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In this article, we used a novel semi-analytical approach, named the optimal auxiliary function method (OAFM), to solve integro-differential equations (IDEs). The OAFM includes an auxiliary function and convergence control parameters, which expedite the convergence of the method. To apply the proposed method, some assumptions regarding small or large parameters in the problem are necessary. We present numerical outcomes acquired via the OAFM alongside those obtained from other numerical techniques in tables. Furthermore, we demonstrate the efficacy and ease of implementing the proposed method for various IDEs using 2D graphs.

In the present work, the new iterative method with a combination of the Laplace transform of the Caputo's fractional derivative has been applied to the generalized (3 + 1) dimensional fractional perturbed Zakharov–Kuznetsov equation in a dusty plasma. The proposed method is applied without any discretization and linearization. The numerical and graphical results show the accuracy of the proposed method for nonlinear differential equations. Moreover, the methods are easy to implement and give the efficient approximate solutions. [ABSTRACT FROM AUTHOR]

In this paper, the implementation of the Fractional Variational iteration method (FVIM) is used to approximate the generalized fractional perturbed Zakharove-Kuznetsov equation. The obtained results show that the suggested approach is very efficient for dealing with various forms of differential equations of fractional order. The numerical comparison for the attained resolution is made through the q-homotopy analysis transmute manner (q-HATM) which confirms the convergence of the devised method. The applicability of the method is observed by applying the proposed technique to two numerical examples. Furthermore, the solutions obtained by the FVIM are completely well-matched with the solutions presented in the literature. [ABSTRACT FROM AUTHOR]

In this article, we discuss an analytical method to solve physical equations occurring in acoustic waves. We acquire an approximate solution to the generalized fractional perturbed Zakharov-Kuznetsov (ZK) equation via the Residual power series method (RPSM). Three problems are considered to demonstrate the validity, efficiency and accuracy of the proposed method. The RPSM solution agrees very well with the results those obtained by q-homotopy perturbation transform method (q-HPTM) and the exact solution. The final results are shown through 3D graphs, 2D graphs and numerical tables. The results demonstrate the quality and power of the presented method. Fractional derivatives in the manuscript are defined in the Caputo’s sense whose order varies from 0 to 1. For mathematical computation Mathematica 10 is used. [ABSTRACT FROM AUTHOR]

Within the context of fractional calculus, we investigate novel mathematical possibilities. In this context, using the fractional dispersion relations for the fractional wave equation, we explore a class of the generalized fractional wave equation numerically. Some important classes of differential equations in the theory of wave studies are Drinfeld–Sokolov–Wilson and Shallow Water equations. In this effort, the natural transform decomposition technique has been implemented to investigate the explicit result of fractional-order coupled schemes of Drinfeld–Sokolov–Wilson and Shallow Water coupled systems. The proposed method is obtained by coupling the Natural transform with the Adomian decomposition process. The current technique significantly works to find the approximate solution without any discretization or constraining parameter assumptions. The obtained numerical and graphical outcomes by the devised technique are compared with the available exact result to verify the convergence of the method. For mathematical calculations, the Mathematica software package is used. [ABSTRACT FROM AUTHOR]

In this article, approximate solutions of some PDE of fractional order are investigated with the help of a new semi-analytical method called the optimal auxiliary function method. The proposed method was tested upon the time-fractional Fisher equation, the time-fractional Fornberg-Whitham equation, and the time-fractional Inviscid Burger equation. The beauty of this method is that there is no need for discretization and assumptions of small or large parameters and provides an approximate solution after only one iteration. The numerical results obtained by the proposed method compared with the other existing methods used in the literature. From the numerical and graphical results, it is clear that the proposed method gives a better solution than existing methods. The MATHEMATICA software package has been used for the huge computational work. [ABSTRACT FROM AUTHOR]

Irregularity indices are graph-theoretic parameters designed to quantify the irregularity in a graph. In this paper, we study the practical applicability of irregularity indices in QSPR modeling of the physicochemical and quantum-theoretic properties of compounds. Our comparative testing shows that the recently introduced I R A index has significant priority in applicability over other irregularity indices. In particular, we show that the correlation potential of the I R A index with certain physicochemical and quantum-theoretic properties such as the enthalpy of formation, boiling point, and π -electron energies is significant. Our QSPR modeling suggests that the regression models with the aforementioned characteristics such as strong curve fitting are, in fact, linear. Considering this the motivation, the I R A index was studied further, and we provide analytically explicit expressions of the I R A index for certain graph operations and compositions. We conclude the paper by reporting the conclusions, implications, limitations, and future scope of the current study. [ABSTRACT FROM AUTHOR]

