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All Journal Epsilon: Jurnal Matematika Murni dan Terapan Desimal: Jurnal Matematika BAREKENG: Jurnal Ilmu Matematika dan Terapan JTAM (Jurnal Teori dan Aplikasi Matematika) Bubungan Tinggi: Jurnal Pengabdian Masyarakat Jurnal Abdimas Prakasa Dakara
Muhammad Ahsar Karim
Program Studi Matematika Fakultas MIPA Universitas Lambung Mangkurat Jl. A. Yani KM. 36, Banjarbaru 70714, Kalimantan Selatan
Humanities Computer Science & IT Control & Systems Engineering Decision Sciences, Operations Research & Management Economics, Econometrics & Finance Education Energy Engineering Mathematics Mechanical Engineering Physics Social Sciences Transportation Other

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Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol Yulida, Yuni; Karim, Muhammad Ahsar
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 3, No 2 (2019): October
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | Full PDF (2719.307 KB) | DOI: 10.31764/jtam.v3i2.1084

Abstract

Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.
ESTIMASI PARAMETER MODEL LOGISTIK DAN RICHARDS PADA PRODUKSI PADI: STUDI KASUS KALIMANTAN SELATAN Yuni Yulida; Faisal Faisal; Muhammad Ahsar Karim; Abdul Hadi; Noorul Wakhdah; Tiara Amaliya
EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN Vol 19, No 2 (2025)
Publisher : Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Lambung Mangkurat

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20527/epsilon.v19i2.16648

Abstract

Padi (Oryza sativa) merupakan komoditas pangan strategis di Indonesia karena beras menjadi sumber karbohidrat utama masyarakat dan berperan penting dalam menjaga ketahanan pangan nasional, termasuk di Kalimantan Selatan yang menjadi salah satu daerah produsen utama. Penelitian ini bertujuan mengestimasi parameter dan membandingkan kinerja model logistik dan Richards dalam memodelkan serta memproyeksikan produksi padi berdasarkan data historis dari Badan Pusat Statistik. Estimasi parameter dilakukan dengan metode nonlinear least squares, sedangkan akurasi model dievaluasi dengan Mean Absolute Percentage Error (MAPE). Hasil penelitian menunjukkan bahwa kedua model mampu merepresentasikan pola pertumbuhan sigmoidal produksi padi, dengan model Richards memberikan hasil yang lebih akurat (MAPE <15%) dibandingkan model logistik. Selain itu, analisis data luas panen mengindikasikan adanya penurunan signifikan sejak 2018, dengan kehilangan kumulatif sekitar 3,54 juta hektar pada periode 2003–2024 dan rata-rata penurunan tahunan sebesar 161 ribu hektar, yang diduga dipengaruhi alih fungsi lahan serta bencana banjir. Proyeksi skenario optimis menunjukkan adanya potensi peningkatan produksi, sedangkan skenario pesimis memperlihatkan kecenderungan stagnasi akibat keterbatasan lahan dan tekanan eksternal lainnya. Dengan demikian, model Richards dinilai lebih representatif dalam memproyeksikan produksi padi di Kalimantan Selatan, dan hasil penelitian ini dapat menjadi dasar dalam perumusan kebijakan perlindungan serta rehabilitasi lahan pertanian guna mendukung ketahanan pangan berkelanjutan.
MODEL SEIDR DALAM STUDI PENYEBARAN TUBERKULOSIS: ANALISIS INTERVENSI DIAGNOSIS DAN DAMPAKNYA TERHADAP DINAMIKA INFEKSI Sila Rizqina; Muhammad Afief Balya; Yuni Yulida; Muhammad Ahsar Karim; Hermei Lissa
EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN Vol 19, No 1 (2025)
Publisher : Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Lambung Mangkurat

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20527/epsilon.v19i1.15641

Abstract

Pulmonary Tuberculosis (TB) is a contagious infectious disease caused by Mycobacterium tuberculosis. A person with tuberculosis serves as a source of transmission to the surrounding population. One way to minimize the transmission is through the implementation of effective diagnostic intervention. One approach to understanding the dynamics of TB spread is through the SEIDR epidemiological mathematical model, which includes susceptible, exposed, infectious, diagnosed, and recovered individuals. This study begins by explaining the construction of the SEIDR model, followed by determining the disease-free equilibrium point, the basic reproduction number, local stability analysis of the disease-free equilibrium, and sensitivity analysis. The final step involves conducting numerical simulations and interpreting the results obtained. There are two equilibrium points derived from the model: the disease-free equilibrium and the endemic equilibrium. The disease-free equilibrium point is asymptotically stable if the basic reproduction number is less than one, while the endemic equilibrium point is determined through simulation. Based on the simulation, it is found that the system experiences an outbreak when the basic reproduction number is greater than one. Sensitivity analysis shows that the birth rate has the highest positive influence on the basic reproduction number, while the natural death rate has the highest negative influence on the basic reproduction number. Numerical simulation results show that when transmission is high, the number of diagnosed individuals increases sharply, while the susceptible and exposed populations decrease drastically; conversely, if transmission is suppressed, active cases decline until they are completely eliminated. Therefore, effective interventions are crucial to reduce transmission and to strengthen diagnostic systems in order to achieve TB elimination at the population level
Co-Authors Abdul Hadi Ana Rizki Mahmudah Arif, Alya Hanifah Azkia Khairal Jamil Azkia Khairal Jamil Dewi Sri Susanti Faisal Faisal Faisal Faisal Faisal Faisal Faisal Faisal Farohatin Na&#039;imah Firman Nurrobi Firmansyah, Audinta Sakti Fitri Nor Annisa Gabriel Henokh Gultom Gian Septiansyah Gian Septiansyah Gita Sari Adriani Haidir Ahsana Hermei Lissa mariyati mariyati Miftahul Jannah Muhammad Afief Balya Muhammad Mahfuzh Shiddiq Muhammad Mahfuzh Shiddiq Muhammad Mefta Eryshady Munaira, Hanna Noorul Wakhdah Nurrobi, Firman Nurul huda Oni Soesanto Pardi Affandi Raihan Nooriman Riska Fitria Rizky Purnama Wulandari Rosyadi, Gusti Muhammad Sila Rizqina Soesanto, Oni Tiara Amaliya Wiranto, Agung Setyo Yuni Yulida