A New Fractional Representation of the Higher Order Taylor Scheme

On the fractional-order logistic equation

Fractional order has the characteristics of memory and non-locality and it is different with integer order. Therefore, fractional differential equations can be used to describe some abnormal natural phenomena. At the same time, how to solve the fractional order partial differential equation and differential equations with fractional order has become a very important research field. Besides analytic solution, it is also important to investigate the numerical methods for fractional differential equations. In the paper, fundamental solution of the time fractional partial differential equation has been deduced, which is derived by Furrier transform and Laplace transform. According to the simulation, there is little difference between numerical solution and the exact solution when the solution is the time variable function. The results show the validity of the method.

In this article, a technique called Haar wavelet-Picard technique is proposed to get the numerical solutions of nonlinear differential equations of fractional order. Picard iteration is used to linearize the nonlinear fractional order differential equations and then Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of Picard iteration, solution is updated by the Haar wavelet method. The results are compared with the exact solution.   Key words: Fractional differential equations, Wavelet analysis, Caputo derivative, Haar wavelets, Picard iteration.

This letter studies some nonlinear fractional differential equations. The sub-equation method is used for finding exact solutions of these equations. Meanwhile, the traveling wave transformation method has been used to convert fractional order partial differential equation to fractional order ordinary differential equation. Calculations in this method are simple and effective mathematical tool for solving fractional differential equations in science and engineering. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.

The purpose of this tutorial workshop is to introduce the fractional calculus and its applications in controller designs. Fractional order calculus, or integration and differentiation of an arbitrary order or fractional order, is a new tools that extends the descriptive power of the conventional calculus. The tools of fractional calculus support mathematical models that in many cases more accurately describe the dynamic response of actual systems in electrical, mechanical, and automatic control applications etc. The theoretical and practical interest of these fractional order operators is nowadays well established, and its applicability to science and engineering can be considered as emerging new topics. The need to digitally compute the fractional order derivative and integral arises frequently in many fields especially in automatic control and digital signal processing. Fractional order proportional-integral-derivative (PID) controllers are based on the fractional order calculus where the derivative or integral can be of a non-integer order. Due to the extra tuning knobs, it is expected that better control performance can be achieved if the fractional order PID controller is used. Fractional calculus has much to offer science and engineering by providing not only new mathematical tools, but more importantly, its application suggests new insights into the system dynamics as well as controls.

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.

Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.

A fractional variational iteration method for solving fractional nonlinear differential equations

The aim of this study is to consider solving an important mathematical model of fractional order ([Formula: see text])-dimensional breaking soliton (Calogero) equation by Khater method. The derivatives are in the local fractional derivative sense. The fractional transformation equation is utilized to convert the proposed nonlinear fractional order differential equation into nonlinear ordinary differential equation. The Khater method is used to construct the closed-form traveling wave solutions of the said fractional differential equation. In addition, many new exact solutions are constructed. This shows that the Khater method is more convenient, powerful, and easy to solve the nonlinear fractional differential equation arising in mathematical physics.

We derive the general prolongation formula for system of fractional partial differential equations (FPDEs) with Caputo derivative involving m dependent variables and n independent variables. A systematic computational method to derive Lie point symmetries of nonlinear FPDEs with Caputo derivative is given, and we illustrate its applicability through system of time fractional diffusion equations and space fractional diffusion equation in Caputo sense. We observe that the set of Lie point symmetries for a given fractional differential equation in Caputo sense is same as the corresponding fractional differential equation in Riemann–Liouville sense. Also we construct their exact solutions in terms of Mittag–Leffler function by using the obtained Lie point symmetries.

Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods

Objective: The aim of this study is to obtain numerical solutions of fractional order mixed KdV Burger’s equation using iterative methods. Methodology: The study is centered on a time fractional mixed KdV Burger’s equation in the Caputo sense. The approximate solutions of this equation are obtained using Adomian decomposition method and Homotopy Perturbation method. Also, a Python code is developed to obtained numerical solutions and simulate graphically. Finding: The obtained solutions are compared with the exact solution and presented graphically using Python program for better analysis. Novelty: These methods shall be used to obtain approximate solution of such type of non-linear fractional order differential equations. Keywords: Fractional Differential Equation, Caputo Derivative, ADM, HPM, KdV Burger equation, Python

In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions. Some examples are given to check the proposed method’s accuracy and graphical representations. Exact and numerical solutions are also compared in the paper for some examples. The efficiency of the method can be enhanced further by increasing the scale level.

The fractional differential equations have been studied by many authors and some effective methods for fractional calculus were appeared in literature, such as the fractional sub-equation method and the first integral method. The fractional complex transform approach is to convert the fractional differential equations into ordinary differential equations, making the solution procedure simple. Recently, the fractional complex transform has been suggested to convert fractional order differential equations with modified Riemann-Liouville derivatives into integer order differential equations, and the reduced equations can be solved by symbolic computation. The present paper investigates for the applicability and efficiency of the exp-function method on some fractional non-linear differential equations.

Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule