Academic literature on the topic 'Adams-Bashforth-Moulton'

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.

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.

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

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.

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.

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.

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.

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.

In this thesis, we construct a new optimal contractivity-preserving (CP) explicit, 2-step, 6-stage, 6-derivative, Hermite--Birkhoff--Obrechkoff method of order 13, denoted by HBO(13) with nonnegative coefficients, for solving nonstiff first-order initial value problems y'=f(t,y), y(t_0)=y_0. This new method is the combination of a CP 2-step, 6-derivative, Hermite--Obrechkoff of order 9, denoted by HO(9), and a 6-stage Runge-Kutta method of order 5, denoted by RK(6,5). The new HBO(13) method has order 13. We compare this new method, programmed in Matlab, to Adams-Bashforth-Moulton method of order 13 in PECE mode, denoted by ABM(13), by testing them on several frequently used test problems, and show that HBO(13) is more efficient with respect to the CPU time, the global error at the endpoint of integration and the relative energy error. We show that the new HBO(13) method has a larger scaled interval of absolute stability than ABM(13) in PECE mode.

Cancer is one of the most serious illnesses in all of the world. Although most of the cancer patients are treated with chemotherapy, radiotherapy and surgery, wide research is conducted related to experimental and theoretical immunology. In recent years, the research on cancer immunotherapy has led to major medical advances. Cancer immunotherapy refers to the stimulation of immune system to deal with cancer cells. In medical practice, it is mainly achieved by using effector cells such as activated T-cells and Interleukin-2 (IL-2), which is the main cytokine responsible for lymphocyte activation, growth and differentiation. A well-known mathematical model, named as Kirschner-Panetta (KP) model, represents richly the dynamics of the interaction between cancer cells, IL-2 and the effector cells. The dynamics of the KP model is described and the solution to which is approximated by using polynomial approximation based methods such as Adomian decomposition method and differential transform method. The rich nonlinearity of the KP model causes these approaches to become so complicated in order to deal with the representation of polynomial approximations. It is illustrated that the approximated polynomials are in good agreement with the solution obtained by common numerical approaches. In the KP model, the growth of the tumour cells can be expressed by a linear function or any limited-growth function such as logistic equation, in which the cancer population possesses an upper bound mentioned as carrying capacity. Effector cells and IL-2 construct two external sources of medical treatment to stimulate immune system to eradicate cancer cells. Since the main goal in immunotherapy is to remove the tumour cells with the least probable medication side effects, an advanced version of the model may include a time dependent external sources of medical treatment, meaning that the external sources of medical treatment could be considered as control functions of time and therefore the optimum use of medical sources can be evaluated in order to achieve the optimal measure of an objective function. With this sense of direction, two distinct strategies are explored. The first one is to only consider the external source of effector cells as the control function to formulate an optimal control problem. It is shown under which circumstances, the tumour is eliminated. The approach in the formulation of the optimal control is the Pontryagin maximum principal. Furthermore the optimal control problem will be dealt with using particle swarm optimization (PSO). It is shown that the obtained results are significantly better than those obtained by previous researchers. The second strategy is to formulate an optimal control problem by considering both the two external sources as the controls. To our knowledge, it is the first time to present a multiple therapeutic protocol for the KP model. Some MATLAB routines are develop to solve the optimal control problems based on Pontryagin maximum principal and also the PSO. As known, fractional differential equations are more appropriate to describe the persistent memory of physical phenomena. Thus, the fractional KP model is defined in the sense of Caputo differentiation operator. An effective method for numerical treatment of the model is described, namely Predictor-Corrector method of Adams-Bashforth-Moulton type. A robust MATLAB routine is coded based on the mentioned approach and the solution obtained will be compared with those of the classical KP model. The code is prepared in such a way to be able to deal with systems of fractional differential equations, in which each equation has its own fractional order (i.e. multi-order systems of fractional differential equations). The theorems for existence of solutions and the stability analysis of the fractional KP model are represented. In this regard, a frequently used method of solving fractional differential equations (FDEs) is described in details, namely multi-step generalized differential transform method (MSGDTM), then it is illustrated that the method neglects the persistent memory property and takes the incorrect approach in dealing with numerical solutions of FDEs and therefore it is unfit to be used in differential equations governed by fractional differentiation operators. The sigmoidal behavior of the solution to the logistic equation caused it to be one of the most versatile models in natural sciences and therefore the fractional logistic equation would be a relevant problem to be dealt with. Thus, a power series of Mittag-Leffer functions is introduced, the behaviour of which is in good agreement with the solution to fractional logistic equation (FLE), and then a fractional integro-differential equation is represented and proved to be satisfied with the power series of Mittag-Leffler function. The obtained fractional integro-differential equation is named as modified fractional differential equation (MFDL) and possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in the thesis, may be appropriately applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics. Inverse problems to FDEs occur in many branches of science. Such problems have been investigated, for instance, in fractional diffusion equation and inverse boundary value problem for semi- linear fractional telegraph equation. The determination of the order of fractional differential equations is an issue, which has been analyzed and discussed in, for instance, fractional diffusion equations. Thus, fractional order estimation has been conducted for some classes of linear fractional differential equations, by introducing the relationship between the fractional order and the asymptotic behaviour of the solutions to linear fractional differential equations. Fractional optimal control problems, in which the system and (or) the objective function are described based on fractional derivatives, are much more complicated to be solved by using a robust and reliable numerical approach. Thus, a MATLAB routine is provided to solve the optimal control for fractional KP model and the obtained solutions are compared with those of classical KP model. It is shown that the results for fractional optimal control problems are better than classical optimal control problem in the sense of the amount of drug administration.

