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"Kalkulus fraksional adalah topik yang secara khusus membahas tentang integral dan turunan fraksional, yakni integral dan turunan berorde bilangan real positif. Kalkulus fraksional telah diterapkan dalam berbagai bidang pemodelan, salah satunya pada model sistem dinamik yang bersifat chaos. Chaos merupakan kondisi saat solusi dari suatu sistem deterministik terus bergerak secara aperiodik, namun terbatas dan bersifat sensitif terhadap perubahan nilai awal. Sistem chaos umumnya dikenal dibangun oleh sistem persamaan diferensial berorde integer, namun berbagai penelitian telah menunjukkan bahwa chaos juga dapat terjadi pada sistem dengan turunan berorde fraksional. Pada skripsi ini dibahas chaos pada sistem Burke-Shaw fraksional berorde sama. Pada model sistem Burke-Shaw fraksional, diperoleh hasil bahwa chaos terjadi saat orde turunan fraksional alpha > alpha*, dengan alpha* suatu konstan. Fenomena chaos pada sistem Burke-Shaw fraksional untuk berbagai orde turunan fraksional alpha diselidiki dengan mengamati potret fase dari solusi sistem dan nilai eksponen Lyapunov yang diperoleh secara numerik. Dari simulasi yang dilakukan untuk beberapa orde turunan fraksional yang berbeda, didapat hasil yang konsisten bahwa sistem bersifat stabil untuk alpha < alpha*, dan bersifat chaos saat alpha > alpha*. Dinamik dari solusi sistem juga diamati untuk melihat sensitivitas solusi terhadap perbedaan nilai awal yang kecil.

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Fractional calculus is a topic that specifically discussed about fractional integral and fractional derivative, which means integral and derivative with order of positive real number. Fractional calculus has been applied to various fields of mathematical modelling, one of them is in the model of chaotic dynamical system. Chaos is a condition where the solution of a deterministic system keep moving in aperiodic state, but is bounded and sensitive with the change of initial conditions. Generally a chaotic system is built by differential equations with integer order, but various work has showed that chaos can be exhibited in system with fractional order. In this thesis chaotic behavior of Burke-Shaw commensurate order fractional system is discussed. From the fractional Burke-Shaw system we obtained the result that chaos will happen when the fractional derivative order alpha > alpha*, where alpha* is a constant. Chaotic phenomena of fractional Burke-Shaw system with various fractional derivative order alpha is investigated by observing the potrait phase of system's solution and Lyapunov exponents by some numerical calculations. By simulation that is done with some different values of fractional order alpha of the derivative, we obtained consistent result that shows the system is stable for alpha < alpha* and is chaotic for alpha > alpha*. The dynamic of the system's solutions is also observed to see the solution's sensitivity with small difference of initial conditions."

"Cacar monyet merupakan jenis penyakit yang ditimbulkan oleh infeksi virus MPOX yang berasal dari monyet kra pemakan kepiting (Macaca fascicularis). Penelitian terkini menunjukkan bahwa virus MPOX memiliki 2 jenis kelas, yaitu Klad I dan Klad II. Klad II memiliki dua subkelas, yaitu Klad IIa dan Klad IIb. Seluruh jenis virus MPOX tidak hanya menyerang hewan primata, tetapi juga hewan pengerat dan manusia. Model epidemi cacar monyet dikonstruksi dengan mempertimbangkan infeksi terhadap manusia dan hewan pengerat. Model dibentuk dengan pendekatan persamaan diferensial fraksional nonlinier berdimensi delapan yang menerapkan orde Atangana-Baleanu. Model terhadap populasi manusia rentan (susceptible), populasi manusia terpapar (exposed), populasi manusia terinfeksi tetapi tidak menimbulkan gejala (asymptomatic), populasi manusia terinfeksi dan menimbulkan gejala (infectious), dan populasi sembuh dan bebas dari cacar monyet (recovered). Di sisi lain, model terhadap populasi hewan pengerat rentan (susceptible), populasi hewan pengerat terpapar (exposed), dan populasi hewan pengerat terinfeksi (infected). Hal penting yang diperhatikan pada model ini adalah proses konstruksi, kepositifan dan keterbatasan solusi, eksistensi and analisis kestabilan solusi, dan simulasi numerik. Model memiliki titik kesetimbangan bebas dan endemik yang stabil secara asimtotik. Estimasi parameter dilakukan dengan studi literatur terhadap model epidemi cacar monyet pada manusia dan hewan pengerat. Simulasi numerik menunjukkan hasil dari masing-masing variabel dengan nilai orde (α = 0,85; 0,9; 0,95; 0,97). Oleh karena itu, aplikasi model fraksional kalkulus dapat menjadi sebuah pandangan baru dalam mengontrol dan menanggulangi dampak negatif dan penyakit cacar monyet.

