The formulation and analysis of two operator Boundary-Domain Integral Equation systems for variable coefficient mixed BVP with in two dimensional domain is discussed. To analyse the two-operator approach, we applied one of its linear versions to the mixed (Dirichlet-Neumann) BVP for a linear second-order scalar elliptic variable-coefficient PDE and reduced it to four different BDIE systems. %The two-operator BDIE systems are nonstandard systems of equations containing integral operators defined on the domain under consideration and potential type and pseudo-differential operators defined on open sub-manifolds of the boundary. Using the results as in CMN09 andAM11, a rigorous analysis of the two-operator BDIE systems is given and it is shown that they are equivalent to the mixed BVP and thus are uniquely solvable, while the corresponding boundary domain integral operators are invertible in the appropriate Sobolev-Slobodetski (Bessel potential) spaces.

Closed form expressions in real Clifford geometric algebras Cl(0,3), Cl(3,0), Cl(1,2), and Cl(2,1) are presented in a coordinate-free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components ai and ajk in a form (ai-ajk)2, i≠ j≠ k . After disentanglement, the obtained formulas simplify to the well-known Moivre-type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.

In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite-time blow-up for the weak solutions with initial energy J ( u 0 ) ≤ d . When the initial energy J ( u 0 ) > d , we find another criterion for the vanishing solution and blow-up solution. We also establish the decay rate of the global solution and estimate the life span of the blow-up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem.

Different ways of representing the group $SU(3)$ within a Geometric Algebra approach are explored. As part of this we consider characteristic multivectors for $SU(3)$, and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime --- the STA. As part of this, a representation of $SU(3)$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by G\"unaydin and G\"ursey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme, and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing $SU(3)$, such as $G_2$ and $SU(8)$, are also considered.

In this article, by using the notion of newly defined $q_{1}q_{2}$-derivatives and integrals some new Simpson’s type inequalities for co-ordinated convex functions are shown. The outcomes raised in this paper are extensions and generalizations of the comparable results in the literature on Simpson’s inequalities for co-ordinated convex functions.

Coronavirus disease 2019 (COVID-2019) is a viral disease which is declared as a pandemic by WHO. This disease is posing a global threat, and almost every country in the world is now affected by this disease. Currently, there is no vaccine for this disease and because of this containing COVID-19 is not an easy task. It is noticed that elderly people got severely affected by this disease specially in Europe. In the present paper, we propose and analyze a mathematical model for COVID-19 virus transmission by dividing whole population in old and young groups. We find disease-free equilibrium and the basic reproduction number ($R_0$). We estimate the parameter corresponding to rate of transmission and rate of detection of COVID-19 using real data from Italy and Spain by least square method. We also perform sensitivity analysis to identify the key parameters which influence the basic reproduction number and hence regulate the transmission dynamics of COVID-19. Finally, we extend our proposed model to optimal control problem to explore the best cost-effective and time-dependent control strategies that can reduce the number of infectives in a specified interval of time.

In this paper, a class of nonautonomous SEIRS epidemic models with vaccination and nonlinear incidence is investigated. Under some quite weak assumptions, a couple of new threshold values in the form of integral, i.e., $R_1$, $R^*_1$, $R_2$ and $R^*_2$ on the extinction and permanence of disease for the model are established. As special cases of our model, the autonomous, periodic and almost periodic circumstances are discussed respectively. The nearly necessary and sufficient criteria of threshold on the extinction and permanence of disease for above cases are obtained as well. Numerical examples and simulations are presented to illustrate the analytic results.

