This paper is devoted to the study of Gronwall–Bellman-type inequalities on an arbitrary time scale T $\mathbb{T}$ . We investigate some new explicit bounds of a certain class of nonlinear retarded dynamic inequalities of Gronwall–Bellman type on time scales. These inequalities extend some known dynamic inequalities on time scales. We also generalize and unify some continuous inequalities and their corresponding discrete analogues. To illustrate the benefits of our work, we present some applications of these results. The main results will be proved by using some analysis techniques and a simple consequence of the Keller’s chain rule on time scales.

In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalitiesof Gronwall{Bellman{Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussedin the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given herecan be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Someexamples are presented to demonstrate the applications of our results.

In 2020, El-Deeb et al. proved several dynamic inequalities. It is our aim in this paper to give the retarded time scales case of these inequalities. We also give a new proof and formula of Leibniz integral rule on time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study boundedness of some delay initial value problems by applying our results as application.

Hybrid nanofluids combine two or more nano properties with a base fluid such as water ethylene. Usually, this helps enhance the heat transfer rate; in this article, using new similarity transformations created by Lie group analysis, the governing nonlinear partial differential equations are transformed into a system of connected nonlinear ordinary differential equations. The resulting design is numerically solved using a BVP4C solver with the shooting method (MATLAB). The magneto hydrodynamic flow of an incompressible fluid and the rate of heat and mass transfer were investigated for two cases, with various nanoparticle shapes including cylindrical, spherical, and platelet. Case 1 was CNT (1%), graphene (1%), and aluminum oxide (1%), and Case 2 was copper (1%), silver (1%), and cobalt ferrite (1%). When the Hartmann number rises, velocity and temperature exhibit inverse behavior: the velocity profile increases, and the temperature profile decreases. When the suction rises, the velocity and temperature profiles both increase. Optimization techniques were used from response surface methodology (RSM) to set factorial variables so that the response met the desired maximum or minimum value. Factorial methods like ANOVA were used to model the response, but they were expanded to simulate the effects in terms of extrapolation.