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Journal articles on the topic 'Adams-Bashforth-Moulton'

Author: Grafiati

Published: 22 February 2023

Last updated: 23 February 2023

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1

Murni, Delvitri, Bukti Ginting, and Narwen . "PENERAPAN METODE ADAMS-BASHFORTH-MOULTON ORDE EMPAT UNTUK MENENTUKAN SOLUSI PERSAMAAN DIFERENSIAL LINIER HOMOGEN ORDE TIGA KOEFISIEN KONSTAN." Jurnal Matematika UNAND 5, no. 2 (May 23, 2016): 21. http://dx.doi.org/10.25077/jmu.5.2.21-25.2016.

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Abstrak. Persamaan diferensial linier homogen orde tiga koesien konstan direduksimenjadi persamaan diferensial biasa orde-1, yaitu y0= f(x; y) dengan syarat awaly(x0) = y. Persamaan diferensial biasa orde-1 diselesaikan menggunakan metodeRunge-Kutta orde empat untuk menentukan nilai pendekatan y01; y2; dan y. Selanjutnya,digunakan metode Adams-Bashforth orde empat untuk menentukan nilai pendekatany; ; dst sebagai prediktor. Nilai yang ditampilkan oleh metode Adams-Bashforthorde empat digunakan pada metode Adams-Moulton orde empat sebagai korektor. Prosesmetode Adams-Bashforth orde empat dan metode Adams-Moulton orde empat dikatakansebagai metode Adams-Bashforth-Moulton orde empat atau metode prediktor-korektor.

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2

Misirli, Emine, and Yusuf Gurefe. "Multiplicative Adams Bashforth–Moulton methods." Numerical Algorithms 57, no. 4 (November 23, 2010): 425–39. http://dx.doi.org/10.1007/s11075-010-9437-2.

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3

Beuken, Loïc, Olivier Cheffert, Aleksandra Tutueva, Denis Butusov, and Vincent Legat. "Numerical Stability and Performance of Semi-Explicit and Semi-Implicit Predictor–Corrector Methods." Mathematics 10, no. 12 (June 11, 2022): 2015. http://dx.doi.org/10.3390/math10122015.

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Semi-implicit multistep methods are an efficient tool for solving large-scale ODE systems. This recently emerged technique is based on modified Adams–Bashforth–Moulton (ABM) methods. In this paper, we introduce new semi-explicit and semi-implicit predictor–corrector methods based on the backward differentiation formula and Adams–Bashforth methods. We provide a thorough study of the numerical stability and performance of new methods and compare their stability with semi-explicit and semi-implicit Adams–Bashforth–Moulton methods and their performance with conventional linear multistep methods: Adams–Bashforth, Adams–Moulton, and BDF. The numerical stability of the investigated methods was assessed by plotting stability regions and their performances were assessed by plotting error versus CPU time plots. The mathematical developments leading to the increase in numerical stability and performance are carefully reported. The obtained results show the potential superiority of semi-explicit and semi-implicit methods over conventional linear multistep algorithms.

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4

Permata, Hendrik Widya, Ari Kusumastuti, and Juhari Juhari. "Solusi Numerik Model Gerak Osilasi Vertikal dan Torsional Pada Jembatan Gantung." Jurnal Riset Mahasiswa Matematika 1, no. 1 (October 20, 2021): 1–13. http://dx.doi.org/10.18860/jrmm.v1i1.13409.

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Model gerak osilasi vertikal dan torsional merupakan model yang menggambarkan gerak osilasi vertikal dan gerak torsional pada batang yang digantung. Gerak osilasi vertikal merupakan gerak naik turun suatu benda yang terjadi terus berulang, dan kemudian pada waktu tertentu akan berhenti atau mengalami redaman. Gerak torsional merupakan getaran sudut dari suatu objek yang mengalami rotasi. Model gerak osilasi dan torsional pada dasarnya merupakan sistem persamaan diferensial orde dua. Tujuan dari penelitian ini adalah untuk mengetahui solusi numerik model gerak osilasi vertikal dan torsional menggunakan metode Adams-Bashforth-Moulton orde empat, lima, dan enam. Model gerak osilasi vertikal dan torsional terlebih dahulu diselesaikan menggunakaan metode Runge-Kutta-Fehlberg orde lima untuk mendapatkan solusi awal kemudian model tersebut diselesaikan menggunakan metode Adams-Bashforth-Moulton orde empat, lima dan enam. Hasil solusi numerik setiap metode Adam-Bashforth-Moulton selanjutnya diuji dengan galat relatif. Hasil simulasi numerik model gerak osilasi vertikal dan torsi diperoleh bahwa gerak osilasi vertikal dan gerak torsional merupakan gerak harmonik teredam dan semakin tinggi orde pada metode Adams-Bashforth-Moulton maka akan lebih cepat galat relatif menuju nilai nol dan sebaliknya

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5

Ma, Shichang, Yufeng Xu, and Wei Yue. "Numerical Solutions of a Variable-Order Fractional Financial System." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/417942.

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The numerical solution of a variable-order fractional financial system is calculated by using the Adams-Bashforth-Moulton method. The derivative is defined in the Caputo variable-order fractional sense. Numerical examples show that the Adams-Bashforth-Moulton method can be applied to solve such variable-order fractional differential equations simply and effectively. The convergent order of the method is also estimated numerically. Moreover, the stable equilibrium point, quasiperiodic trajectory, and chaotic attractor are found in the variable-order fractional financial system with proper order functions.

