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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 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 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]

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]