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

THE GARCH MODEL VOLATILITY OF SHARIA STOCKS ASSOCIATED CAUSALITY WITH MARKET INDEX

The purpose of this paper is to examine the volatility of Islamic stocks related to the causality of the composite stock price index (CSPI). The aim is to investigate the causality of several levels of stock returns with the movement of the CSPI, and determine its volatility as a measure of risk. To determine the causality relationship is done by using the granger causality test method, with Vector Autoregressive (VAR) modeling. Whereas to determine the volatility is done using the Generalized Autoregressive Conditional Heteroscedastisiy (GARCH) model approach. The results of the causality test show that there is a direct relationship that affects and is influenced by the CSPI, and the relationship that affects each other between the company's stock market and the movement of the CSPI. While the volatility follows the GARCH model (1, 1). Based on the results of this study are expected to be used as consideration in making investment decisions in the analyzed stocks

Convergence Analysis from the Solution of Riccati’s Fractional Differential Equation by Using Polynomial Least Squares Method

Riccati's Fractional Differential Equation (RFDE) has become a topic of study for researchers because RFDE can model variety of phenomenon in science such as random processes, optimal control and diffusion problems. Phenomena that can be modeled in a mathematical form can make it easier for humans to analyze several things from that phenomenon. RFDE generally does not have an exact solution, therefore a numerical approach solution is needed, one of the methods that gives good accuracy to the actual or exact solution is Polynomial Least Squares, where the errors calculated based on mean absolute percentage error (MAPE) produce a percentage below 1%. In addition, the convergence of a sequence from approximate solutions indicates that the sequence will converge to a solution

Aproksimasi Solusi Persamaan Diferensial Osilator Fraksional Menggunakan Metode Analisis Homotopi Laplace

Persamaan diferensial pecahan telah menarik banyak ahli untuk meneliti lebih dalam karena sangat membantu dalam pemodelan berbagai masalah, seperti persamaan diferensial osilator pecahan. Telah banyak metode yang digunakan untuk menyelesaikan masalah tersebut, diantaranya Metode Analisis Homotopi Laplace yang merupakan gabungan dari Transformasi Laplace dan Metode Analisis Homotopi. Penulis menggunakan metode ini untuk mencari solusi persamaan diferensial osilator pecahan nonlinier. Selanjutnya dapat diamati hubungan konvergensi antara orde persamaan diferensial osilator pecahan dan urutan fungsi solusi persamaan diferensial osilator pecahan

Sistem Chaos Model Risiko Keuangan: Analisis Dinamik

Chaos phenomena appear in dynamic, nonlinear and deterministic systems. One model that is being intensively researched is financial risk. This model has system variables such as interest rate, investment demand, and stock price index. This study shows that the new financial system has interesting characteristics including multistability equilibrium points, Lyapunov exponents and bifurcation diagrams. The results of this study use MATLAB for phase diagrams of the financial system. The Lyapunov exponent and analysis of the Bifurcation diagram have been generated showing the chaotic phenomena in the intervals 0 a 15 and 0 b 0.25. The resulting Kaplan-Yorke dimension is 2.2506. The results of this study can be used to predict financial risk chaos

Analisis Pengaruh Tingkat Suku Bunga Dan Nilai Tukar Terhadap 2 Harga Saham Syariah Dengan Pendekatan Error Correction Model (ECM)

Saham merupakan sekuritas yang memiliki tingkat risiko yang tinggi. Risiko atau kerugian tidak dapat dihilangkan dalam berinvestasi, namun dapat diminimalkan. Oleh sebab itu, untuk meminimalkan tingkat risiko perlu diketahui faktor-faktor apa saja yang memengaruhinya, dan seberapa besar pengaruhnya. Penelitian ini membahas tentang analisis pengaruh variabel tingkat suku bunga dan nilai tukar terhadap harga saham syariah dengan pendekatan Error Correction Model (ECM). Harga saham syariah yang digunakan adalah harga saham syariah Astra Agro Lestari Tbk. (AALI). Hasil penelitian menunjukkan bahwa variabel tingkat suku bunga SBI dan nilai tukar memiliki pengaruh yang signifikan terhadap harga saham syariah AALI. Variabel kurs dan suku bunga SBI memengaruhi variabel saham syariah AALI sebesar 19,75 %

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems

Supply Chain Strategy for Managing Risk for Health Insurance: An Application of Bayesian Model

Abstract- One important matter that should be investigated is the estimation of the risk distribution model of claims for each age group of the insured in health insurance because it is beneficial to prevent the occurrence of losses for insurance companies in the future. Supply chain strategy can be used in health insurance industry to manage the risks. In this paper, the research done is about the risk distribution model estimation on health insurance claims using Bayesian. The objective is to derive a health insurance risk model and determine the amount of net premium for each insured age group in health insurance. The sample of this study is the participant of health insurance in the Bandung area, Indonesia, especially for the insured who live in flood-prone areas. The estimation of the Poisson and Gamma distribution parameter is performed using the Bayesian method, which OpenBUGS involves Markov Chain Monte Carlo (MCMC) the simulation technique. The estimation results show that the frequency of claims significantly follows the Poisson distribution, whereas the amount of claims substantially follows the Gamma distribution. With the result of the analysis, the estimated frequency distribution of claims and the amount of claims, a health insurance risk model may be established. Thus the net premium of health insurance for every age group, for the insured who live in the area prone to floods can be determined

SOLUSI PENDEKATAN PERSAMAAN GELOMBANG FRAKSIONAL NON LINEAR MENGGUNAKAN NEW VERSION OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD

Non-linear differential equations with fractional derivative order are mathematical models that are widely used in modeling physical phenomena, one of the applications of these models is non-linear fractional wave equations. Many methods for solving non-linear fractional partial differential equations, one of which is the New Version of Optimal Homotopy Asymptotic Method which is developed by Liaqat Ali in 2016. The author will use this method to solve non-linear fractional wave equations predetermined, so that the convergence of function of the approximation solution non-linear fractional wave equation can be observed and it can be observed that the function of approximation solution of non-linear fractional wave equation solution using the New Version of Optimal Homotopy Asymptotic Method is simple and has a value error using Mean Absolute Percentage Error which is categorized very wel

Determining Flood Protection Strategy with Uncertain Parameter Using Adjustable Robust Counterpart Methodology

Flooding is a natural disaster that often occurs, it is not surprising that floods are one of the problems that must be resolved in various countries, one of which is Indonesia. Flood is very detrimental to the public because the impact could be the loss of material and non-material. A flood protection system is needed and must be managed properly. This aims in management of flood protection systems often requires efficient cost control strategies that are the lowest possible long-term costs, but still meets the flood protection standards imposed by regulators in all plans. In this paper a flood protection strategy is modeled using Adjustable Robust Optimization. In this approach, there are two kinds of variables that must be decided, i.e., adjustable and non-adjustable variables. A numerical simulation is presented using Scilab Software. Keywords: Flood Protection Strategy, Uncertainty, Adjustable Robust Optimization, Scilab Software