In the present paper, the optimal auxiliary function method (OAFM) has been extended for the first time to fractional-order partial differential equations (FPDEs) with convergence analysis. To find the accuracy of the OAFM, we consider the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations as a test example. The proposed technique has auxiliary functions and convergence control parameters, which accelerate the convergence of the method. The other advantage of this method is that there is no need for a small or large parameter assumption, and it gives an approximate solution after only one iteration. Further, the obtained results have been compared with the exact solution through different graphs and tables, which shows that the proposed method is very effective and easy to implement for different FPDEs. [ABSTRACT FROM AUTHOR]

In this paper, a comparative study of natural transform decomposition method and new iterative method is presented. The proposed methods are tested upon nonlinear fractional order foam drainage problem and fractional order modified regularized long-wave equation. The solutions obtained by the proposed methods have been compared with the classical solutions and the solution obtained by Adomian decomposition method. Furthermore, the efficiency and reliability of the proposed methods are shown with the help of numerical and graphical results. The fractional order derivatives are defined in Caputo's sense whose order belongs to the closed interval [0,1]. The results reveal that the methods are quickly convergent and yield encouraging results. [ABSTRACT FROM AUTHOR]

In this paper, the Optimal Homotopy Asymptotic Method is extended to derive the approximate solutions of fractional order two-dimensional partial differential equations. The fractional order Zakharov–Kuznetsov equation is solved as a test example, while the time fractional derivatives are described in the Caputo sense. The solutions of the problem are computed in the form of rapidly convergent series with easily calculable components using Mathematica. Reliability of the proposed method is given by comparison with other methods in the literature. The obtained results showed that the method is powerful and efficient for determination of solution of higher-dimensional fractional order partial differential equations. [ABSTRACT FROM AUTHOR]

An experiment having four replications was designed in Randomized complete block design with split plot arrangements (RCBD in split plot) and conducted at Research Farm of Agronomy, The University of Agriculture Peshawar, during kharif 2015. Row spacing (45, 55, 65 and 75 cm) were allotted to main plots while nitrogen placement methods (control, single side of the row, between the rows and both sides of the row,) were assigned to sub plots. Maximum cob height from ground (81.1 cm), ear weight at silking (166.2 g), grain yield (2226.2 kg ha-1)and thousand grain weight (194.5 g) were obtained from row spacing of 65 cm and biological yield (15058.1 kg ha-1) was observed at row spacing of 45 cm while maximum weeds dry weight (90.5g m-2) was examined at row spacing of 75 cm. Similarly maximum cob height from ground (81.6 cm), ear weight at silking (1661.8 g), grain yield (2184.9 kg ha-1), biological yield (14821.9 kg ha-1) and thousand grain weight (199 g) were obtained from applying nitrogen to both sides of the row while maximum weeds dry weight (79.2 g) was noted in control plots. All the interactions (RS×NP) were found non-significant for all the parameters. It is concluded that maize should be sown at 65 cm row spacing with nitrogen applied to both sides of the rows to get maximum yield in Peshawar, Pakistan. [ABSTRACT FROM AUTHOR]

In the present work, the natural transform iterative method (NTIM) is implemented to solve the biological population model (BPM) of fractional order. The method is tested for three nonlinear examples. The NTIM is a combination of a new iterative method and natural transform. We see that the solution pattern converges to the exact solution in a few iterations. The method handles an extensive range of differential equations of both fractional and integer order. The fractional order derivative is considered in Caputo's sense. For mathematical computation, Mathematica 10 is used. [ABSTRACT FROM AUTHOR]

In this study, the new iterative method has been applied to a coupled system of fractional-order Drinfeld–Sokolov–Wilson (FDSW) and fractional shallow water (FSW) equations. The fractional-order derivatives are taken in the Caputo sense whose order is between 0 and 1. The suggested method is capable to handle the FDEs without any transformation and discretization. The obtained results have been compared with the exact solution and with the q-homotopy analysis transform method. The outcomes show the efficiency and effectiveness of NIM by comparing through tables and graphs. [ABSTRACT FROM AUTHOR]

In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractional-order Kawahara and modified Kawahara equations. The method convergence is controlled by a flexible function known as the auxiliary function. The values of the unknown arbitrary constants in the auxiliary function are computed using the Caputo derivative fractional-order and the well-known approach of least squares. Fractional-order derivatives are taken in the Caputo sense with numerical values in the closed interval 0 , 1 . The suggested method is directly applied to fractional-order Kawahara and modified Kawahara equations, with no need for small or large parameter assumptions. The numerical results obtained by the proposed method are compared to the new iterative method (NIM). Results reveal that the proposed method converges faster to the exact solution than other methods in the literature. [ABSTRACT FROM AUTHOR]