A semi-numerical solution to Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at leading order (LO), next-to-leading order (NLO) and next-to-nextto-leading order (NNLO) in the small-x limit is presented.Here we have used Taylor series expansion method to solve the evolution equations and, t-and x-evolutions of the singlet structure functions have been obtained with such solution. We have also calculated t-and x-evolutions of deuteron structure functions F d 2 , and the results are compared with the E665 data and NMC data. The results are also compared to those obtained by the fit to F d 2 produced by the NNPDF collaboration based on the NMC and BCDMS data.

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In this paper we have solved the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu (GLR-MQ) evolution equation for gluon distribution function G(x, Q 2 ) and studied the effects of the nonlinear GLR-MQ corrections to the Leading Order (LO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations. Here we incorporate a Regge like behaviour of gluon distribution function to obtain the solution of GLR-MQ evolution equation. We have also investigated the Q 2dependence of gluon distribution function from the solution of GLR-MQ evolution equation. Moreover it is interesting to observe from our results that nonlinearities increase with decreasing correlation radius (R) between two interacting gluons. Results also confirm that the steep behavior of gluon distribution function is observed at R = 5 GeV −1 , whereas it is lowered at R = 2 GeV −1 with decreasing x as Q 2 increases. In this work we have also checked the sensitivity of λ G in our calculations. Our computed results are compared with those obtained by the global DGLAP fits to the parton distribution functions viz. GRV, MRST, MSTW and with the EHKQS model.

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The present paper reports the effect of nonlinear corrections to the behaviour of gluon distribution in the Gribov-Levin-Ryskin-Mueller-Qiu (GLR-MQ) approach. Here the nonlinear GLR-MQ evolution equation is solved analytically in order to determine the x and Q 2 dependence of gluon distribution function. We observe that, the gluon distribution increases with increasing Q 2 and decreasing x, which is in agreement with the perturbative QCD fits at small-x, however this particular behaviour of the gluon distribution function is tamed by the shadowing effects of gluon recombination. The obtained results of the nonlinear gluon distribution at the hot spot are compared with different global parton analysis as well as with those obtained from the solution of the modified-DGLAP (MD-DGLAP) equation and are found to be quite compatible.