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6

Segarra, Jaime. "MÉTODOS NUMÉRICOS RUNGE-KUTTA Y ADAMS BASHFORTH-MOULTON EN MATHEMATICA." Revista Ingeniería, Matemáticas y Ciencias de la Información 7, no. 14 (July 15, 2020): 13–32. http://dx.doi.org/10.21017/rimci.2020.v7.n14.a81.

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n este estudio, el objetivo principal es realizar el análisis de los métodos numéricos Runge-Kutta y Adams Bashforth-Moulton. Para cumplir con el objetivo se utilizó el sistema de ecuaciones diferenciales del modelo Lotka-Volterra y se usó el software matemático Wolfram Mathematica. En los resultados se realiza la comparación de los métodos RK4, AB4 y AM4 con el comando NDSolve utilizando el modelo Lotka-Volterra. Los resultados obtenidos en los diagramas de fase y la tabla de puntos de la iteración indicaron que el método RK4 tiene mayor precisión que los métodos AB4 y AM4.

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7

Liao, Hongyun, Yipeng Ding, and Ling Wang. "Adomian Decomposition Algorithm for Studying Incommensurate Fractional-Order Memristor-Based Chua’s System." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850134. http://dx.doi.org/10.1142/s0218127418501341.

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Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical approximate solution of the incommensurate fractional-order memristor-based Chua’s system is investigated. Dynamical characteristics of the proposed system are studied by using phase diagram, bifurcation analysis and power spectrum. Research results show that compared with the Adams–Bashforth–Moulton algorithm, the Adomian decomposition algorithm yields more accurate results and its solution generally converges more rapidly. Compared with 3.776 achieved by the Adams–Bashforth–Moulton algorithm, the minimum order of the incommensurate fractional-order memristor-based Chua’s system solved by using Adomian decomposition algorithm is 1.76, which is much smaller. A reliable and efficient binary test for chaos, called “0–1 test”, is utilized to detect the presence of chaotic attractors in the system dynamics.

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8

Peinado, J., J. Ibáñez, E. Arias, and V. Hernández. "Adams–Bashforth and Adams–Moulton methods for solving differential Riccati equations." Computers & Mathematics with Applications 60, no. 11 (December 2010): 3032–45. http://dx.doi.org/10.1016/j.camwa.2010.10.002.

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9

Al-Sulami, Hamed, Moustafa El-Shahed, Juan J. Nieto, and Wafa Shammakh. "On Fractional Order Dengue Epidemic Model." Mathematical Problems in Engineering 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/456537.

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This paper deals with the fractional order dengue epidemic model. The stability of disease-free and positive fixed points is studied. Adams-Bashforth-Moulton algorithm has been used to solve and simulate the system of differential equations.

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10

Tutueva, Aleksandra, Timur Karimov, and Denis Butusov. "Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods." Mathematics 8, no. 5 (May 13, 2020): 780. http://dx.doi.org/10.3390/math8050780.

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Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

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11

Farah Liyana Azizan, Saratha Sathasivam, Muraly Velavan, Nur Rusyidah Azri, and Nur Iffah Rafhanah Abdul Manaf. "Prediction of Drug Concentration in Human Bloodstream using Adams-Bashforth-Moulton Method." Journal of Advanced Research in Applied Sciences and Engineering Technology 29, no. 2 (January 31, 2023): 53–71. http://dx.doi.org/10.37934/araset.29.2.5371.

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Pharmaceutical drugs are chemicals intended to avoid, assess, heal, or cure a disease. It is also commonly referred to as medication. When medicine is taken, it gets absorbed into the bloodstream, spreads throughout the body, and achieves its maximum concentration. Following this, the medication level gradually decreases as it is removed from the body. The drug concentration according to the time can be predicted using mathematical concepts and pharmacokinetic models. The compartmental model is a fundamental type of model used in pharmacokinetics. The number of compartments required to describe the drug's action in the body is one-compartment, two-compartment, and multicompartment. These models can forecast medication concentrations in the body over time. This paper will focus on the one-compartment model and Adams Bashforth-Moulton method. Adams Method is one of the linear multistep techniques applied to solve numerical ordinary differential equations that contain the predictor method (Adams Bashforth) and corrector method (Adams Moulton). The integrated development environment used for the computation and graphing is MATLAB. The expected result of this report is that we can predict the concentration of the chosen drugs over time and how long a particular person needs to wait before donating blood safely.

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12

Azhar, Aurizan Himmi, Sugiyanto Sugiyanto, Muhammad Wakhid Musthofa, and Muhamad Zaki Riyanto. "Transformasi Fourier Multiplikatif Dan Aplikasinya Pada Persamaan Diferensial Multiplikatif." Jurnal Derivat: Jurnal Matematika dan Pendidikan Matematika 8, no. 2 (December 20, 2021): 149–60. http://dx.doi.org/10.31316/j.derivat.v8i2.1996.

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This research is a development of multiplicative calculus. This study is about the Fourier multiplicative transformation and its application to the multiplicative differential equation. This study aims to determine the Fourier multiplicative transformation as well as the multiplicative differential equation. This study contains numerical simulations to solve the problem of ordinary multiplicative differential equations of the first order. The methods used in this research are descriptive research methods through the study of literature. The results of this study are the application of multiplicative Fourier transformations to multiplicative differential equations and numerical solutions of ordinary multiplicative differential equations with the Adam Bashforth-Moulton multiplicative method. Keywords: Multiplicative Calculus, Fourier Multiplicative Transformation, Multiplicative Differential Equations, Adams Bashforth Moulton Multiplicative Method

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13

Danca, Marius-F., and Nikolay Kuznetsov. "Matlab Code for Lyapunov Exponents of Fractional-Order Systems." International Journal of Bifurcation and Chaos 28, no. 05 (May 2018): 1850067. http://dx.doi.org/10.1142/s0218127418500670.

