan lebih dari satu variabel dependen. Pada paper ini analisis korelasi kanonik digunakan untuk menganalisis kaitan antara perilaku belajar terhadap prestasi belajar yaitu prestasi akademik siswa SMPN I Sukasari Purwakarta. Terdapat tiga kriteria yang digunakan untuk melakukan interpretasi fungsi kanonik yaitu bobot kanonik, muatan kanonik, dan muatan silang kanonik. Hasil analisis data menunjukkan bahwa  didapat dua fungsi kanonik. Untuk interpretasi dipilih fungsi kanonik pertama karena dinilai lebih layak berdasarkan uji signifikansi fungsi kanonik secara bersama maupun individu. Hasil interpretasi menunjukkan bahwa perilaku belajar dan prestasi belajar mempunyai kaitan yang cukup kuat. Kontribusi terbesar yang diberikan oleh variabel independen dan dependen adalah intensitas belajar mandiri di luar jam sekolah dan nilaiIPA.       Kata kunci:   Fungsi Kanonik, Bobot Kanonik. Muatan Kanonik, Muatan Silang Kanoni

Parameter Estimation STAR (1;1) Model Using Binary Weight

Space Time Auto Regressive(1;1)  Model or STAR(1;1)  model is a form of model that involves location and time. The STAR(1;1)  model is a stationary space time model in mean and variance. The STAR model can be used to forecast future observations at these locations by involving the effects of observations at other nearby locations in spatial lag 1 and lag time 1 [2]. The STAR model can be written as a linear model assuming that error is normally distributed with zero mean and constant variance. In this research, the parameter estimation procedure for STAR model using binary weight, MKT method and STAR model for the estimation of petroleum production in 3 wells is assumed to be in a homogeneous reservoir