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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.

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]