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In this paper, the Benettin–Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. First, it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams–Bashforth–Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams–Bashforth–Moulton method, is utilized. Four representative examples are considered.

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14

Arianti, Liatri, Rusli Hidayat, and Kosala Dwija Purnomo. "PENYELESAIAN MODIFIKASI MODEL PREDATOR PREY LESLIE-GOWER DENGAN SEBAGIAN PREY TERINFEKSI MENGGUNAKAN ADAMS BASHFORTH MOULTON ORDE EMPAT." Majalah Ilmiah Matematika dan Statistika 19, no. 2 (September 2, 2019): 53. http://dx.doi.org/10.19184/mims.v19i2.17268.

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Eco-epidemiology is a science that studies the spread of infectious diseases in a population in an ecosystem where two or more species interact like a predator prey. In this paper discusses about how to solve modification Leslie Gower of predator prey models (with Holling II response function) with some prey infected using fourth order Adams Bashforth Moulton method. This paper used a simple disease-spreading model that is Susceptible-Infected (SI). The model is divided into three populations: the sound prey (which is susceptible), the infected prey and predator population. Keywords: Adams Basforth Moulton, Eco-epidemiology Holling Tipe II, Local stability, Leslie-Gower, Predator-Prey model

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15

Kumar, Sunil, Ali Ahmadian, Ranbir Kumar, Devendra Kumar, Jagdev Singh, Dumitru Baleanu, and Mehdi Salimi. "An Efficient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets." Mathematics 8, no. 4 (April 10, 2020): 558. http://dx.doi.org/10.3390/math8040558.

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In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that arises in medical science to study epidemiology and medical care for the injured. Operational matrices merged with the collocation method are used to convert fractional-order problems into algebraic equations. The Adams–Bashforth–Moulton predictor correcter scheme is also discussed for solving the same. We have compared the solutions with the Adams–Bashforth predictor correcter scheme for the accuracy and applicability of the Bernstein wavelet method. The convergence analysis of the Bernstein wavelet has been also discussed for the validity of the method.

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16

Maioli, Gabrielle, and Suzete Maria Silva Afonso. "Aplicação de métodos numéricos no estudo da decomposição de matéria orgânica antropogênica em um rio." Jornal Eletrônico de Ensino e Pesquisa de Matemática 2, no. 1 (July 1, 2018): 1–44. http://dx.doi.org/10.4025/jeepema.v2.n1.art1.

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Neste artigo exploraremos os métodos numéricos de passo único, denominados métodos de Euler e Runge - Kutta, e os métodos de passos múltiplos, denominados métodos de Adams-Bashforth e Adams-Moulton, para encontrar soluções aproximadas de problemas de valor inicial para equações diferenciais ordinárias de primeira ordem. Aplicaremos os métodos numéricos abordados para estudar a decomposição de matéria orgânica antropogênica em um rio, cujo processo pode ser modelado por uma equação diferencial ordinária.

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17

Suescun-Díaz, Daniel, Diego A. Rasero-Causil, and Jaime H. Lozano-Parada. "Neutron Density Calculation Using the Generalised Adams-Bashforth-Moulton Method." Universitas Scientiarum 24, no. 3 (November 20, 2019): 543–63. http://dx.doi.org/10.11144/javeriana.sc24-3.ndcu.

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This paper presents a numerical solution to the equations of point kinetics for nuclear power reactors, a set of seven coupled differential equations that describe the temporal variation of neutron density and the concentration of delayed neutron precursors. Due to the nature of the system, we propose to numerically solve the point kinetics equations by implementing the Adams-Bashforth and Adams-Moulton methods, which are predictor-corrector schemes with their respective modifiers to increase precision. The proposed method was tested computationally for different forms of reactivity with up to six groups of delayed neutron precursors. This method was used in a recent publication to solve the inverse problem of finding the reactivity. In this work, it is shown that it can also be used for the calculation of nuclear power, that it is simple and easy to implement, and that it produces good results when compared with those in the literature for neutron population density and concentration of delayed neutron precursors.

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18

Kuzairi, Kuzairi, Tony Yulianto, and Lilik Safitri. "APLIKASI METODE ADAMS BASHFORTH-MOULTON (ABM) PADA MODEL PENYAKIT KANKER." Jurnal Matematika "MANTIK" 2, no. 1 (October 30, 2016): 14. http://dx.doi.org/10.15642/mantik.2016.2.1.14-21.

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Cancer is a deadly disease that is characterized by the growth of abnormal cells, the growth is ongoing, forming a tumor. Tumors are divided into two parts, namely benign and malignant tumors. Malignant tumors are a general term for cancer. The disease of cancer has a mathematical model in the form of a system of differential equations, for it required a method to obtain the solution of the system of differential equations. The method used is the method of numerical methods Bashforth Adams Moulton (ABM) order one, two, three, and four. From the results of this study concluded that the method ABM order three better than the method ABM first order, second order and fourth order at issue models of cancer, It can be seen in the graphic simulation using ABM order three, it shows that increasing time population of immune effector cells (E) and a population of effector molecules (C) increased and then stabilized. The population of immune effector cells (E) stabilized at 33.3336, while the population of the effector molecule (C) is stable in the scope of the numbers 33,333, 33,333 are said to be in scope for changes in population effector molecule (C) can not be known with certainty. While the population of cancer cells (T) remains at 0 at each iteration (stable) remains in a state that is free of cancer

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19

Tutueva, Aleksandra, and Denis Butusov. "Stability Analysis and Optimization of Semi-Explicit Predictor–Corrector Methods." Mathematics 9, no. 19 (October 3, 2021): 2463. http://dx.doi.org/10.3390/math9192463.

