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Ema Carnia
Departemen Matematika, FMIPA, Universitas Padjadjaran
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Kajian Perbandingan Turunan dan Integral Fraksional Riemann–Liouville, Caputo, dan Konformabel serta Aplikasinya Gunawan, Gani; Rusyaman, Endang; Sukono, Sukono; Carnia, Ema
Jurnal Matematika Integratif Vol 21, No 2: Oktober 2025
Publisher : Department of Matematics, Universitas Padjadjaran

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24198/jmi.v21.n2.68569.245-258

Abstract

Penelitian ini membandingkan tiga definisi utama turunan fraksional, yaitu Riemann–Liouville, Caputo, dan Konformabel, dalam pemodelan sistem dinamis yang melibatkan efek memori. Populasi kajian adalah operator turunan fraksional sebagai generalisasi kalkulus klasik berorde bukan bilangan bulat. Intervensi dilakukan melalui pendekatan analitik dan numerik, meliputi kajian sifat dasar operator, analisis eksistensi solusi, dan penerapannya pada persamaan diferensial fraksional linear sederhana. Perbandingan difokuskan pada kesesuaian terhadap sifat kalkulus klasik, bentuk solusi, dan karakter peluruhan sistem. Hasil menunjukkan bahwa turunan Riemann–Liouville dan Caputo menghasilkan solusi berbasis fungsi Mittag–Leffler dengan peluruhan lambat, sehingga sesuai untuk fenomena dengan memori jangka panjang, sedangkan turunan Konformabel menghasilkan solusi eksponensial dengan peluruhan lebih cepat dan efisiensi komputasi yang lebih tinggi. Rentang waktu kajian difokuskan pada orde fraksional α = 1/2 sebagai representasi perilaku umum ketiga definisi. Temuan ini menegaskan bahwa pemilihan operator fraksional harus disesuaikan dengan tujuan analisis dan konteks aplikasi, khususnya antara kebutuhan representasi efek memori dan efisiensi analitik
On the Structural Relationship Between the Characteristic and Minimal Polynomials of a Linear Operator} Suharto, Istiqomah; Kurniadi, Edi; Carnia, Ema
Jurnal Matematika Integratif Vol 22, No 1: April 2026
Publisher : Department of Matematics, Universitas Padjadjaran

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24198/jmi.v22.n1.68420.37-58

Abstract

In this paper, we study the relationship between the characteristic and minimal polynomial of a linear operator, with a focus on figuring out under what conditions that the two polynomials equal each other. We emphasize that the characteristic and minimal polynomial of a linear operator are the same if and only if every eigenvalue has a geometric multiplicity of 1, which is equivalent to having only one Jordan block per eigenvalue. We provide an alternative proof for such a theory. For such matrices, we also show that the minimal polynomial can be easily derived from the normalized linear dependence of the Krylov sequence $\{v, Av, A^2v, \dots, A^{n-1}v\}$ for any generic vector $v$. We apply these algorithms to analyze the nilpotent and companion matrices. The results algorithmically verify that for a companion matrix $C$, its characteristic and minimal polynomials are identical and equal to its generating polynomial, $p_C(X)=m_C(X)=f(X)$. For a nilpotent matrix $N$ with index $k$, we confirm that its minimal polynomial is $m_N(X)=X^k$.
Co-Authors Abdallah, Yusuf Ibrahim Agus Supriatna Ariesandy, Sena Arjun Hasibuan Asep K. Supriatna Asep K. Supriatna Asep Kuswandi Supriatna Basuki , Basuki Basuki Dwindi Agryanti Johar Edi Kurniadi Edi Kurniadi Elah Dewia Endang Rusyaman Faldiyan, Muhammad Faurani, Nuraesa Nufus Gani Gunawan Hafizhah, Nur Hasibuan, Arjun Henti Henti Henti Henti Henti, Henti Herlina Napitupulu Irawati Irawati Isah . Aisah Isah Aisah Isah Aisah Kankan Parmikanti Maryati, Ati Muhammad Deni Johansyah Muhammad Faldiyan Nuraesa Nufus Faurani Nursanti Anggriani Permatasari, Ananda Ayu Putra, Ricardo Eka Setiadji, Setiadji Sisilia Sylviani Sri Wahyuni Suharto, Istiqomah Sukono Sukono Supriatna, Agus Valerie ​Valerie Valerie ​Valerie Yurio Windiatmoko Yurio Windiatmoko, Yurio