Approximate solutions for time-fractional two-component evolutionary system of order $2$ using coupled fractional reduced differential transform method

In this paper, Coupled Fractional Reduced Differential Transform method is extended to apply to the generalized time-fractional two-component evolutionary system of order 2. By using this method, the solutions in the form of a generalized Taylor series are obtained. The graphics of numerical solutions together with the error analysis demonstrate that the present method is effective and accurate for obtaining approximate solutions of fractional coupled equations. Moreover, the results also indicate that the solutions obtained by residual power series method in previous literature (M. Alquran, Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method, J. Appl. Anal. Comput.,5(2015)(4), 589-599.) contain errors.     

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method

In this paper, by introducing the fractional derivative in the sense of Caputo, the generalized two-dimensional differential transform method (DTM) is directly applied to solve the coupled Burgers equations with space- and time-fractional derivatives. The presented method is a numerical method based on the generalized Taylor series formula which constructs an analytical solution in the form of a polynomial. Several illustrative examples are given to demonstrate the effectiveness of the generalized two-dimensional DTM for the equations.     

Analysis of Formal and Analytic Solutions for Singularities of the Vector Fractional Differential Equations

In this article,we study on the existence of solution for a singularities of a system of nonlinear fractional differential equations (FDE).We construct a formal power series solution for our considering FDE and prove convergence of formal solutions under conditions.We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function.     

Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

广义泰勒定理:"同伦分析方法"之有效性的一个数理逻辑证明

推导了复变函数一个广义意义上的泰勒级数表达式,证明了有关的收敛性定理,大大增大摄动级数解的收敛区域。定理的证明亦为一种新的、求解非线性问题的解析方法（即“同伦分析方法”）的有效性奠定了一个坚实的数理逻辑基础。     

Group classifications, optimal systems and exact solutions to the generalized Thomas equations

In this paper, the complete group classifications are performed on the types of Thomas equations (TEs), which arise in the study of chemical exchange progress, etc., all of the vector fields of the equations are presented. Then, the optimal system of the general Thomas equation is given, and all of the symmetry reductions and exact solutions generated from the optimal system are investigated. Furthermore, the exact analytic solutions to the Thomas equations are obtained by the generalized power series method.     

Lie symmetry analysis, conservation laws and exact solutions of fourth-order time fractional Burgers equation

In this paper, the fourth-order time fractional Burgers equation has been investigated, which can be used to describe gas dynamics and traffic flow. By employing the Lie group analysis method, the invariance properties of the equation are provided. With the aid of the sub-equation method, a new type of explicit solutions are well constructed with a detailed derivation. Furthermore, based on the power series theory, we investigate its approximate analytical solutions. Finally, its conservation laws with two kinds of independent variables are performed by making use of the nonlinear self-adjointness method.     

Efficient analytic method for solving nonlinear fractional differential equations

In this paper, we propose a new approach of the generalized differential transform method (GDTM) for solving nonlinear fractional differential equations. In GDTM, it is a key to derive a recurrence relation of generalized differential transform (GDT) associated with the solution in the given fractional equation. However, the recurrence relations of complex nonlinear functions such as exponential, logarithmic and trigonometry functions have not been derived before in GDTM. We propose new algorithms to construct the recurrence relations of complex nonlinear functions and apply the GDTM with the proposed algorithms to solve nonlinear fractional differential equations. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed technique is robust and accurate for solving fractional differential equations.     

The method of approximate particular solutions to simulate an anomalous mobile-immobile transport process

In the current work, a generalized mathematical model based on the Coimbra time fractional derivative of variable order, which describes an anomalous mobile-immobile transport process in complex systems is investigated numerically. A robust numerical technique based on the meshfree strong form method combined with an efficient time-stepping scheme is performed to compute the approximate solution of the problem with high accuracy. For this purpose, firstly, an effective implicit time discretization approach is used for discretizing the variable-order time fractional problem in the time direction. Then a global meshless technique based on the method of approximate particular solutions is performed to fully discretize the model in the spatial domain. The validity and performance of the procedure to numerically simulate the proposed generalized solute transport model on regular and irregular domains are demonstrated through some numerical examples.     

Oscillatory behavior of a fractional partial differential equation

In this paper, a fractional partial differential equation subject to the Robin boundary condition is considered. Based on the properties of Riemann-Liouville fractional derivative and a generalized Riccati technique, we obtained sufficient conditions for oscillation of the solutions of such equation. Examples are given to illustrate the main results.     

Universal Abelian covers of rational surface singularities and multi-index filtrations

In previous papers, the authors computed the Poincaré series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincaré series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincaré series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincaré series.     

关于分数幂算子(-△)~(α/2)的一个非线性估计

利用广义Young不等式给出了分数幂算子(一△)~(α/2)(α1)的一个加权非线性估计,为非线性分数幂耗散方程在加权了空间中讨论解的存在性提供帮助.     

Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations

In this paper, the Lie symmetry analysis is performed on the fifth-order KdV types of equations which arise in modeling many physical phenomena. The similarity reductions and exact solutions are obtained based on the optimal system method. Then the exact analytic solutions are considered by using the power series method.     

An exact solution for variable coefficients fourth-order wave equation using the Adomian method

This paper presents a novel approach for the analysis of a fourth-order parabolic equation dealing with vibration of beams by using the decomposition method. In this regard, a general approach based on the generalized Fourier series expansion is applied. The obtained analytic solution is simplified in terms of a given set of orthogonal basis functions. The result is compared with the classical modal analysis technique which is widely used in the field of structural dynamics. It is shown that the result of the decomposition method leads to an exact closed-form solution which is equivalent to the result obtained by the modal analysis method. Some examples are given to demonstrate the validity of the present study.     

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.     

On solving a system of singular Volterra integral equations of convolution type

This paper presents approximate analytical solutions for a system of singular Volterra integral equations of convolution type by using the fractional differential transform method. The solutions are calculated in the form of convergent series with easily computable terms and also the exact solutions can be achieved by well-known series solutions. Several examples are given to demonstrate reliability and performance of the presented method.     

一类四阶偏微分方程的对称分析及级数解

研究了一类四阶偏微分方程的李对称,构造了方程所容许的李对称的优化系统,进行了对称约化,得到了精确解.进一步,基于幂级数理论,得到了这类四阶偏微分方程的幂级数解.     

The heat transfer problem for inhomogeneous materials in photoacoustic applications and spectral parameter power series

A method for studying the one‐dimensional heat transfer process within an inhomogeneous spatially bounded medium in the presence of an external heat source is presented. It is based on a recently introduced technique for solving problems related to Sturm–Liouville equations that consists in the representation of solutions in the form of a spectral parameter power series. We consider a heat transfer model linked to photoacoustic and show that the introduced method, besides its relative simplicity and analytical nature, offers an efficient numerical algorithm as well as a convenient way to work separately with different physically meaningful components of the temperature distribution function. Detailed explanations and numerical examples are given. Copyright © 2013 John Wiley & Sons, Ltd.     

一个与广义二项级数有关的随机游动问题

利用广义二项级数本文给出了随机格点最终到达给定边界的概率.