A New Approach to Population Growth Model Involving a Logistic Differential Equation of Fractional Order.

On the fractional-order logistic equation

In This paper, we propose a numerical algorithm for solving nonlinear fractional-order Logistic differential equation (FLDE) by using Sumudu decomposition method (SDM). This method is a combination of the Sumudu transform method and decomposition method. We have apply the concepts of fractional calculus to the well known population growth modle inchaotic dynamic. The fractional derivative is described in the Caputosense. The numerical results shows that the approach is easy to implement and accurate when applied to various fractional differentional equations.

This paper deals with the approximate and analytical solutions of non linear fractional differential equations namely, Lorenz System of Fractional Order and the obtained results are compared with the results of Homotopy Perturbation method and Variational Iteration method in the standard integer order form. The reason for using fractional order differential equations is that, fractional order differential equations are naturally related to systems with memory which exists in most systems and also they are closely related to fractals which are abundant in systems. The derived results are more general in nature. The solution of such equations spread at a faster rate than the classical differential equations and may exhibit asymmetry. A few numerical methods for the solution of fractional differential equation models have been discussed in the literature. However many  of such methods are used for very specific types of differential equations, often just linear equations or even smaller classes, but this method shows the high accuracy and efficiency of the approach. Special cases involving the Mittag-Leffler function and exponential function are also considered.   Keywords: Generalized Mittag-Leffler function, Caputo fractional derivative, Lorenz system.   AMS 2010 Subject Classification: 26A33, 33E12.

Solution of a fractional logistic ordinary differential equation

In many cases, the order of a differential equation is a natural number. However, in some applications, this order can be in the form of a fractional number, so that the equation is then called a fractional differential equation. In this paper, we study the numerical solution of the fractional logistic differential equation with order α, where 0 < α ≤ 1. The equation can be considered as one of the fractional Riccati differential equations. The numerical methods we use are the Adomian decomposition method (ADM) and the variational iteration method (VIM). We use the Caputo derivative to find the solution. The effect of the fractional-order into the transient solution is studied graphically to find the interpretation in the logistic population growth model.

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

This paper presents an algorithm to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo-fractional derivatives. The proposed differentiation matrices named fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev–Gauss–Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. Fractional Chebyshev collocation method as a framework for solving fractional differential equations with multi-order Caputo derivatives is also presented. The numerical stability of spectral methods for linear fractional-order differential equations (FDEs) is studied by using the proposed framework. Furthermore, the proposed fractional Chebyshev differentiation matrices obtain the fractional-order derivative of a function with spectral convergence. Therefore, they can be used in various spectral collocation methods to solve a system of linear or nonlinear multi-ordered FDEs. To illustrate the true advantages of the proposed fractional Chebyshev differentiation matrices, the numerical solutions of a linear FDE with a highly oscillatory solution, a stiff nonlinear FDE, and a fractional chaotic system are given. In the first, second, and forth examples, a comparison is made with the solution obtained by the proposed method and the one obtained by the Adams–Bashforth–Moulton method. It is shown the proposed fractional differentiation matrices are highly efficient in solving all the aforementioned examples.

2 - Basics of fractional calculus and fractional order differential equations

The computation of fractional order differential and integration equations is highly utilized in numerous fields, such as mathematics in biology, ecology, physics, chemistry, and so on. This article aims at presenting methods for analytical and numerical solutions, Lie’s symmetry analysis, computing conservation laws for Burger’s fractional order differential equation (FODE), and reducing time-fractional cylindrical-Burgers equation order. Hence, we have employed a generalized (G′/G) expansion method for analytical solutions and a non-standard finite difference scheme for numerically solving Burger’s FODE. Additionally, the fractional derivative generalization of Noether's theorem has been utilized to compute the equation’s conservation laws. Numerical results have been reported here to approve theoretical results acquired in non-standard finite difference schemes. The proposed generalized (G′/G) expansion method is a simple, accurate, and practical approach to problem solving. Additionally, this method can be employed to execute non-linear wave equations. Programs including MATLAB and Maple have been utilized to simplify the process of solving complex equations.

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation.

Stability, bifurcation and a new chaos in the logistic differential equation with delay

Differential equation models are encountered in various fields of study such as science, technology, engineering, economics, demography, among many others. Population growth models in the form of ordinary differential equations (ODEs) are vastly adopted to investigate the dynamics of growth rate. In this paper, two growth models are considered namely Malthus model and Verhulst model, which describe exponential and logistic growth, respectively. Various approximate methods have been adopted to solve these models, however the solutions obtained by these existing approaches could still be improved when considering the absolute error. For this reason, the models are solved in this article by employing multistep block methods which possess improved accuracy due to the implementation of the schemes as simultaneous integrators for the solution of the growth models. Asides the block methods showing impressive accuracy compared to existing approaches in literature, many studies arbitrarily choose step-length values for block methods and there is a need to justify what informs the choice of step-length values. Hence, varying step-length values are chosen in this article for the developed multistep block methods and compared with existing approaches to demonstrate the improved accuracy of the block methods, and also determine the best step-length value to use when aiming for a specific level of accuracy. From the numerical results, the article clearly shows that the block methods obtain better accuracy and provides information on step-length choices for block methods when solving Malthus and Verhulst Growth Models.

The article, being a continuation of the first one [A.A. Kilbas and J.J. Trujillo (2001). Differential equations of fractional order. Methods, results and problems, I. Applicable Analysis , 78 (1-2), 153-192.], deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. The methods and the results in the theory of such fractional differential equations are presented including the Dirichlet-type problem for ordinary fractional differential equations, studying such equations in spaces of generalized functions, partial fractional differential equations and more general abstract equations, and treatment of numerical methods for ordinary and partial fractional differential equations. Problems and new trends of research are discussed.

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

The differential equations were considered as fractional order differential equations in literature. Homotopy Analysis Method was used to obtain analytical solutions of these equations. We applied reverse processes to analytical solutions of some fractional order differential equations, and observed that solutions could not satisfy the corresponding equations. Due to this case, we proposed a new approach for fractional order derivative and it was verified by using this new approach that any differential equations cannot be converted into fractional order differential equations so simply.