Therapeutic Activity of Partially Purified Fractions of Emblica officinalis （Syn, Phyllanthus embfica） Dried Fruits against Trypanosoma evansi

Emblica officinalis （E. oJficinalis） dried fruits were evaluated for its antitrypanosomal activity and cytotoxic effects. Vero cell line maintained in DMEM （Dubecco＇s Modified Eagle Medium） and incubated with Trypanosoma evansi for more than 12 h. MPE was added to the Vero cell culture medium at different concentrations （250-1,000 μg/mL） with trypanosomes concentration （1 × 106 trypanosomes/mL in each ELISA plate well） and incubated at appropriate conditions for 72 h. In-vitro cytotoxieity of MPE of E. officinalis was determined on Vero cells at concentrations （（1.56-100 ~tg/mL）. Acute toxicity and in-vivo infectivity tests were done in mice. Obtained MPE ofE. officinalis underwent process of purification via column chromatography, preparative chromatography and HPLC （higher performance liquid chromatography） with bioassay at different strata on Alsever＇s medium. In-vivo assay for trypanocidal activity, MPE and PPFs （partially purified fractions） of E. officinalis with two sets of mice, each mouse was inoculated with 1 × 104/mL oftrypanosomes and treated （48 h post inoculation） at concentrations （12.5, 25, 50, 100 and 200 mg/kg body weight） were administered at dose rate of 100 [tL per mouse via intraperitoneal route （in treating parassitemic mice） to different groups of mice, 6 mice per concentration. HPLC of partially purified fractions ofE. officinalis was carried out with mobile phase ofacetonitdle： water （40：60） in gradient mode. In vitro, MPE induced immobilization and killing of the parasites in concentration-time dependent manner. Significant reduction of trypanosomes counts from concentration of 250μg/mL and complete killing of trypanosomes at 5th hour of observation, which was statistically equivalent to 4th hour of Diminazine Aceturate （Berenil）, standard reference drug used. HPLC of the partially purified fractions revealed two major prominent peaks at retention time of 1-4 min. In vivo, both MPE and PPFs of test material did prolong lives of mice by 6-9 days but could not cure them. At concentration of 2,000 kg/kg body weight of MPE in acute test, all mice survived. For in-vivo infectivity test, mice injected with immobilized trypanosomes developed parasitemia and died while, the other group survived. MPE, PPFs and Diminazine Aceturate were toxic to Vero cells at all concentrations exception of 1.56, 1.56-3.13 and 1.56-6.25 μg/mL, respectively. From this report, PPFs ofE. officinalis dried fruits demonstrated potential pathway for a new development oftrypanocide in near future if additional investigations are put in place.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

Crank-Nicolson ADI Galerkin Finite Element Methods for Two Classes of Riesz Space Fractional Partial Differential Equations

In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.

Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials

A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations

The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin （EFG） method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α（0 〈 α≤ 1）. The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.

Adaptive Single Piecewise Interpolation Reproducing Kernel Method for Solving Fractional Partial Differential Equation

It is well-known that using the traditional reproducing kernel method(TRKM) for solving the fractional partial differential equation(FPDE) is very intractable. In this study, the adaptive single piecewise interpolation reproducing kernel method(ASPIRKM) is determined to solve the FPDE. This improved method not only improves the calculation accuracy, but also reduces the waste of time. Two numerical examples show that the ASPIRKM is a more time-saving numerical method than the TRKM.

Partial Fraction Decomposition by Repeated Synthetic Division

We present an efficient and elementary method to find the partial fraction decomposition of a rational function when the denominator is a product of two highly powered linear factors.

On The Cauchy Problem For Some Parabolic Fractional Partial Differential Equations With Time Delays

The Cauchy problem for some parabolic fractional partial differential equation of higher orders and with time delays is considered. The existence and unique solution of this problem is studied. Some smoothness properties with respect to the parameters of these delay fractional differential equations are considered.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems.