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The increasing complexity of advanced devices and systems increases the scale of mathematical models used in computer simulations. Multiparametric analysis and study on long-term time intervals of large-scale systems are computationally expensive. Therefore, efficient numerical methods are required to reduce time costs. Recently, semi-explicit and semi-implicit Adams–Bashforth–Moulton methods have been proposed, showing great computational efficiency in low-dimensional systems simulation. In this study, we examine the numerical stability of these methods by plotting stability regions. We explicitly show that semi-explicit methods possess higher numerical stability than the conventional predictor–corrector algorithms. The second contribution of the reported research is a novel algorithm to generate an optimized finite-difference scheme of semi-explicit and semi-implicit Adams–Bashforth–Moulton methods without redundant computation of predicted values that are not used for correction. The experimental part of the study includes the numerical simulation of the three-body problem and a network of coupled oscillators with a fixed and variable integration step and finely confirms the theoretical findings.

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20

Marasi, H. R., M. H. Derakhshan, A. Soltani Joujehi, and Pushpendra Kumar. "Higher-order fractional linear multi-step methods." Physica Scripta 98, no. 2 (January 17, 2023): 024004. http://dx.doi.org/10.1088/1402-4896/acad42.

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Abstract In this paper, we propose two arrays, containing the coefficients of fractional Adams-Bashforth and Adams-Moulton methods, and also recursive relations to produce the elements of these arrays. Then, we illustrate the application of these arrays in a suitable way to construct higher-order fractional linear multi-step methods in general form, with extended stability regions. The effectiveness of the new method is shown in comparison with some available previous results in an illustrative test problem.

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21

Rosu, Florin. "Parallel Algorithm for Numerical Methods Applied to Fractional-order System." Scalable Computing: Practice and Experience 21, no. 4 (December 20, 2020): 701–7. http://dx.doi.org/10.12694/scpe.v21i4.1837.

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A parallel algorithm is presented that approximates a solution for fractional-order systems. The algorithm isimplemented in CUDA, using the specific GPU capabilities. The numerical methods used are Adams-Bashforth-Moulton (ABM) predictor-corrector scheme and Diethelm’s numerical method. A comparison is done between these numerical methods that adapts the same algorithm for the approximation of the solution.

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22

Riyadi, Mohamad, and Daswa Daswa. "SOLUSI HAMPIRAN PERSAMAAN LOGISTIK NON-AUTONOMOUS." Jurnal Edukasi dan Sains Matematika (JES-MAT) 5, no. 1 (April 21, 2019): 63. http://dx.doi.org/10.25134/jes-mat.v5i1.1745.

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The aim of this study is to derive the approximation solution of the non-autonomous logistic equation with a non-constant carrying capacity. The solution is found via predictor-corrector method (Adams-Bashforth-Moulton method, Milne method and Hamming method). The approximation solution that obtained, then, is compared to the exact solution. The results show that, for small step size, the approximation solution approximate the exact solution is in good agreement.

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Rchid Sidi Ammi, Moulay, Mostafa Tahiri, and Delfim F. M. Torres. "Local existence and uniqueness for a fractional SIRS model with Mittag–Leffler law." General Letters in Mathematics 10, no. 2 (June 2021): 61–71. http://dx.doi.org/10.31559/glm2021.10.2.7.

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In this paper, we study an epidemic model with Atangana-Baleanu-Caputo fractional derivative. We obtain a special solution using an iterative scheme via Laplace transformation. Uniqueness and existence of a solution using the Banach fixed point theorem are studied. A detailed analysis of the stability of the special solution is presented. Finally, our generalized model in the derivative sense is solved numerically by the Adams-Bashforth-Moulton method.

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Yépez-Martínez, H., and J. F. Gómez-Aguilar. "Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel." Mathematical Modelling of Natural Phenomena 13, no. 1 (2018): 13. http://dx.doi.org/10.1051/mmnp/2018002.

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Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

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Jeng, Siow W., and Adem Kilicman. "Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods." Symmetry 12, no. 6 (June 5, 2020): 959. http://dx.doi.org/10.3390/sym12060959.

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Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough volatility. Currently, fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form and therefore, we must rely on numerical methods to obtain a solution. In this paper, we will be giving a short introduction to option pricing theory (Black–Scholes model, classical Heston model and its characteristic function), an overview of the current advancements on the rough Heston model and numerical methods (fractional Adams–Bashforth–Moulton method and multipoint Padé approximation method) for solving the fractional Riccati equation. In addition, we will investigate on the performance of multipoint Padé approximation method for the small u values in D α h ( u − i / 2 , x ) as it plays a huge role in the computation for the option prices. We further confirm that the solution generated by multipoint Padé (3,3) method for the fractional Riccati equation is incredibly consistent with the solution generated by fractional Adams–Bashforth–Moulton method.

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Awati, Vishwanath B., Krishna B. Chavaraddi, and Priya M. Gouder. "Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid." Nonlinear Engineering 8, no. 1 (January 28, 2019): 39–45. http://dx.doi.org/10.1515/nleng-2018-0031.

