A numerical study of fractional order population dynamics model
TL;DR: In this paper, the population dynamics model including the predator-prey problem and the logistic equation are generalized by using fractional operator in term of Caputo-Fabrizio derivative (CF-derivative).
Abstract: In this research, the population dynamics model including the predator-prey problem and the logistic equation are generalized by using fractional operator in term of Caputo-Fabrizio derivative (CF-derivative). The models under study include of fractional Lotka-Volterra model (FLVM), fractional predator-prey model (FPPM) and fractional logistic model of population growth (FLM-PG) with variable coefficients. After that a numerical scheme is presented to obtain numerical solutions of these fractional models. These solutions are made using three-step Adams-Bashforth scheme. To show the efficiency and the accuracy of the present scheme, a few examples are evaluated. The numerical simulations of the results are depicted the accuracy of the present scheme.
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TL;DR: In this paper, a mathematical model of brain tumor growth and diffusion is presented, which is an extension of a simple two-dimensional mathematical model derived from fractional operator in terms of Caputo which is called the fractional Burgess equations (FBEs).
Abstract: In this paper, we present a mathematical model of brain tumor. This model is an extension of a simple two-dimensional mathematical model of glioma growth and diffusion which is derived from fractional operator in terms of Caputo which is called the fractional Burgess equations (FBEs). To obtain a solution for this model, a numerical technique is presented which is based on operational matrix. First, we assume the solution of the problem under the study is as an expansion of the Bernoulli polynomials. Then with combination of the operational matrix based on the Bernoulli polynomials and collocation method, the problem under the study is changed to a system of nonlinear algebraic equations. Finally, the proposed technique is simulated and tested on three types of the FBEs to confirm the superiority and accuracy.
29 citations
TL;DR: In this paper, a numerical operational matrix approach based on Euler wavelets is proposed to solve the nonlinear pantograph Volterra delay integro-differential equation of fractional order.
12 citations
TL;DR: In this article , a numerical operational matrix approach based on Euler wavelets is proposed to solve the pantograph Volterra delay integro-differential equation of fractional order.
9 citations
TL;DR: In this paper , a numerical algorithm is presented to obtain approximate solution of distributed order integro-differential equations, the approximate solution is expressed in the form of a polynomial with unknown coefficients and in place of differential and integral operators, they make use of matrices that they deduce from the shifted Legendre polynomials.
Abstract: In this paper, a numerical algorithm is presented to obtain approximate solution of distributed order integro-differential equations. The approximate solution is expressed in the form of a polynomial with unknown coefficients and in place of differential and integral operators, we make use of matrices that we deduce from the shifted Legendre polynomials. To compute the numerical values of the polynomial coefficients, we set up a system of equations that tallies with the number of unknowns, we achieve this goal through the Legendre–Gauss quadrature formula and the collocation technique. The theoretical aspects of the error bound are discussed. Illustrative examples are included to demonstrate the validity and applicability of the method.
9 citations
01 May 2022
TL;DR: In this article , the authors developed accurate collocation techniques for solving a fractional fractional-order Lotka-Volterra (LV) predator-prey model, which has a wide variety of applications in biology to simulate the interaction between two different species.
Abstract: This paper aims to develop accurate novel collocation techniques for solving a fractional fractional-order Lotka–Volterra (LV) predator–prey model. This type of equation has a wide variety of applications in biology to simulate the interaction between two different species. The fractional-order operator is defined in the Caputo sense. We introduce some new polynomials for solving this type of equation named the Morgan-Voyce polynomials. These new techniques named direct and quasilinear Morgan-Voyce techniques are presented and then used to solve the LV system revealing some of the new features of the model. Some of the advantages of the application of these techniques are that they are straightforward along with high accuracy. Detailed error and convergence analysis for the proposed methods are provided revealing the upper bound for these techniques. To test the accuracy of the proposed methods, the methods are tested on several examples with different values of the parameters and fractional orders and also compared with each other and with other related methods from the literature. The results are demonstrated through tables and figures and conclude that the provided technique is highly accurate compared to the other methods. The results revealed the agreement between the obtained approximate solution and the dynamics of the described model. These methods can be considered as promising to adapt to other similar models with application to different areas of engineering and biology. • Introducing two accurate methods to solve the fractional-order Lotka–Volterra model. • Utilizing the generalized novel bases called the Morgan-Voyce functions. • Testifying the accurateness of the methods via defining the residual error functions. • Showing the superiority of the developed techniques through numerical simulations.
9 citations
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2,657 citations
TL;DR: In this article, a new fractional derivative with non-local and no-singular kernel was proposed and applied to solve the fractional heat transfer model, and some useful properties of the new derivative were presented.
Abstract: In this manuscript we proposed a new fractional derivative with non-local and
no-singular kernel. We presented some useful properties of the new derivative
and applied it to solve the fractional heat transfer model.
2,364 citations
01 Jan 2015
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.
1,972 citations
Posted Content•
TL;DR: In this paper, a new fractional derivative with non-local and no-singular kernel was proposed and applied to solve the fractional heat transfer model, and some useful properties of the new derivative were presented.
Abstract: In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.
1,372 citations
TL;DR: The authors propose des relations constitutives generalisees for les materiaux viscoelastiques dans lesquelles les derivees d'ordre entier par rapport au temps sont remplacees par des derivees de ordre fractionnaire.
Abstract: On propose des relations constitutives generalisees pour les materiaux viscoelastiques dans lesquelles les derivees d'ordre entier par rapport au temps sont remplacees par des derivees d'ordre fractionnaire. De tels modeles decrivent bien le comportement des materiaux reels
1,070 citations