The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems.

Abstrak: Aplikasi Model Pertumbuhan Logistik dalam Menentukan Proyeksi Penduduk di Kabupaten Banyumas. Kabupaten Banyumas adalah salah satu kabupaten di Jawa Tengah yang memiliki kepadatan penduduk terbanyak ketiga dengan luas wilayah seluas 1.328 km². Berdasarkan data dari BPS jumlah penduduk di Kabupaten Banyumas setiap tahunnya semakin bertambah, maka diperlukan solusi untuk mengurangi dampak negatif agar tidak terjadi adanya ledakan populasi. Solusi yang dapat digunakan untuk memproyeksi penduduk di Kabupaten Banyumas yaitu dengan menggunakan model pertumbuhan logistik. Model ini digunakan untuk menghitung nilai laju pertumbuhan dan daya dukung lingkungan (carrying capacity) dengan menggunakan data jumlah penduduk di Kabupaten Banyumas tahun 2015 sampai tahun 2021. Hasil yang diperoleh menunjukan bahwa daya dukung lingkungan (carrying capacity) yang membatasi penduduk di Kabupaten Banyumas adalah  dengan laju pertumbuhan relatif pertahunnya sebesar 7,75%. Model ini juga memproyeksikan jumlah penduduk di Kabupaten Banyumas dari tahun 2022 sampai tahun 2030. Dengan perhitungan menggunakan Microsoft Excel diperoleh proyeksi jumlah penduduk di Kabupaten Banyumas pada tahun 2022 berjumlah 1.811.059 jiwa hingga tahun 2030 diperkirakan berjumlah 1.944.653 jiwa. Kata Kunci: Kabupaten Banyumas, Model Pertumbuhan Logistik, carrying capacity

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation.

This research aims to obtaining the solutions (if any) from the Non Linear Diophantine equation of 11 ^x + 13 ^y = z ². There are 3 possibilities to obtain the solutions (if any) from the Non Linear Diophantine equation, namely single, multiple, and no solution. This research is conducted in two stages: (1) by utilizing simulation to obtain the solutions (if any) from the Non Linear Diophantine equation of 11 ^x + 13 ^y = z ² and (2) by utilizing congruency theory with its characteristics proven that the Non Linear Diophantine equation has no solution for non negative whole numbers (integers) of x, y, z.

Paper ini akan dibahas solusi dari persamaan Black-Scholes fraksional yang merupakan bentuk umum dari persamaan Black-Scholes dan kebaruan penelitian tentang persamaan Black-Scholes yang dimodifikasi. Adapun metode-metode untuk mencari solusi dari persamaan Black Scholes Fraksional sudah banyak ditulis dalam banyak jurnal internasional. Solusi persamaan Black Scholes fraksional dalam hal ini ditinjau dengan pendekatan Kalkulus Fraksional. Dengan pendekatan Kalkulus Fraksional proses penyelesaian dalam mencari solusi persamaan Black Scholes Fraksional menjadi lebih efisien. Beberapa Metode yang digunakan untuk mencari solusi Persamaan Black Scholes Fraksional diantaranya adalah metode transformasi Sumudu, metode ekspansi deret, metode perturbasi Homotopi, metode dekomposisi Adomian, dan metode dekomposisi Laplace Adomian.

This research discusses the formation of the Fractional Logistic model and its solution analysis, as well as its simulation to predict the Human Development Index in Cilacap Regency using the Fractional Logistic growth model. This study uses secondary data obtained from the official website of the Central Statistics Agency of Cilacap Regency. Based on these data, the growth of the Human Development Index in Cilacap Regency has increased relatively. This shows the Cilacap community's good quality of life. Based on the environmental carrying capacity value of 73, a relative growth rate per year of 0.14273269 is obtained. This model predicts the Human Development Index in Cilacap Regency in 2024 and 2025. The prediction results for 2024 of 71.47 and 2025 of 71.67 are achieved when the fractional derivative order is one. The best approximate solution is obtained when the fractional derivative order is 1, 0.95 0.90, 0.85, 0.80 and 0.75.