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Abstract The boundary roughness effects on nonlinear saturation of Rayleigh-Taylor instability (RTI) in couple-stress fluid have been studied using numerical technique on the basis of stability of interface between two fluids of the system. The resulting fourth order ordinary nonlinear differential equation is solved using Adams-Bashforth predictor and Adams-Moulton corrector techniques numerically. The various surface roughness effects and surface tension effects on nonlinear saturation of RTI of two superposed couple-stress fluid and fluid saturated porous media are well investigated. At the interface, the surface tension acts and finally stability of the problem is discussed in detail.

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27

Çelik, Vedat. "Bifurcation Analysis of Fractional Order Single Cell with Delay." International Journal of Bifurcation and Chaos 25, no. 02 (February 2015): 1550020. http://dx.doi.org/10.1142/s0218127415500200.

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This paper presents the bifurcation analysis of fractional order model of delayed single cell which is proposed for delayed cellular neural networks with respect to the time delay τ. The bifurcation points, time delay τc, are determined by modified Mikhailov stability criterion for a range of fractional delayed cell order 0.3 ≤ q < 1. Numerical results obtained from Adams–Bashforth–Moulton method demonstrate that the supercritical Hopf bifurcation occurs in the system.

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Suescún-Díaz, D., D. A. Rasero Causil, and J. H. Figueroa-Jimenez. "Adams-Bashforth-Moulton method with Savitzky-Golay filter to reduce reactivity fluctuations." Kerntechnik 82, no. 6 (December 18, 2017): 674–77. http://dx.doi.org/10.3139/124.110842.

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29

Shabana, Ahmed A., Dayu Zhang, and Gengxiang Wang. "TLISMNI/Adams algorithm for the solution of the differential/algebraic equations of constrained dynamical systems." Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 232, no. 1 (July 31, 2017): 129–49. http://dx.doi.org/10.1177/1464419317718658.

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This paper examines the performance of the 3rd and 4th order implicit Adams methods in the framework of the two-loop implicit sparse matrix numerical integration method in solving the differential/algebraic equations of heavily constrained dynamical systems. The variable-step size two-loop implicit sparse matrix numerical integration/Adams method proposed in this investigation avoids numerical force differentiation, ensures satisfying the nonlinear algebraic constraint equations at the position, velocity, and acceleration levels, and allows using sparse matrix techniques for efficiently solving the dynamical equations. The iterative outer loop of the two-loop implicit sparse matrix numerical integration/Adams method is aimed at achieving the convergence of the implicit integration formulae used to solve the independent differential equations of motion, while the inner loop is used to ensure the convergence of the iterative procedure used to satisfy the algebraic constraint equations. To solve the independent differential equations, two different implicit Adams integration formulae are examined in this investigation; a 3rd order implicit Adams-Moulton formula with a 2nd order explicit predictor Adams Bashforth formula, and a 4th order implicit Adams-Moulton formula with a 3rd order explicit predictor Adams Bashforth formula. A standard Newton–Raphson algorithm is used to satisfy the nonlinear algebraic constraint equations at the position level. The constraint equations at the velocity and acceleration levels are linear, and therefore, there is no need for an iterative procedure to solve for the dependent velocities and accelerations. The algorithm used for the error check and step-size change is described. The performance of the two-loop implicit sparse matrix numerical integration/Adams algorithm developed in this investigation is evaluated by comparison with the explicit predictor-corrector Adams method which has a variable-order and variable-step size. Simple and heavily constrained dynamical systems are used to evaluate the accuracy, robustness, damping characteristics, and effect of the outer-loop iterations of the proposed implicit schemes. The results obtained in this investigation show that the two-loop implicit sparse matrix numerical integration methods proposed in this study can be more efficient for stiff systems because of their ability to damp out high-frequency oscillations. Explicit integration methods, on the other hand, can be more efficient in the case of non-stiff systems.

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Xin, Baogui, and Yuting Li. "0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/876298.

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A new integer-order chaotic financial system is extended by introducing a simple investment incentive into a three-dimensional chaotic financial system. A four-dimensional fractional-order chaotic financial system is presented by bringing fractional calculus into the new integer-order financial system. By using weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.

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He, Shaobo, Hayder Natiq, Santo Banerjee, and Kehui Sun. "Complexity and Chimera States in a Network of Fractional-Order Laser Systems." Symmetry 13, no. 2 (February 20, 2021): 341. http://dx.doi.org/10.3390/sym13020341.

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By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation diagram. Numerical simulation outcomes demonstrate that the system generates chaos with the decreasing of q. Moreover, this paper designs the coupled fractional-order network of laser systems and subsequently obtains its numerical solution using ABM. These solutions have demonstrated chimera states of the proposed fractional-order laser network.

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Setiawan, Laurensius Ian, and Sudi Mungkasi. "PENYELESAIAN MODEL EPIDEMI SIR MENGGUNAKAN METODE RUNGE-KUTTA ORDE EMPAT DAN METODE ADAMS-BASHFORTH-MOULTON." Komputasi: Jurnal Ilmiah Ilmu Komputer dan Matematika 18, no. 2 (July 29, 2021): 55–61. http://dx.doi.org/10.33751/komputasi.v18i2.3623.

