The application of nanoparticles in the base fluids strongly influences the presentation of cooling as well as heating techniques. The nanoparticles improve thermal conductivity by fluctuating the heat characteristics in the base fluid. The expertise of nanoparticles in increasing heat transference has captivated several investigators to more evaluate the working fluid. This study disputes the investigation of convection flow for magnetohydrodynamics second-grade nanofluid with an infinite upright heated flat plate. The fractional model is obtained through Fourier law by exploiting Prabhakar fractional approach along with graphene oxide ( GO ) ({\rm{GO}}) and molybdenum disulfide ( Mo S 2 ) ({\rm{Mo}}{{\rm{S}}}_{2}) nanoparticles and engine oil is considered as the base fluid. The equations are solved analytically via the Laplace approach. The temperature and momentum profiles show the dual behavior of the fractional parameters ( α , β , γ ) (\alpha ,\beta ,\gamma ) at different times. The velocity increases as Grashof number {\rm{Grashof\; number}} increases and declines for greater values of magnetic parameter and Prandtl number. In the comparison of different numerical methods, the curves are overlapped, signifying that our attained results are authentic. The numerical investigation of governed profiles comparison shows that our obtained results in percentages of 0.2 0.2 ≤ temperature ≤ 4.36 4.36 and velocity 0.48 ≤ 7.53 0.48\le 7.53 are better than those of Basit et al. The development in temperature and momentum profile, due to engine oil–GO is more progressive, than engine oil–MoS2.

In this article, a well-known technique, the variational iterative method with the Laplace transform, is used to solve nonlinear evolution problems of a simple pendulum and mass spring oscillator, which represents the duffing equation. In the variational iteration method (VIM), finding the Lagrange multiplier is an important step, and the variational theory is often used for this purpose. This paper shows how the Laplace transform can be used to find the multiplier in a simpler way. This method gives an easy approach for scientists and engineers who deal with a wide range of nonlinear problems. Duffing equation is solved by different analytic methods, but we tackle this for the first time to solve the duffing equation and the nonlinear oscillator by using the Laplace-based VIM. In the majority of cases, Laplace variational iteration method (LVIM) just needs one iteration to attain high accuracy of the answer for linearization anddiscretization, or intensive computational work is needed. The convergence criteria of this method are efficient as compared with the VIM. Comparing the analytical VIM by Laplace transform with MATLAB’s built-in command Simulink that confirms the method’s suitability for solving nonlinear evolution problems will be helpful. In future, we will be able to find the solution of highly nonlinear oscillators.

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The Painlevé equations and their solutions occur in some areas of theoretical physics, pure and applied mathematics. This paper applies natural decomposition method (NDM) and Laplace decomposition method (LDM) to solve the second-order Painlevé equation. These methods are based on the Adomain polynomial to find the non-linear term in the differential equation. The approximate solution of Painlevé equations is determined in the series form, and recursive relation is used to calculate the remaining components. The results are compared with the existing numerical solutions in the literature to demonstrate the efficiency and validity of the proposed methods. Using these methods, we can properly handle a class of non-linear partial differential equations (NLPDEs) simply. Novelty One of the key novelties of the Painlevé equations is their remarkable property of having only movable singularities, which means that their solutions do not have any singularities that are fixed in position. This property makes the Painlevé equations particularly useful in the study of non-linear systems, as it allows for the construction of exact solutions in certain cases. Another important feature of the Painlevé equations is their appearance in diverse fields such as statistical mechanics, random matrix theory and soliton theory. This has led to a wide range of applications, including the study of random processes, the dynamics of fluids and the behaviour of non-linear waves.

The significance of thermal conductivity, convection, and heat transportation of hybrid nanofluids (HNFs) based on different nanoparticles has enhanced an integral part in numerous industrial and natural processes. In this article, a fractionalized Oldroyd-B HNF along with other significant effects, such as Newtonian heating, constant concentration, and the wall slip condition on temperature close to an infinitely vertical flat plate, is examined. Aluminum oxide (Al2O3) and ferro-ferric oxide (Fe3O4) are the supposed nanoparticles, and water (H2O) and sodium alginate (C6H9NaO7) serve as the base fluids. For generalized memory effects, an innovative fractional model is developed based on the recently proposed Atangana–Baleanu time-fractional (AB) derivative through generalized Fourier and Fick’s law. This Laplace transform technique is used to solve the fractional governing equations of dimensionless temperature, velocity, and concentration profiles. The physical effects of diverse flow parameters are discussed and exhibited graphically by Mathcad software. We have considered 0.15≤α≤0.85,2≤Pr⁡≤9,5≤Gr≤20,0.2≤ϕ1,ϕ2≤0.8,3.5≤Gm≤8, 0.1≤ Sc ≤0.8, and 0.3≤λ1,λ2≤1.7. Moreover, for validation of our present results, some limiting models, such as classical Maxwell and Newtonian fluid models, are recovered from the fractional Oldroyd-B fluid model. Furthermore, comparing the results between Oldroyd-B, Maxwell, and viscous fluid models for both classical and fractional cases, Stehfest and Tzou numerical methods are also employed to secure the validity of our solutions. Moreover, it is visualized that for a short time, temperature and momentum profiles are decayed for larger values of α, and this effect is reversed for a long time. Furthermore, the energy and velocity profiles are higher for water-based HNFs than those for the sodium alginate-based HNF.