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Monkey pox is a type of disease caused by infection of the MPOX virus from crab-eating kra monkeys (Macaca fascicularis). Recent research shows that the MPOX virus has 2 types of classes, Clad I and Clad II. Clad II has two subclasses, Clad IIa and Clad IIb. All classes of MPOX viruses affect not only primates, but also rodents and humans. Monkey pox epidemic model is constructed by considering human and rodent infections. The model is formed by an eight-dimensional nonlinear fractional differential equation by applying the Atangana-Baleanu order. Population of model is containing susceptible human population, exposed human population, infected but asymptomatic human population, infected and symptomatic human population, and recovered and free from monkey pox. On the other hand, model the susceptible rodent population, exposed rodent population, and infected rodent population. Important study in this model are the model construction process, positivity and boundedness of solution, existence and stability analysis of solution, and numerical simulation. The model has asymptotically stable free and endemic equilibrium points. Parameter estimation is determined by literature study on the monkey pox epidemic model in humans and rodents. Numerical simulation shows the result of each variable with an order value (α = 0.85, 0.9, 0.95, 0.97). Therefore, fractional calculus model is a further application in controlling and mitigating the disease and its negative impact of monkey pox."

"The book tries to briefly introduce the diverse literatures in the field of fractional order signal processing which is becoming an emerging topic among an interdisciplinary community of researchers. The readers should have preliminary knowledge about basic signal processing techniques. Prerequisite knowledge of fractional calculus is not essential and is exposited at relevant places in connection to the appropriate signal processing topics. Basic signal processing techniques like filtering, estimation, system identification, etc. in the light of fractional order calculus are presented along with relevant application areas. The readers can easily extend these concepts to varied disciplines like image or speech processing, pattern recognition, time series forecasting, financial data analysis and modeling, traffic modeling in communication channels, optics, biomedical signal processing, electrochemical applications and many more. Available MATLAB tools to simulate FOSP theories are also introduced."

"This book describes complex random signals, characterized by the presence of a heavy-tailed distribution or non-negligible dependence between distant observations. Coverage includes simulation, fractional-order modeling and filtering fractional-order systems."

"Introduces a rigorous investigation of q-difference operators in standard and fractional settings. This monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi."

"Fractional dynamics and control provides a comprehensive overview of recent advances in the areas of nonlinear dynamics, vibration and control with analytical, numerical, and experimental results. This book provides an overview of recent discoveries in fractional control, delves into fractional variational principles and differential equations, and applies advanced techniques in fractional calculus to solving complicated mathematical and physical problems.Finally, this book also discusses the role that fractional order modeling can play in complex systems for engineering and science."

"This book introduces a series of problems and methods insufficiently discussed in the field of Fractional Calculus – a major, emerging tool relevant to all areas of scientific inquiry. The authors present examples based on symbolic computation, written in Maple and Mathematica, and address both mathematical and computational areas in the context of mathematical modeling and the generalization of classical integer-order methods. Distinct from most books, the present volume fills the gap between mathematics and computer fields, and the transition from integer- to fractional-order methods."

"This book contains a brief historical introduction and state of the art in fractional calculus. The author introduces some of the so-called special functions, in particular, those which will be directly involved in calculations. The concepts of fractional integral and fractional derivative are also presented. Each chapter, except for the first one, contains a list of exercises containing suggestions for solving them and at last the resolution itself. At the end of those chapters there is a list of complementary exercises. The last chapter presents several applications of fractional calculus."

"Topics in fractional differential equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena, in particular the modeling of memory-dependent and complex media such as porous media. "