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Model epidemi SIR (Susceptible-Infected-Recovered) telah diterapkan secara luas untuk simulasi penyebaran penyakit menular. Makalah ini menyajikan skema numeris metode Runge-Kutta orde empat dan metode Adams-Bashforth-Moulton untuk menyelesaikan model SIR. Lebih lanjut, makalah ini menyajikan penyelesaian model SIR yang dihasilkan dengan simulasi komputer. Hasil simulasi atas kedua metode tersebut memberikan rata-rata nilai mutlak selisih yang sangat kecil. Dengan demikian, skema numeris dan hasil simulasi dalam makalah ini dapat dipercaya kebenarannya. Dalam melakukan simulasi penyebaran penyakit menular, penggunaan dua metode yang berbeda disarankan agar hasil simulasi diyakini benar.

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Narender, G., G. Sreedhar Sarma, and K. Govardhan. "The Impact of Radiation Effect on MHD Stagnation-Point Flow of a Nanofluid over an Exponentially Stretching Sheet in the Presence of Chemical Reaction." International Journal of Applied Mechanics and Engineering 24, no. 4 (November 1, 2019): 125–39. http://dx.doi.org/10.2478/ijame-2019-0053.

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Abstract The present study is to investigate the effect of the chemical reaction parameter on stagnation point flow of magnetohydrodynamics field past an exponentially stretching sheet by considering a nanofluid. The problem is governed by governing coupled nonlinear partial differential equations with appropriate boundary conditions. The transformed non-dimensional and coupled governing ordinary differential equations are solved numerically using the fourth order Adams-Bashforth Moulton method. The effects of various dimensionless parameters on velocity, temperature and concentration fields are studied and then the results are presented in both tabular and graphical forms.

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Wang, Zhen. "A Numerical Method for Delayed Fractional-Order Differential Equations." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/256071.

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A numerical method for nonlinear fractional-order differential equations with constant or time-varying delay is devised. The order here is an arbitrary positive real number, and the differential operator is with the Caputo definition. The general Adams-Bashforth-Moulton method combined with the linear interpolation method is employed to approximate the delayed fractional-order differential equations. Meanwhile, the detailed error analysis for this algorithm is given. In order to compare with the exact analytical solution, a numerical example is provided to illustrate the effectiveness of the proposed method.

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Suescún-Díaz, D., M. Narváez-Paredes, and J. H. Lozano-Parada. "Calculation of nuclear reactivity using the generalised Adams-Bashforth-Moulton predictor corrector method." Kerntechnik 81, no. 1 (March 16, 2016): 86–93. http://dx.doi.org/10.3139/124.110591.

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Wicker, Louis J. "A Two-Step Adams–Bashforth–Moulton Split-Explicit Integrator for Compressible Atmospheric Models." Monthly Weather Review 137, no. 10 (October 1, 2009): 3588–95. http://dx.doi.org/10.1175/2009mwr2838.1.

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Abstract Split-explicit integration methods used for the compressible Navier–Stokes equations are now used in a wide variety of numerical models ranging from high-resolution local models to convection-permitting climate simulations. Models are now including more sophisticated and complicated physical processes, such as multimoment microphysics parameterizations, electrification, and dry/aqueous chemistry. A wider range of simulation problems combined with the increasing physics complexity may place a tighter constraint on the model’s time step compared to the fluid flow’s Courant number (e.g., the choice of the integration time step based solely on advective Courant number considerations may generate unacceptable errors associated with the parameterization schemes). The third-order multistage Runge–Kutta scheme has been very successful as the split-explicit integration method; however, its efficiency arises partially in its ability to use a time step that is 20%–40% larger than more traditional integration schemes. In applications in which the time step is constrained by other considerations, alternative integration schemes may be more efficient. Here a two-step third-order Adams–Bashforth–Moulton integrator is stably split in a similar manner as the split Runge–Kutta scheme. For applications in which the large time step is not constrained by the advective Courant number it requires less computational effort. Stability is demonstrated through eigenvalue analysis of the linear coupled one-dimensional velocity–pressure equations, and full two-dimensional nonlinear solutions from a standard test problem are shown to demonstrate solution accuracy and efficiency.

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Pan, Yongjun, Yansong He, and Aki Mikkola. "Accurate real-time truck simulation via semirecursive formulation and Adams–Bashforth–Moulton algorithm." Acta Mechanica Sinica 35, no. 3 (February 2, 2019): 641–52. http://dx.doi.org/10.1007/s10409-018-0829-1.

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Utami, Febry Noorfitriana, Ari Kusumastuti, and Juhari Juhari. "Simulasi Numerik Model Matematika Vibrasi Dawai Flying Fox Menggunakan Metode Adams-Bashforth-Moulton." Jurnal Riset Mahasiswa Matematika 2, no. 1 (November 1, 2022): 307–15. http://dx.doi.org/10.18860/jrmm.v1i7.14512.

