44 research outputs found
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
Multi-step Homotopy Analysis Method for Solving Malaria Model
In this paper, we consider the modified epidemiological malaria model proposed by Abadi and Harald. The multi-step homotopy analysis method (MHAM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. We illustrated the profiles of the solutions of each of the compartments. Figurative comparisons between the MHAM and the classical fourth-order reveal that this method is very effective
A new analytic numeric method solution for fractional modified epidemiological model for computer viruses
Computer viruses are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue, and we present one such here. We consider the modified epidemiological model for computer viruses (SAIR) proposed by J. R. C. Piqueira and V. O. Araujo. This model includes an antidotal population compartment (A) representing nodes of the network equipped with fully effective anti-virus programs. The multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MSGDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that this method is very effective. Mathematica 9 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution
Solution of the SIR models of epidemics using MSGDTM
Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we use the multi-step generalized differential transform method (MSGDTM) to approximate the numerical solution of the SIR model and numerical simulations are presented graphically
Enhanced Multistage Differential Transform Method: Application to the Population Models
We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).open4
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Multistep generalized transformation method applied to solving equations of discrete and continuous time-fractional enzyme kinetics
In this paper, Caputo based Michaelis–Menten kinetic model based on Time Scale Calculus (TSC) is proposed. The main reason for its consideration is a study of tumor cells population growth dynamics. In the particular case discrete-continuous time kinetics, Michaelis–Menten model is numerically treated, using a new algorithm proposed by authors, called multistep generalized difference transformation method (MSGDETM). In addition numerical simulations are performed and is shown that it represents the upgrade of the multi-step variant of generalized differential transformation method (MSGDTM). A possible conditions for its further development are discussed and possible experimental verification is described.This is the peer-reviewed version of the article: Vosika, Z., Mitić, V.V., Vasić, A., Lazović, G., Matija, L., Kocić, L.M., 2017. Multistep generalized transformation method applied to solving equations of discrete and continuous time-fractional enzyme kinetics. Communications in Nonlinear Science and Numerical Simulation 44, 373–389. [https://doi.org/10.1016/j.cnsns.2016.08.024
Fractional calculus: numerical methods and SIR models
Fractional calculus is ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. The idea of generalizing differential operators to a non-integer order, in particular to the order 1/2, first appears in the correspondence of Leibniz with L’Hopital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. In the following three hundred years a lot of mathematicians contributed to the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand and Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), and many others. Yet, it is only after the First Conference on Fractional Calculus and its applications that the fractional calculus becomes one of the most intensively developing areas of mathematical analysis. Recently, many mathematicians and applied researchers have tried to model real processes using the fractional calculus. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time which can be successfully achieved by using fractional calculus. In other words, the nature of the definition of the fractional derivatives have provided an excellent instrument for the modeling of memory and hereditary properties of various materials and processes