A High Order Formula to Approximate the Caputo Fractional Derivative

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diff erential equations.In comparison with the previous formulae,the main superiority of the new formula is its order of accuracy which is 4−α,while the order of accuracy of the previous ones is less than 3.It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost.The eff ectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples.Moreover,an application of the new formula in solving some fractional partial diff erential equations is presented by constructing a fi nite diff erence scheme.A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

(G′/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics

In this paper, the (G′/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.

Moving finite element methods for time fractional partial differential equations

With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuni- form meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations （FPDEs）. The unconditional stability and convergence rates of 2 -a for time and r for space are proved when the method is used for the linear time FPDEs with a-th order time derivatives. Numerical exam-ples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.

Temperature of Prograde Metamorphism, Decompressional Partial Melting and Subsequent Melt Fractional Crystallization in the Weihai Migmatitic Gneisses,Sulu UHP Terrane:Constraints from Ti-in-Zircon Thermometer

In order to constrain temperature during subduction and subsequent exhumation of fel- sic continental crust, we carried out a Ti-in-zircon thermometer coupled with zircon internal structure and U-Pb age on migmatitic gneisses from the Weihai region in the Sulu ultra-high pres- sure （UHP） metamorphic terrane, eastern China. The Weihai migmatitic gneisses are composed of in- tercalated compositional layers of melanosome and plagioclase （Pl）-rich lencosome and K-feldspar （Kfs）-rich pegmatite veins. Four stages of zircon growth were recognized in the Weihai migmatitic gneisses. They successively recorded informations of protolith, prograde metamorphism, decompres- sional partial melting during early stage exhumation and subsequent fractional crystallization of pri- mary melt during later stage cooling exhumation. The inherited cores in zircon from the melanosome and the Pl-rich leucosome suggest that the pro- tolith of the migmatitic gneiss is Mid- Neoproterozoic （-780 Ma） magmatic rock. Metamorphic zircons with concordant ages ranging from 243 to 256 Ma occur as over- growth mantles on the protolith magmatic zir- con cores. The estimated growth temperatures （625-717 ＂C） of the metamorphic zircons have a negative correlation with their ages, indicating a progressive metamorphism in HP eciogite-facies condition during subduction. Zircon recrystal- lized rims （228-2 Ma） in the PI-rich ieucosome layers provide the lower limit of the decompress-sional partial melting time during exhumation. The ages from 228^-2 to 219~2 Ma recorded in the Pl-rich leucosome and the Kfs-rich pegmatite vein, respectively, suggest the duration of the fractional crystallization of primary melt during exhumation. The calculated growth temperatures of the zircon rims from the Pl-rich leucosome range from 858 to 739 , and the temperatures of new growth zircon grains （219±2 Ma） in Kfs-rich vein are between 769 and 529 . The estimated temperatures have a positive correlation with ages from the Pl-rich leucosome to the Kfs-rich pegmatite vein, strongly indi- cating that a process of fractional crystallization of the partial melt during exhumation.

An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative

Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.

Adomian's Method Applied to Solve Ordinary and Partial Fractional Differential Equations

This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.

consistent Riccati expansion fractional partial differential equation Riccati equation modified Riemann–Liouville fractional derivative exact solution

In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie＇s modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.

The Averaging Principle for Stochastic Fractional Partial Differential Equations with Fractional Noises

The purpose of this paper is to establish an averaging principle for stochastic fractional partial differential equation of order a.>1 driven by a fractional noise.We prove the existence and uniqueness of the global mild solution for the considered equation by the fixed point principle.The solutions for SPDEs with fractional noises can be approximated by the solution for the averaged stochastic systems in the sense of p-moment under some suitable assumptions.

SOME PROBLEMS OF FRACTIONAL PARTIAL DIFFERENCE DIFFUSION EQUATIONS

This paper mainly discusses the problems of fractional variational problems and fractional diffusion problems using fractional difference and summation. Through the Euler finite difference method we get a variational formulation of the variation problem and the discrete solution to the time-fractional and space-fractional difference equation using separating variables method and two-side Z-transform method.

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

This paper deals with the existence,uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions.The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.