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This study discusses numerical simulation using the Adams-Bashforth-Moulton (ABM) method of order 4 in the flying fox string mathematical model which is in the form of ordinary differential equations depending on time, consisting of two equations, namely the equation of the flying fox string y(t) and the angular equation of the flying fox string θ(t). This mathematical model is a model that has been constructed by Kusumastuti, et al (2017) and has been validated by comparing analytical solutions to its numerical solutions by Sari (2018). The analysis of the behavior of the Kusumastuti 2017 model conducted by Makfiroh (2020) shows that the phase portrait graph is in the form of a spiral with eigenvectors pointing towards the equilibrium point so that the mathematical model of the flying fox string vibration can be concluded as a valid mathematical model that is close to the actual situation. This study attempts to determine the numerical simulation of the deflection of the flying fox string y(t) and the numerical simulation of the angle of the flying fox string θ(t). The Runge-Kutta method of order 4 was used to generate 3 initial values for order 4 ABM. Next, a comparison of the y(t) and θ(t) solution graphs of order 4 ABM with the solution graph with Runge-Kutta of order 4 was performed in Sari 2018. The first simulation was carried out when h=1, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 fluctuated in the range of [0,0.09] with almost the same graphic profile, and the difference in the value of θ(t) ABM of order 4, and Runge-Kuta order 4 which is quite large with different graphic profiles. The second simulation was carried out when h=0.01, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 was fluctuating which also ranged from [0.0.09] with the same graphic profile, and the difference in the values of θ(t) ABM of order 4 and Runge -Kutta order 4 fluctuates in the range of [0,1] with the same graphic profile. So concluded that when h=0.01 comparison of ABM of order 4 and Runge-Kutta of order 4 is the best for displaying the graph profiles of y(t) and θ(t). Further research can explore numerical solutions using other methods.

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Utami, Febry Noorfitriana, Ari Kusumastuti, and Juhari Juhari. "Simulasi Numerik Model Matematika Vibrasi Dawai Flying Fox Menggunakan Metode Adams-Bashforth-Moulton." Jurnal Riset Mahasiswa Matematika 2, no. 1 (November 1, 2022): 10–18. http://dx.doi.org/10.18860/jrmm.v2i1.14512.

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This study discusses numerical simulation using the Adams-Bashforth-Moulton (ABM) method of order 4 in the flying fox string mathematical model which is in the form of ordinary differential equations depending on time, consisting of two equations, namely the equation of the flying fox string y(t) and the angular equation of the flying fox string θ(t). This mathematical model is a model that has been constructed by Kusumastuti, et al (2017) and has been validated by comparing analytical solutions to its numerical solutions by Sari (2018). The analysis of the behavior of the Kusumastuti 2017 model conducted by Makfiroh (2020) shows that the phase portrait graph is in the form of a spiral with eigenvectors pointing towards the equilibrium point so that the mathematical model of the flying fox string vibration can be concluded as a valid mathematical model that is close to the actual situation. This study attempts to determine the numerical simulation of the deflection of the flying fox string y(t) and the numerical simulation of the angle of the flying fox string θ(t). The Runge-Kutta method of order 4 was used to generate 3 initial values for order 4 ABM. Next, a comparison of the y(t) and θ(t) solution graphs of order 4 ABM with the solution graph with Runge-Kutta of order 4 was performed in Sari 2018. The first simulation was carried out when h=1, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 fluctuated in the range of [0,0.09] with almost the same graphic profile, and the difference in the value of θ(t) ABM of order 4, and Runge-Kuta order 4 which is quite large with different graphic profiles. The second simulation was carried out when h=0.01, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 was fluctuating which also ranged from [0.0.09] with the same graphic profile, and the difference in the values of θ(t) ABM of order 4 and Runge -Kutta order 4 fluctuates in the range of [0,1] with the same graphic profile. So concluded that when h=0.01 comparison of ABM of order 4 and Runge-Kutta of order 4 is the best for displaying the graph profiles of y(t) and θ(t). Further research can explore numerical solutions using other methods.

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McHugh, J. P., and D. Barkey. "Nonlinear evolution of a thin anodic film." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2202 (June 2017): 20160930. http://dx.doi.org/10.1098/rspa.2016.0930.

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The formation of pores in anodic aluminium oxide films is treated with a model equation. The model treats the oxide layer as a thin viscous liquid in two dimensions. Surface tension on the top boundary, electrostriction due to the external electric field and mass flow through the bottom boundary due to oxide formation are all included. Viscous flow is treated with the creeping flow assumption. The model equation is solved numerically using a Fourier spectral method in space and Adams–Bashforth/Adams–Moulton methods in time. Initial conditions include sinusoidal shapes as well as random shapes. The results show that pores form at the trough of the initial sinusoidal shape. Random shapes get smoothed before forming pore structures with spacing different than predicted by linear theory.

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Govardhan, K., G. Narender, and G. Sreedhar Sarma. "Heat and Mass transfer in MHD Nanofluid over a Stretching Surface along with Viscous Dissipation Effect." International Journal of Mathematical, Engineering and Management Sciences 5, no. 2 (April 1, 2020): 343–52. http://dx.doi.org/10.33889/ijmems.2020.5.2.028.

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A study of viscous dissipation effect of magnetohydrodynamic nanofluid flow passing over a stretched surface has been analyzed numerically. The formulated highly nonlinear equations for the above-mentioned flow are converted into first order ODEs. Utilizing the shooting technique along with the Adams-Bashforth Moulton Method is used to solve the BVP by using the computational software FORTRAN. The numerical results are computed by choosing different values of the involved physical parameters and compared with the earlier published results. The graphical numerical results of different physical quantities of interest are presented to analyze their dynamics under the varying physical quantities.

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Özdemir, Necati, Sümeyra Uçar, and Beyza Billur İskender Eroğlu. "Dynamical Analysis of Fractional Order Model for Computer Virus Propagation with Kill Signals." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 3-4 (May 26, 2020): 239–47. http://dx.doi.org/10.1515/ijnsns-2019-0063.

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AbstractThe kill signals are alert about possible viruses that infect computer network to decrease the danger of virus propagation. In this work, we focus on a fractional-order SEIR-KS model in the sense of Caputo derivative to analyze the effects of kill signal nodes on the virus propagation. For this purpose, we first prove the existence and uniqueness of the model and give qualitative analysis. Then, we obtain the numerical solution of the model by using the Adams–Bashforth–Moulton algorithm. Finally, the effects of model parameters are demonstrated with graphics drawn by MATLAB program.

