In this study, the viscosity of MgO-Water nanofluid in a different volume fraction of nanoparticles, temperatures, and shear rates has been predicted by Artificial Neural Networks (ANNs) and surface methods. In the ANN method, an algorithm is proposed to select the best neuron number for the hidden layer. In the fitting method, a surface is proposed for each volume fraction of nanoparticles, and finally, the results of ANN and surface fitting method have been compared. It can be observed that, increasing the volume fraction from 0.07% to 1.25% at temperatures of 25, 30, 40, 50, and 60 °C resulted in about two-fold increase in viscosity. Also, the best network has 24 neurons in the hidden layer. It can be seen that for a network with 24 neurons in the hidden layer has the best overall correlation, and this coefficient is 0.999035. The mean absolute value of errors in ANN and fitting method are 0.0118 and 0.0206, respectively.

In this paper, the Laplace transform combined with the local discontinuous Galerkin method is used for distributed-order time-fractional diffusion-wave equation. In this method,at first, we convert the equation to some time-independent problems by Laplace transform.Then we can solve these stationary equations by the local discontinuous Galerkin method to discretize diffusion operators at the same time. Then, by using a numerical inversion of the Laplace transform we can find the solutions of the original equation. One of the advantages of this procedure is its parallel implementation. Another advantage of this approach is that the number of stationary problems that should be solved is much less than that are needed in time-marching methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the method.

This study is concerned with the optimal time rates of weak solutions for the 2D magneto-micropolar equations with only micro-rotational dissipation and magnetic diffusion. Due to some new observations, we obtain the optimal time decay rates of weak solutions $\|\nabla u(t)\|_{L^2}+\|\nabla w(t)\|_{L^2}\le C(1+t)^{-2}$ and $ \|\nabla b(t)\|_{L^{p}}\le C(1+t)^{-\frac12-(1-\frac1p)}$ with $p\in[2,\infty)$.

We consider the following quasilinear Keller-Segel system \left\{ {l} u_t = \Delta u - \nabla (u \nabla v) + g(u), \quad &(x,t)\in \Omega \times [0,T_{max}) ,\\[6pt] 0= \Delta v - v + u, \qquad \quad &(x,t)\in \Omega \times [0,T_{max}), \\[6pt] \right. on a ball $\Omega \equiv B_R(0)\subset^n$, $n\geq 3$, $R>0$, under homogeneous Neumann boundary conditions and non negative initial data. The source term $g(u)$ is superlinear and of logistic type i.e. $g(u)=\lambda u - \mu u^k,\ k>1, \ \mu >0$, $\lambda \in $ and $T_{max}$ is the blow-up time.\\ The solution $(u,v)$ may or may not blow up in finite time. Under suitable conditions on data, we prove that the function $u$, which blows up in $L^{\infty} (\Omega)$-norm , blows up also in $L^p(\Omega)$-norm for some $p>1$. Moreover a lower bound of the lifespan (or blow-up time when it is finite) $T_{max}$ is derived. \\ In addition, if $\Omega \subset ^3$ a lower bound of $T_{max}$ is explicitly computable.

The present paper is focused on the analysis on the existence of positive solutions of a second order differential system with two delays \left \{{lcr} x_1''(t)-a_1(t)x_1(t)+m_1(t)f_1(t,x(t),x_\tau(t))=0,\ \ t>0,\\ x_2''(t)-a_2(t)x_2(t)+m_2(t)f_2(t,x(t),x_\tau(t))=0,\ \ t>0,\\ x_1(t)=0,\ \ -\tau_1\ \leq t \leq 0,\;\; x_1(t)=0,\\ x_2(t)=0,\ \ -\tau_2\ \leq t \leq 0,\;\; x_2(t)=0 \right. by using two well-known fixed point theorems.

By using some real analysis techniques, We study the structural characteristics of a multi-parameter Hilbert-type integral inequality with the hybrid kernel, and obtain some equivalent conditions for this inequality. We also consider the operator expression of the equivalent inequalities. The conclusions not only integrates some results of references, but also nds some new Hilbert-type integral inequalities with simple form by choosing suitable parameter values.

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives .

We prove the existence and uniqueness of global strong solution to the free boundary problem in one dimensional compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when the viscosity and heat conductivity depend on temperature in power law of Chapman-Enskog and the data is in the neighborhood of some background solution at initial time. We also study the large time behavior of the solution and obtain its decay property.

In this paper, we prove the well-posedness of the solution to the three-dimensional Boussinesq system with nonlinear damping term. Existence, uniqueness and continuous dependence with respect to initial data of the weak solution are proved, under minimum regularity requirement in Sobolev and Lebegue spaces. Energy method, compactness method, and appropriate technical lemmas are used.

