Approximate analytical solution to the Falkner–Skan wedge flow with the permeable wall of uniform suction

Based on a new analytical method, namely homotopy analysis method (HAM), an approximate analytical solution to the Falkner–Skan wedge flow; with the boundary conditions of $f(0)=\gamma >0\text{,}{f}^{\prime }(0)=0\text{,}{f}^{\prime }(+\infty )=0$, i.e. the permeable wall mass transfer conditions of uniform suction, is given. The comparisons are also made between the results of the present work and numerical method by 4th-order Runge–Kutta method combined with Newton–Raphson technique. It is found that the results of the present work agree well with those by numerical method, which verifies the validity of the present work.     

An approximate solution of the MHD Falkner–Skan flow by Hermite functions pseudospectral method

Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner–Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.     

Magnetoviscous potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer

A linear analysis of the Kelvin–Helmholtz instability of interface between two viscous and magnetic fluids has been carried out where there was heat and mass transfer across the interface while the fluids have been subjected to a constant magnetic field parallel to the streaming direction. The viscous potential flow theory has been used for the investigation. A dispersion relation has been obtained and a stability criterion is given by a critical value of relative velocity as well as the critical value of magnetic field. The resulting plots show the effect of various physical parameters such as wave number, viscosity ratio, ratio of magnetic permeabilities and heat transfer coefficient. It has been observed that heat and mass transfer has a destabilizing effect whereas the horizontal magnetic field stabilizes the system.     

Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model

This letter presents a research for coupled flow and heat transfer of an upper-convected Maxwell fluid above a stretching plate with velocity slip boundary. Unlike most classical works, the new heat flux model, which is recently proposed by Christov, is employed. Analytical solutions are obtained by using the homotopy analysis method (HAM). The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed. A comparison of Fourier’s Law and the Cattaneo–Christov heat flux model is also presented.     

An algebraic identity and the Jacobi–Trudi formula

The first Jacobi–Trudi identity expresses Schur polynomials as determinants of matrices, the entries of which are complete homogeneous polynomials. The Schur polynomials were defined by Cauchy in 1815 as the quotients of determinants constructed from certain partitions. The Schur polynomials have become very important because of their close relationship with the irreducible characters of the symmetric groups and the general linear groups, as well as due to their numerous applications in combinatorics. The Jacobi–Trudi identity was first formulated by Jacobi in 1841 and proved by Nicola Trudi in 1864. Since then, this identity and its numerous generalizations have been the focus of much attention due to the important role which they play in various areas of mathematics, including mathematical physics, representation theory, and algebraic geometry. Various proofs of the Jacobi–Trudi identity, which are based on different ideas (in particular, a natural combinatorial proof using Young tableaux), have been found. The paper contains a short simple proof of the first Jacobi–Trudi identity and discusses its relationship with other well-known polynomial identities.     

High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.     

An analytical solution for dynamic behavior of a beam–column frame with a tip body

This paper analyzes the vibration characteristics of a beam-column frame, typical examples of which are often found in optical pickup actuators of optical disc drives (ODDs) and many architectural structures. The dynamic behaviour of this beam structure is predicted by solving mathematically its vibration characteristics governed by beam configurations. For practical applications and simplicity in the analysis, the vibration analysis for the structure is limited to lateral and longitudinal directions of the beams. As a result, mode and modal frequencies are obtained from mathematical expressions. The accuracy of vibration characteristics, which is mathematically induced, is demonstrated by a finite element (FE) analysis. Finally, it is shown that mode shapes are modified by using design values with the mathematical expressions.     

Concurrent simulation of multibody systems coupled with stress–strain and heat transfer solvers

In the present work multibody 3-axis truck vehicle model coupled with submodels of suspension bodies is considered. Coupling the main model with submodels allows to perform simulation of multibody dynamics and non-stationary heat conduction and stress distribution processes in bodies simultaneously. High efficiency of parallel simulation on computer cluster is achieved using developed software based on MPI library.     

Global existence and decay of solution for the nonisentropic Euler–Maxwell system with a nonconstant background density

In this paper, the Cauchy problem for the nonisentropic Euler–Maxwell system with a nonconstant background density is studied. The global existence of classical solution is constructed in three space dimensions provided the initial perturbation is sufficiently small. The proof is mainly based on classical energy estimate and the techniques of symmetrizer. And the time decay of the solution is also established by combining the decay estimate of the Green’s function with some time-weighted estimate.     

Extension of the Lamé–Gadolin problem for large deformations and its analytical solution

The formulation and solution of the axisymmetric static problem of the stress-strain state of two hollow circular elastic cylinders, one of which is predeformed and inserted into the other cylinder, is considered in the case of large plane deformations (an extension of the Lamé–Gadolin problem to large deformations). Using the theory of the superposition of large deformations, an exact analytical solution of the static problem for cylinders made of incompressible Treloar and Bartenev–Khazanovich materials is obtained, including the case when the cylinders are made of dissimilar materials. An analytical solution is obtained in parametric form for a compressible Blatz–Ko material. Non-linear effects are investigated.     