The explicit numerical integration method, introduced and proposed in the paper given by Chiou and Wu [1], is further developed. The method is based on the relationship that m-step Adams-Moulton method is linear convex combination of the (m − 1)-step Adams-Moulton and m-step Adams-Bashforth method with a fixed weighting coefficients. The general form taken from Chiou and Wu [1] is used to evaluate the recurrence expressions using the different number of previous mesh points. The explicit expressions are given for modified 3-step predictor-corrector method. The numerical algorithms are given for first and second-order nonlinear initial value problems and for system of ordinary differential equations. Some numerical examples, for different kind of problems, are used to demonstrate the efficiency and the accuracy of the proposed numerical method. The calculated numerical solutions show superiority of presented modified predictor-corrector method to standard Adams-Bashforth-Moulton predictor-corrector method.

Abstract This paper details a real-time simulation of an articulating wheel loader, which is comprised of a multibody system modeling the chassis and the bucket assembly and a set of subsystems. The hydraulic subsystem is modeled by a set of ODE’s which represent the oil pressure fluctuations in the system. An Adams-Bashforth-Moulton integration algorithm has been implemented using the Nordsieck form to develop a constant step-size multirate integration scheme, modeling the interaction between the hydraulic subsystem and multibody dynamics models. An example illustrating the simulation of a wheel loader bucket operation is presented at the end of the paper.

Abstract This paper details a constant stepsize, multirate integration scheme which has been proposed for multibody dynamic analysis. An Adams-Bashforth Moulton integration algorithm has been implemented, using the Nordsieck form to store internal integrator information, for multirate integration. A multibody system has been decomposed into several subsystems, treating inertia coupling effects of subsystem equations of motion as the inertia forces. To each subsystem, different rate Nordsieck form of Adams integrator has been applied to solve subsystem equations of motion. Higher order derivative information from the integrator provides approximation of inertia force computation in the decomposed subsystem equations of motion. To show the effectiveness of the scheme, simulations of a vehicle multibody system that consists of high frequency suspension motion and low frequency chassis motion have been carried out with different tire excitation forces. Efficiency of the proposed scheme has been also investigated.

In this paper, the adaptive synchronization of a fractional-order complex T system with a random parameter is analyzed. Firstly, the Laguerre polynomial approximation method is applied to investigate the fractional-order system with a random parameter which obeys an exponential distribution. Based on this method, the stochastic system is reduced into the equivalent deterministic one. The improved Adams-Bashforth-Moulton algorithm with the predictor-correctors scheme is used to solve the approximately deterministic system numerically. Based on the stability theory of fractional-order systems, the synchronization for the deterministic system with unknown parameters is realized by designing appropriate synchronization controllers and estimation law for uncertain parameters. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.

An efficient numerical method, the fractional Chebyshev collocation method, is proposed for obtaining the solution of a system of linear fractional order delay-differential equations (FDDEs). It is shown that the proposed method overcomes several limitations of current numerical methods for solving linear FDDEs. For instance, the proposed method can be used for linear incommensurate order fractional differential equations and FDDEs, has spectral convergence (unlike finite differences), and does not require a canonical form. To accomplish this, a fractional differentiation matrix is derived at the Chebyshev-Gauss-Lobatto collocation points by using the discrete orthogonal relationship of the Chebyshev polynomials. Then, using two proposed discretization operators for matrix functions results in an explicit form of solution for a system of linear FDDEs with discrete delays. The advantages of using the fractional Chebyshev collocation method are demonstrated in two numerical examples in which the proposed method is compared with the Adams-Bashforth-Moulton method.

Abstract Hydraulic fracture/reservoir properties and fluid-in-place can be quantified by using rate-transient analysis (RTA) techniques applied to flow rates/pressures gathered from multi-fractured horizontal wells (MFHWs) completed in unconventional reservoirs. These methods are commonly developed for the analysis of production data from single wells without considering communication with nearby wells. However, in practice, wells drilled from the same pad can be in strong hydraulic communication with each other. This study aims to develop the theoretical basis for analyzing production data from communicating MFHWs completed in single-phase shale gas reservoirs. A simple and practical semi-analytical method is developed to quantify the communication between wells drilled from the same pad by analyzing online production data from the individual wells. This method is based on the communicating tanks model and employs the concepts of macroscopic material balance and the succession of pseudo-steady states. A set of nonlinear ordinary differential equations (ODEs) are generated and solved simultaneously using the efficient Adams-Bashforth-Moulton algorithm. The accuracy of the solutions is verified against robust numerical simulation. In the first example provided, a MFHW well-pair is presented where the wells are communicating through primary hydraulic fractures with different communication strengths. In the subsequent examples, the method is extended to consider production data from a three-well and a six-well pad with wine-rack-style completions. The developed model is flexible enough to account for asynchronous wells that are producing from distinct reservoir blocks with different fracture/rock properties. For all the studied cases, the semi-analytical method closely reproduces the results of fully numerical simulation. The results demonstrate that, in some cases, when new wells start to produce, the production rates of existing wells can drop significantly. The amount of productivity loss is a direct function of the communication strengths between the wells. The new method can accurately quantify the communication strength between wells through transmissibility multipliers between the hydraulic fractures that are adjusted to match individual well production data. In this study, a new simple and efficient semi-analytical method is presented that can be used to analyze online production data from multiple wells drilled from a pad simultaneously with minimal computation time. The main advantage of the developed method is its scalability, where additional wells can be added to the system very easily.