Fangzhu, which has been lost for thousands of years, is an ancient device for water collection from air, its mechanism is unknown yet. Here we elucidate its possible surface-geometric and related physical properties by the oldest the Yin-Yang contradiction. In view of modern nanotechnology, we reveal that Fangzhu’s water-harvesting ability is obtained through a hydrophilic-hydrophobic hierarchy of the surface, mimicking spider web’s water collection, lotus or desert beetle’s water intake. The convex-concave hierarchy of Fangzhu’s textured surface enables it to have low wettability(high geometric potential) to attract water molecules from air through the nano-scale convex surface and transfer the attracted water along the concave surface to the collector. A mathematical model is established to reveal three main factors affecting its effectiveness, i.e., the air velocity, the surface temperature and surface structure. The lost technology can play an extremely important role in modern architecture, ocean engineering, transportation and others to catch water from air for everyday use.

The interior Dirichlet boundary value problem for the diffusion equation in non-homogeneous media is reduced to a system of Boundary-Domain Integral Equations (BDIEs) employing the parametrix obtained in different from . We further extend the results obtained in for the mixed problem in a smooth domain with L²(Ω) right hand side to Lipschitz domains and PDE right-hand in the Sobolev space H−1(Ω), where neither the classical nor the canonical co-normal derivatives are well defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.

Nanofluid as a special thermal transporting medium has recently received unprecedented attention due to its improved heat transfer performance compared to conventional fluids. Numerous researches have been conducted on the natural convection characteristics of different nanofluids in various configurations of cavities due to the important applications of natural convection in environmental, petrochemical, medical, aviation and space technology, industrial and many more areas. The emergence of a magnetic field as a tool for the manipulation of convective flow and heat transfer behaviours of nanofluids in non-square enclosures has been extensively reviewed. The influence of several variables such as controlling parameters, heat distribution methods, thermal and concentration boundary conditions, magnetic field types, numerical methods, correlation types, nanofluid types, heaters types, numbers and length, and slip conditions, etc., on the magnetohydrodynamic (MHD) natural convection flow and heat transfer behaviours of nanofluid in non-square cavities has been given great attention and brought to the spotlight for discussion. The concepts of bioconvection, micro-polar nanofluid, bio-nanofluid (green nanofluid), ionic nanofluid, and hybrid nanofluid have also been discussed for the first time in relation to natural convection. Special cases of MHD natural convection in non-square cavities involving hybrid nanofluids and micro-polar nanofluids are also presented herein. The application of several numerical methods (which is the major approach studied so far) to investigate the hydromagnetic behaviours of nanofluids in non-square cavities is the focus of this work.

We apply the Bielecki metric on the space C([a, b]), to analyze the different types of stabilities of non-linear fractional integral equation corresponding to fractional boundary value problems. Sufficient conditions are obtained to prove stability results for fractional non-linear Volterra and Fredholm integral equations, given by Ulam, Hyer and Rassias. We extend the respective stability results to the fractional integral equations where the domain of integration is an unbounded interval. We provide numerical examples which asserts our stability results.

We establish nonexistence of nontrivial solutions (including sign-changing ones) for some partial differential inequalities of elliptic and parabolic type containing nonlinear terms that depend on the positive and negative part of the sought function in different ways. Systems of elliptic inequalities with similar structure are also considered. The proofs are based on the test function method.

In this paper we investigate the size of a bi-dimensional fragmentation process. A rectangle of dimensions x and y is considered, it is split into four sub-rectangles with some probability that depends on x and y, we iterate until the stop of the process. The total number of the all the obtained rectangles at the end of the process satisfies some equality in distribution which is resolved , using some tools on integral equations, via the contraction method.

The turbulent characteristics of heat transfer and flow have been determined by applying the Van Driest model of the eddy diffusivity for water and ethylene glycol-based nanofluids. The properties of CuO, Al2O3 and SiO2 nanofluids in two base liquids viz., water and EG-water mixture with the ratio of 60:40 are considered for various concentrations and bulk temperature ranges. Based on the observations, it is concluded that numerical outcomes are validated with experimental measurements for heat transfer properties. It is monitored that SiO2 reaches a higher temperature gradient in comparison to CuO at a similar temperature and concentration in EG-water with the mixture of 60: 40. The gradients are greater for the EGW mixture compared to water-based nanofluids. However, the water-based nanofluids have higher heat transfer coefficients compared to EG-water nanofluid at identical flow velocities.

In the present paper we provide the conditions guaranteeing the generalization of so-called {\it diauxic} behaviour of solutions of ODEs. This behaviour was described by Monod in 1949 in the context of bacterial growth. Then a similar behaviour was observed and described referring to dynamic of CDK1 protein during cell cycle. The diauxic behaviour is described in terms of inflection points.

In this paper, we investigate a fractional p-Laplacian system with different critical Sobolev-Hardy exponents. By variational methods and the Mountain-Pass lemma, positive minimizers of the related best Sobolev constants and the existence of positive solutions of the system are found.