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Rakkiyappan, R., R. Sivasamy, and Ju H. Park. "Synchronization of fractional-order different memristor-based chaotic systems using active control." Canadian Journal of Physics 92, no. 12 (December 2014): 1688–95. http://dx.doi.org/10.1139/cjp-2013-0671.

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In this article, synchronization of two different fractional-order memristor-based chaotic systems is considered. To achieve synchronization, an active control technique is used. The main proof is concerned with the problem of synchronization of memristor-based Lorenz systems with memristor-based Chua’s circuits. Numerical simulations of fractional-order memristor-based chaotic systems are performed by using the Caputo version and a predictor–corrector algorithm for fractional-order differential equations, which is a generalization of the Adams–Bashforth–Moulton method. From the simulations, it will be verified that the proposed control method is effective in achieving synchronization.

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Zayernouri, Mohsen, and Anastasios Matzavinos. "Fractional Adams–Bashforth/Moulton methods: An application to the fractional Keller–Segel chemotaxis system." Journal of Computational Physics 317 (July 2016): 1–14. http://dx.doi.org/10.1016/j.jcp.2016.04.041.

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Chiou, J. C., and S. D. Wu. "On the generation of higher order numerical integration methods using lower order Adams–Bashforth and Adams–Moulton methods." Journal of Computational and Applied Mathematics 108, no. 1-2 (August 1999): 19–29. http://dx.doi.org/10.1016/s0377-0427(99)00096-5.

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46

Tul Ain, Qura, T. Sathiyaraj, Shazia Karim, Muhammad Nadeem, and Patrick Kandege Mwanakatwe. "ABC Fractional Derivative for the Alcohol Drinking Model using Two-Scale Fractal Dimension." Complexity 2022 (June 6, 2022): 1–11. http://dx.doi.org/10.1155/2022/8531858.

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Drinking kills a significant proportion of individuals every year, particularly in low-income communities. An impulsive differential equation system is used to explore the effectiveness of activated charcoal in detoxifying the body after methanol poisoning. Our impression of activated charcoal is shaped by the fractional dynamics of the problem, which leads to speedy and low-cost first aid. The adsorption capacity of activated charcoal is investigated using impulsive differential equations. The ABC fractional operator’s findings paint a more realistic image of first aid in public and primary health centers, which can assist to reduce methanol poisoning deaths. Numerical simulations are provided using generalized Adams–Bashforth–Moulton method (GABMM).

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Zhu, Darui, Ling Liu, and Chongxin Liu. "Adaptive Pinning Synchronization Control of the Fractional-Order Chaos Nodes in Complex Networks." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/936985.

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Adaptive pinning synchronization control is studied for a class of fractional-order complex network systems which are constructed depending on small-world network algorithm. Based on the fractional-order stability theory, the suitable adaptive control scheme is designed to guarantee global asymptotic stability of all the nodes in complex network systems and the node selected algorithm is given. In numerical implementation, it is shown that the numerical solution of the fractional-order complex network systems can be obtained by applying an improved version of Adams-Bashforth-Moulton algorithm. Furthermore, simulation results are provided to confirm the validity and synchronization performance of the advocated design methodology.

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Zhu, Darui, Chongxin Liu, and Bingnan Yan. "Drive-Response Synchronization of a Fractional-Order Hyperchaotic System and Its Circuit Implementation." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/815765.

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A novel fractional-order hyperchaotic system is proposed; the theoretical analysis and numerical simulation of this system are studied. Based on the stability theory of fractional calculus, we propose a novel drive-response synchronization scheme. In order to achieve this synchronization control, the Adams-Bashforth-Moulton algorithm is studied. And then, a drive-response synchronization controller is designed to realize the synchronization of the drive and response system, and the simulation results are given. At last, the fractional oscillator circuit of the new fractional-order hyperchaotic system is designed based on the EWB software, and it is verified that the simulation results of the fractional-order oscillator circuit are consistent with the numerical simulation results through circuit simulation.

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Parovik, Roman Ivanovich. "Studies of the Fractional Selkov Dynamical System for Describing the Self-Oscillatory Regime of Microseisms." Mathematics 10, no. 22 (November 10, 2022): 4208. http://dx.doi.org/10.3390/math10224208.

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A non-linear fractional Selkov dynamic system for mathematical modeling of microseismic phenomena is proposed. This system is a generalization of the previously known Selkov system, which has self-oscillatory modes and is used in biology to describe glycolytic vibrations of the substrate and product. The Selkov fractional dynamical system takes into account the influence of heredity and is described using derivative fractional orders. The article investigates the Selkov fractional dynamic model using the Adams–Bashforth–Moulton numerical method, constructs oscillograms and phase trajectories, and studies the equilibrium points. Based on the spectra of the maximum Lyapunov exponents, it is shown that in the fractional dynamic model there can be relaxation and damped oscillations.

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Tutueva, A. V., D. N. Butusov, E. E. Kopets, V. G. Rybin, and A. G. Davidchuk. "Semi-explicit multistep Adams-Bashforth-Moulton methods for solving stiff systems of ordinary differential equations." Izvestiâ vysših učebnyh zavedenij. Priborostroenie 64, no. 8 (August 31, 2021): 599–607. http://dx.doi.org/10.17586/0021-3454-2021-64-8-599-607.

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