Bioconvection mechanism using third-grade nanofluid flow with Cattaneo–Christov heat flux model and Arrhenius kinetics

This paper aims to study the effects of activation energy and thermal radiation in the bioconvection flow of nanofluid (third-grade nanofluid) containing swimming microorganisms in the presence of a heat source-sink past a stretching sheet. Brownian movement and thermophoresis diffusion are used in mathematical modeling. The given flow phenomenon is modeled in the form of governing partial differential equations. Furthermore, appropriate dimensionless transformation is used to transfer the governing system of PDEs into an ordinary one. The remodeled systems of ODEs are tackled numerically by bvp4c on Matlab with a shooting scheme in computational tool MATLAB. The bearing of prominently involved parameters on the numerical solution of velocity, temperature distribution, nanoparticles concentration and concentration of microorganisms is comprehensively discussed and elaborated through figures. It is established that velocity can be improved with a mixed convection aspect. Furthermore, the temperature and concentration of nanoparticles reduce against Prandtl number, also, large Peclet number declines the microorganisms field. The work contained in this paper has applications in nanotechnology, electrical and mechanical engineering, biomedicine, biotechnology, drug delivery, cancer treatment, food processing and various industries. No such work is yet reported, and it is good for the research in applied sciences.

Effects of chemically reactive and thermally radiative MHD Prandtl nanofluid by a vertically heated stretchable surface with convective conditions

The existing investigations purpose to disclose the interaction effects of transverse magnetic and hydrodynamic flow of Prandtl nanofluid subjected to convective boundary conditions over a vertical heated stretching surface. Developing a fundamental flow model, a boundary layer approximation is done, which yields momentum, concentration, and energy expressions. Moreover, Brownian effect and thermophoresis influence are also taken into the account. The constitutive flow laws of nonlinear (PDEs) is altered into ordinary one via similarity transformation variables. The dimensionless nonlinear systems of (ODEs) are then solved through bvp4c numerical algorithm. Consequences of innumerable flow factors on steam wise velocity, thermal field, and concentration of nanoparticle are explicitly debated and plotted graphically. The drag force coefficient and heat transference rate are assumed and deliberated accordingly. It has been perceived that f higher estimation of thermophoresis parameter upsurges the internal thermal energy of the nanofluid and nanoparticles concentration field.

A Liouville’s Formula for Systems with Reflection

In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators.

On reaction network implementations of neural networks

This paper is concerned with the utilization of deterministically modelled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent) and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of ‘networks’ used in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each network’s properties.

An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

EXPLICIT TRANSFORMATION OF THE RICCATI EQUATION AND OTHER POLYNOMIAL ODES TO SYSTEMS OF LINEAR ODES

The purpose of this work is to propose and demonstrate a way to explicitly transform polynomial ODE systems to linear ODE systems. With the help of an additional first integral, the one-dimensional Riccati equation is transformed to a linear system of three ODEs with variable coefficients. Solving the system, we can find a solution to the original Riccati equation in the general form or only to the Cauchy problem. The Riccati equation is one of the most interesting nonlinear first order differential equations. It is proved that there is no general solution of the Riccati equation in the form of quadratures; however, if at least one particular solution is known, then its general solution is also found. Thus, it is enough only to find a particular solution of the linear system of ODEs. The applied transformation method is a special case of the method described in our work [Zaytsev M. L., Akkerman V. B. (2020) On the identification of solutions to Riccati equation and the other polynomial systems of ODEs // preprint, Research Gate. DOI: 10.13140 / RG.2.2.26980.60807]. This method uses algebraic transformations and transition to new unknowns consisting of products of the original unknowns. The number of new unknowns becomes less than the number of equations. For the multidimensional Riccati equations, we do not present the corresponding linear system of ODEs because of the large number of linear equations obtained (more than 100). However, we present the first integral with which this can be done. In this paper, we also propose a method for finding the first integral, which can be used to reduce a search for the solution of any polynomial systems of ODEs to a search of solutions to linear systems of ODEs. In particular, if the coefficients in these equations are constant, then the solution is found explicitly.