A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

The <i>m</i>-Point Quaternary Approximating Subdivision Schemes

In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.

New Applications to Solitary Wave Ansatz

In this article, the solitary wave and shock wave solitons for nonlinear Ostrovsky equation and Potential Kadomstev-Petviashvili equations have been obtained. The solitary wave ansatz is used to carry out the solutions.

Engineering of responsive polymer based nano-reactors for facile mass transport and enhanced catalytic degradation of 4-nitrophenol

Silver nanoparticles with average diameter of 10 ± 3 nm were synthesized within the sieves of poly（N-isopropylacrylamide-2-hydroxyethylmethacrylate-acrylic acid）（p（NIPAAm-HEMA-AAc））polymer microgels. Free radial emulsion polymerization was employed for synthesis of p（NIPAAm-HEMA-AAc） polymer microgels. Silver nanoparticles were introduced within the microgels sphere by in situ reduction method. Microgels and hybrid microgels were characterized by Fourier transform infrared spectroscopy, ultra violet-visible spectroscopy,transmission electron microscopy and dynamic light scattering measurements. Catalytic activity of Ag-p（NIPAAm-HEMA-AAc） hybrid microgels was studied using catalytic reduction of 4-nitrophenol（4-NP） as a model reaction in aqueous media. The influence of sodium borohydride（Na BH4） concentration, catalyst dose and 4-NP concentration on catalytic reduction of 4-NP was investigated. A linear relationship was found between catalyst dose and apparent rate constant（kapp）. The mechanism of catalysis by hybrid microgels was explored for further development in this area. The deep analysis of catalytic process reveals that the unique combination of NIPAAm, HEMA and AAc does not only stabilize silver nanoparticles in polymer network but it also enhances the mass transport of hydrophilic substrate like 4-NP from outside to inside the polymer network.

On global behavior for complex soliton solutions of the perturbed nonlinear Schrödinger equation in nonlinear optical fibers

In this research article,the perturbed nonlinear Schrödinger equation(P-NLSE)is examined by utilizing two analytical methods,namely the extended modified auxiliary equation mapping and the generalized Riccati equation mapping methods.Consequently,we establish several sorts of new families of complex soliton wave solutions such as hyperbolic functions,trigonometric functions,dark and bright solitons,periodic solitons,singular solitons,and kink-type solitons wave solutions of the P-NLSE.Using the mentioned methods,the results are displayed in 3D and 2D contours for specific values of the open parameters.The obtained findings demonstrate that the implemented techniques are capable of identifying the exact solutions of the other complex nonlinear evolution equations(C-NLEEs)that arise in a range of applied disciplines.