Efficient algebraic solution of reaction–diffusion systems for the cardiac excitation process

In this paper, we deal with the problem of solving the large algebraic linear system arising in the numerical solution of a reaction–diffusion (R–D) system associated with myocardial excitation process modeling. We show that an ad hoc preconditioning technique can be devised so as to efficiently and simultaneously handle the differential equations of the R–D system, with no additional memory requirements.Two different formulations are commonly considered for the theoretical and numerical analyses, respectively. We observe that the formulation employed for the theoretical analysis of the problem actually yields the best numerical performance, when compared with the usual numerical scheme.     

Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.     

On the security of the Feng–Liao–Yang Boolean functions with optimal algebraic immunity against fast algebraic attacks

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g, such that fg = 0. There is an increasing interest in construction of Boolean functions that possess optimal algebraic immunity, combined with other characteristics, like balancedness, high nonlinearity, and high algebraic degree. In this paper, we investigate a recently proposed infinite class of balanced Boolean functions with optimal algebraic immunity, optimum algebraic degree and much better nonlinearity than all the previously introduced classes of Boolean functions with maximal algebraic immunity. More precisely, we study the resistance of the functions against one of the new algebraic attacks, namely the fast algebraic attacks (FAAs). Using the special characteristics of the family members, we introduce an efficient method for the evaluation of their behavior against these attacks. The new algorithm is based on the well studied Berlekamp–Massey algorithm.     

Unsteady flow and heat transfer in a thin film of Ostwald–de Waele liquid over a stretching surface

In this paper, the effects of viscous dissipation and the temperature-dependent thermal conductivity on an unsteady flow and heat transfer in a thin liquid film of a non-Newtonian Ostwald–de Waele fluid over a horizontal porous stretching surface is studied. Using a similarity transformation, the time-dependent boundary-layer equations are reduced to a set of non-linear ordinary differential equations. The resulting five parameter problem is solved by the Keller–Box method. The effects of the unsteady parameter on the film thickness are explored numerically for different values of the power-law index parameter and the injection parameter. Numerical results for the velocity, the temperature, the skin friction and the wall-temperature gradient are presented through graphs and tables for different values of the pertinent parameter. One of the important findings of the study is that the film thickness increases with an increase in the power-law index parameter (as well as the injection parameter). Quite the opposite is true with the unsteady parameter. Furthermore, the wall-temperature gradient decreases with an increase in the Eckert number or the variable thermal conductivity parameter. Furthermore, the surface temperature of a shear thinning fluid is larger compared to the Newtonian and shear thickening fluids. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena.     

Computational examples of reaction–convection–diffusion equations solution under the influence of fluid flow: First example

This paper presents several numerical tests on reaction–diffusion equations in the Turing space, affected by convective fields present in incompressible flows under the Schnakenberg reaction mechanism. The tests are performed in 2D on square unit, to which we impose an advective field from the solution of the problem of the flow in a cavity. The model developed consists of a decoupled system of equations of reaction–advection–diffusion, along with the Navier–Stokes equations of incompressible flow, which is solved simultaneously using the finite element method. The results show that the pattern generated by the concentrations of the reacting system varies both in time and space due to the effect exerted by the advective field.     

An explicit form of the solution of the Cauchy problem for the Barenblatt–Zheltov–Kochina equation

A solution of the Cauchy problem in an isotropic medium and in an anisotropic medium with clearly expressed vertical or horizontal permeability is constructed for the Barenblatt–Zheltov–Kochina model representation of the seepage of a fluid in fissured porous rock by reducing the seepage problems under consideration to solving an abstract Cauchy problem in a Banach space.     

Boolean functions with maximum algebraic immunity: further extensions of the Carlet–Feng construction

The algebraic immunity of Boolean functions is studied in this paper. More precisely, having the prominent Carlet–Feng construction as a starting point, we propose a new method to construct a large number of functions with maximum algebraic immunity. The new method is based on deriving new properties of minimal codewords of the punctured Reed–Muller code \(\mathrm{RM}^{\star }(\lfloor \frac{n-1}{2}\rfloor ,n)\) for any n, allowing—if n is odd—for efficiently generating large classes of new functions that cannot be obtained by other known constructions. It is shown that high nonlinearity, as well as good behavior against fast algebraic attacks, is also attainable.     

Semidiscrete and asymptotic approximations for the nonstationary radiative–conductive heat transfer problem in a periodic system of grey heat shields

We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε ^3/4) for asymptotic approximations. Bibliography: 9 titles.