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Energy restoration is the prime issue for the researcher and they have tried to develop advanced techniques and resources for renewable energy. The nanofluid is one of the resources for the restoration of energy which depends on the dynamic thermophysical properties of metal nanoparticles. The recent study is concerned with the thin film flow of carbon nanotubes (CNTs) water-based nanofluid for the improvement of heat transfer applications. The flow of two types of CNTs nanofluid was studied, comprising single-walled carbon nanotube (SWCNT) and multi-walled carbon nanotube (MWCNT) over the surface of a thin stirring needle. The study has been carried out in the presence of a magnetic effect and viscous dissipation. The BVP 2.0 package has been used for the solution of the modeled problem. The effect of the physical constraints like Prandtl number, magnetic field and Eckert number vs the momentum and thermal boundary layers has been analyzed. The sum of the residual errors has been obtained up to the 20th order estimates to settle the strong convergence of the problem. The obtained results show that the thin film has a quick response to the increasing of heat transfer rate over the surface of a thin needle as compared to the thick boundary layers. [ABSTRACT FROM AUTHOR]

In the present article, we give a Bézier variant of Păltăanea operators, which involves Gould-Hopper polynomials. First, we investigate the rate of convergence by using Ditzian-Totik modulus of smoothness, weighted modulus of continuity and also for a class of Lipschitz function. Furthermore, we obtain the quantitative Voronovskaja type theorem in terms of Ditzian-Totik modulus of smoothness. In the last section, we study the rate of point-wise convergence for the functions having a derivative of bounded variation. [ABSTRACT FROM AUTHOR]

We obtain the general solution of the following functional equation f, k ≥ 2. We establish the Hyers-Ulam-Rassias stability of the above functional equation in the fuzzy normed spaces. More precisely, we show under suitable conditions that a fuzzy q-almost affine mapping can be approximated by an affine mapping. Further, we determine the stability of same functional equation by using fixed point alternative method in fuzzy normed spaces. [ABSTRACT FROM AUTHOR]

We propose a Kantorovich variant of (p,q)-analogue of Szász-Mirakjan operators. We establish the moments of the operators with the help of a recurrence relation that we have derived and then prove the basic convergence theorem. Next, the local approximation and weighted approximation properties of these new operators in terms of modulus of continuity are studied. [ABSTRACT FROM AUTHOR]

In this article, we propose a different generalization of (p , q) -BBH operators and carry statistical approximation properties of the introduced operators towards a function which has to be approximated where (p , q) -integers contains symmetric property. We establish a Korovkin approximation theorem in the statistical sense and obtain the statistical rates of convergence. Furthermore, we also introduce a bivariate extension of proposed operators and carry many statistical approximation results. The extra parameter p plays an important role to symmetrize the q-BBH operators. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new generalization of Bleimann-Butzer-Hahn operators by using [Formula: see text]-integers which is based on a continuously differentiable function μ on [Formula: see text]. We establish the Korovkin type approximation results and compute the degree of approximation by using the modulus of continuity. Moreover, we investigate the shape preserving properties of these operators., Competing Interests: The authors declare that they have no competing interests.