###### Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

this work is organised as follows. In Section 2 , we introduce the definition of conformable fractional derivative [ 32 ] and its properties which will be utilised to reduce FDE into an ODE. Section 3 contains the description of variable separated ODE method and the technique of implementing it to ODEs. In Section 4 , the proposed method will be applied to construct the solitons and periodic wave solutions of space-time fractional mKdV–ZK equation. Then, the behaviour of some obtained solutions is displayed graphically. Finally, our discussions and conclusions

###### Oscillatory flow of a Casson fluid in an elastic tube with variable cross section

elastic tube with application to blood flow. Ramachandra Rao and Devanathan (1973) studied the pulsatile flow in tubes of varying cross section. Taylor and Gerrard (1977) presented a mathematical model to analyze the blood flow through arteries and expressed the different pressure radius relationships for elastic tube. Kaimal (1981) analyzed viscoelastic properties on oscillatory flow with consideration of pulsatile nature of tube wall. Furthermore, the significant effect of elasticity of the fluid and pulsation of the tube wall on flow characteristics was

###### Structural optimization under overhang constraints imposed by additive manufacturing processes: an overview of some recent results

( 1 ). To circumvent this difficulty, we rely on the so-called ‘ersatz-material’ approximation which consists in filling the void D /Ω̄ with a very soft material with Hooke's tensor εA (in practice ε = 10 ‒3 ), thus transferring a system posed on Ω into an approximate one posed on D ; see for instance [ 5 ]. In all examples, the Young’s modulus of the considered elastic material is normalized as E = 1 and the Poisson’s ratio is set to v = 0.33. 5.2 Validation of the approximations of Section 4.3 Our first numerical example aims to assess the

###### Multigrid method for the solution of EHL line contact with bio-based oils as lubricants

etc., employing the Gauss-Seidel relaxation with small under-relaxation factors which illustrates that the Gauss-Seidal relaxation is a stable relaxation scheme with good smoothing properties. EHL problems have also been solved by many other different methods as in [ 15 , 16 ], etc. Although many mathematical models of ordinary differential equations and partial differential equations [ 17 , 18 ], etc., have been solved for its exact solution. In this paper, we employ FAS for the solution of EHL line contact problem. This paper is organized as follows. Section

###### The Triaxiality Role in the Spin-Orbit Dynamics of a Rigid Body

= ± 1 / 3 ) $\begin{array}{} \displaystyle (\cos\iota=\pm\sqrt{1/3}) \end{array}$ and cos ι leads to the body-perpendicular and body-inclined types. In our analysis, we find families that depart from the classical equilibria of the free rigid body and the classical equilibria reported in [ 16 , 21 ]. Though, some of these equilibria are recovered in our setting. This paper is organized as follows. Section 2 is devoted to introduce the triaxial intermediary into the Hamiltonian formalism and to describe the canonical variables in which it is expressed. In

###### Complex variables approach to the short-axis-mode rotation of a rigid body

close to the axis of maximum (or minimum) inertia. Therefore, in practice one must to take some care when applying the solution to this kind of motion, which is precisely the case in which the separation of the free rigid body rotation into the main problem of SAM rotation and a perturbation applies. On the other hand, it will be shown in Section 2.3 that the perturbative arrangement of the free rigid body Hamiltonian in the case of SAM rotation is immediately disclosed when using non-singular variables of Poincaré type, cf. [ 30 ]. In these variables, the free

###### Centers: their integrability and relations with the divergence

that ∆( x , y ) ≥ 0 or ∆( x , y ) ≤ 0 for all ( x , y ) ∈ ℝ 2 . For more details on characteristic directions see for instance [ 2 ]. 5 Poincaré–Liapunov constants Suppose that the analytic differential system 1 has a monodromic singular point at the origin O . Let Σ be an analytic transversal section at O , that is, an analytic arc transverse to the flow of the system such that O ∈ ∂ Σ, the boundary of Σ. We consider a parameter ρ of Σ such that ρ = 0 corresponds to the origin of coordinates and Σ is parameterized by the interval (0, ρ *) with ρ

###### On problems of Topological Dynamics in non-autonomous discrete systems

of the absolute value of their terms. Proposition 6 A sequence of real numbers ( x n ) n = 0 ∞ $\begin{array}{} (x_{n})_{n=0}^{\infty} \end{array} $ is B − Benford, if and only if, the sequence ( log | x n | ) n = 0 ∞ $\begin{array}{} (\log|x_{n}|)_{n=0}^{\infty} \end{array} $ is uniformly distributed modulus 1. Using the above result and others from uniformly distribution, in many examples can be proved the property of trajectories of dynamical systems starting in an initial point x 0 or simply general sequences of real numbers