A Discretization Approach for the Nonlinear Fractional Logistic Equation

The present study aimed to develop and investigate the local discontinuous Galerkin method for the numerical solution of the fractional logistic differential equation, occurring in many biological and social science phenomena. The fractional derivative is described in the sense of Liouville-Caputo. Using the upwind numerical fluxes, the numerical stability of the method is proved in the L∞ norm. With the aid of the shifted Legendre polynomials, the weak form is reduced into a system of the algebraic equations to be solved in each subinterval. Furthermore, to handle the nonlinear term, the technique of product approximation is utilized. The utility of the present discretization technique and some well-known standard schemes is checked through numerical calculations on a range of linear and nonlinear problems with analytical solutions.     

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space—time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.     

分数阶对流扩散方程的特征有限元方法

讨论非线性分数阶对流扩散方程的特征有限元方法.利用特征线法和分数阶有限元框架,构建一种基于特征方向的全离散有限元格式.模拟物理问题,并在数值上与常规有限元格式进行比较,计算结果表明：该方法能准确地捕捉到控制方程的精确解,即使是在对流效应占优时,也具有稳定性好和逼近精度高等特征.     

Numerical Solutions of Fractional Boussinesq Equation

Based upon the Adomian decomposition method, a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition, which is introduced by replacing some order time and space derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations. The solutions of our model equation are calculated in the form of convergent series with easily computable components.     

Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.     

Strict Positivity of a Solution to a One-Dimensional Kac Equation Without Cutoff

We consider the solution of a one-dimensional Kac equation without cutoff built by Graham and Méléard. Recalling that this solution is the density of a Poisson driven nonlinear stochastic differential equation, we develop Bismut's approach of the Malliavin calculus for Poisson functionals in order to prove that this solution is strictly positive on ]0,[x.     

Fractional Stochastic Differential Equation Approach for Spreading of Diseases

The nonlinear fractional stochastic differential equation approach with Hurst parameter H within interval H∈(0,1) to study the time evolution of the number of those infected by the coronavirus in countries where the number of cases is large as Brazil is studied. The rises and falls of novel cases daily or the fluctuations in the official data are treated as a random term in the stochastic differential equation for the fractional Brownian motion. The projection of novel cases in the future is treated as quadratic mean deviation in the official data of novel cases daily since the beginning of the pandemic up to the present. Moreover, the rescaled range analysis (RS) is employed to determine the Hurst index for the time series of novel cases and some statistical tests are performed with the aim to determine the shape of the probability density of novel cases in the future.     

Laguerre Wavelet Approach for a Two-Dimensional Time–Space Fractional Schrödinger Equation

This article is devoted to the determination of numerical solutions for the two-dimensional time–spacefractional Schrödinger equation. To do this, the unknown parameters are obtained using the Laguerre wavelet approach. We discretize the problem by using this technique. Then, we solve the discretized nonlinear problem by means of a collocation method. The method was proven to give very accurate results. The given numerical examples support this claim.     

A Review of Fractional Order Entropies

Fractional calculus (FC) is the area of calculus that generalizes the operations of differentiation and integration. FC operators are non-local and capture the history of dynamical effects present in many natural and artificial phenomena. Entropy is a measure of uncertainty, diversity and randomness often adopted for characterizing complex dynamical systems. Stemming from the synergies between the two areas, this paper reviews the concept of entropy in the framework of FC. Several new entropy definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. However, FC is not yet well disseminated in the community of entropy. Therefore, new definitions based on FC can generalize both concepts in the theoretical and applied points of view. The time to come will prove to what extend the new formulations will be useful.     

Numerical Method for the Time Fractional Fokker-Planck Equation

In this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.     

Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method

In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.     

A Collocation Method for Solving Fractional Riccati Differential Equation

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation. The fractional derivatives are described in the Caputo sense. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, the systems of nonlinear algebraic equation are derived. We further solve the system of nonlinear algebraic equation using Maple 13 and thus obtain the coefficients of the generalized Taylor expansion. Illustrative examples are presented to demonstrate the effectiveness of the proposed method.     

On Some Modified Methods on Fractional Delay and Nonlinear Integro-Differential Equation

The fundamental objective of this work is to construct a comparative study of some modified methods with Sumudu transform on fractional delay integro-differential equation. The existed solution of the equation is very accurately computed. The aforesaid methods are presented with an illustrative example.     

He’s Homotopy Perturbation Method and Fractional Complex Transform for Analysis Time Fractional Fornberg-Whitham Equation

In this article, time fractional Fornberg-Whitham equation of He’s fractional derivative is studied. To transform the fractional model into its equivalent differential equation, the fractional complex transform is used and He’s homotopy perturbation method is implemented to get the approximate analytical solutions of the fractional-order problems. The graphs are plotted to analysis the fractional-order mathematical modeling.     

Conformable Fractional Nikiforov-Uvarov Method

We introduce conformable fractional Nikiforov-Uvarov (NU) method by means of conformable fractional derivative which is the most natural definition in non-integer calculus. Since, NU method gives exact eigenstate solutions of Schrödinger equation (SE) for certain potentials in quantum mechanics, this method is carried into the domain of fractional calculus to obtain the solutions of fractional SE. In order to demonstrate the applicability of the conformable fractional NU method, we solve fractional SE for harmonic oscillator potential, Woods-Saxon potential, and Hulthen potential.     

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

In this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover, the use of Legendre wavelet is found to be accurate, simple and fast.     

Group Analysis,Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation

The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular,an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.     

A Physical Axiomatic Approach to Schrodinger's Equation

The Schrodinger equation for non-relativistic quantum systems is derived from some classical physics axioms within an ensemble hamiltonian framework. Such an approach enables one to understand the structure of the equation, in particular its linearity, in intuitive terms. Furthermore it allows for a physically motivated and systematic investigation of potential generalisations which are briefly discussed. Pacs: 03.65.-w; 04.20.-q; 03.30.+ p; 11.10. Lm