Studying spread of epidemics and diseases are world wide problems especially during the current time where the whole world is suffering from COVID-19 pandemic. Antimicrobial resistance (AMR) and waning vaccination are classified as world wide problems. Both depend on the exposure time to antibiotic and vaccination. Here, a simple model for competition between drug resistant and drug sensitive bacteria is given. Conditions for local stability are investigated which agree with observation. Existence of positive solution in the AMR complex networks is proved. Dynamics of the identical AMR models are explored with different topologies of complex networks such as global, star, line and unidirectional line networks coupled through their susceptible states. Chaotic attractors are shown to be existed as the AMR models are located on all the indicted topologies of complex networks. Thus, it is found that the dynamics of the AMR model become more complicated as it is located on either integer-order or fractional-order complex networks. Furthermore, a discretized version of the fractional AMR model is presented. Complex dynamics such as existence of Neimark–Sacker, flip bifurcations, coexistence of multi attractors, homoclinic connections and multi closed invariant curves are investigated. Basin sets of attraction are also computed. Finally, the discretized system is located on complex networks with different topologies which also show rich variety of complex dynamics. Also, 0-1 test is used to verify the existence of unpredictable dynamics. So, studying the dynamics of AMR models on complex networks is very helpful to understand the mechanism of spread of diseases.

The classical Stefan problem, concerning mere heat-transfer during solid-liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba-Jaumann corotational time derivatives, linearized by exploiting the additive Green-Naghdi’s decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase-field model, exploiting a concept of slightly (so-called “semi”) compressible materials. The $L^1$-theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid-liquid phase transition. A rigorous proof of existence of week solutions is provided for an incomplete melting, exploiting a time-discretisation approximation.

Formulas to calculate multivector exponentials in a basis-free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra , . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalised to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.

Through reducing the problem to rational orthogonal system (Takenaka-Malmquist system), this note gives a proof for existence of n-best rational approximation to functions in the Hardy H^2(D) space by using pseudohyperbolic distance.

We study the dependence of the eigenvalues of time-harmonic Maxwell's equations in a cavity upon variation of its shape. The analysis concerns all eigenvalues both simple and multiple. We provide analyticity results for the dependence of the elementary symmetric functions of the eigenvalues splitting a multiple eigenvalue, as well as a Rellich-Nagy-type result describing the corresponding bifurcation phenomenon. We also address an isoperimetric problem and characterize the critical cavities for the symmetric functions of the eigenvalues subject to isovolumetric or isoperimetric domain perturbations and prove that balls are critical. We include known formulas for the eigenpairs in a ball and calculate the first one.

In this paper, speed selection of the time periodic traveling waves for a three species time-periodic Lotka-Volterra competition system is studied via the upper-lower solution method as well as the comparison principle. Through constructing specific types of upper and lower solutions to the system, the speed selection of the minimal wave speed can be determined under some sets of sufficient conditions composed of the parameters in the system.

In this paper we consider the propagation of waves in an open waveguide in R^2 where the index of refraction is a local perturbation of a function which is periodic along the axis of the waveguide and equal to one outside a strip of finite width. Motivated by the limiting absorption principle (proven in an ealier paper by the author for the case of an open waveguide in the half space) we formulate a radiation condition which allows the existence of propagating modes and prove uniqueness, existence, and stability of a solution. In the last part we investigate the decay properties of the radiating part in the direction of periodicity and orthogonal to it.

The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether’s theorem.

The nano/microelectromechanical systems (N/MEMS) have been caught much attention in the past few decades for their attractive properties such as small size, high reliability, batch fabrication, and low power consumption. The dynamic oscillatory behavior of these systems is very complex due to strong nonlinearities in these systems. The basic aim of this manuscript is to investigate the nonlinear vibration property of N/MEMS oscillators by the homotopy perturbation method coupled with Laplace transform (also called as He-Laplace method in literature). A generalized N/MEMS oscillator is systematically studied, and a fairly accurate analytic solution is obtained. Three special cases for electrically actuated MEMS, multi-walled Carbon nanotubes-based MEMS, and MEMS subjected to van der Waals attraction are considered for comparison, and a good agreement with exact solutions is observed.

The substantial, tempered, and shifted fractional derivatives are introduced in a unified framework. Their properties are studied and, in the light of the strict sense criterion for derivative definitions, they are characterized and assessed. In the scope of the framework, new tempered linear sytems and transfer functions are introduced.

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H^4 spaces using the fixed point technique where the elliptic equation contains fourth order differential operators with and without Fredholm property, generalizing the results of the preceding work [22].