This paper considers minimizers of the following inhomogeneous $L^2$-subcritical energy functional \[E(u):=\int_{\R^N}|\nabla u|^{2}dx-\frac{2}{p+1}\int_{\R^N}m(x)|u|^{p+1}dx,%\ u\in H^{1}(\R^N), \] under the mass constraint $\|u\|^{2}_{2}=M$. Here $N\geq1$, $p\in(1,1+\frac{4}{N})$, $M>0$ and the inhomogeneous term $m(x)$ satisfies $0

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.

This article aims to investigate the numerical study of electroosmotic flow of the Eyring Powell fluid under the peristaltic mechanism with the influence of the porous medium in the micro-channel. The modified system is applied externally to an electrical field in the horizontal direction and to a magnetic field in the transverse direction. The flow of nanofluids is considered in the computation. The governing equations in the nano-fluid flow are modulated. Influence of lubrication theory approximation longequations are shortened. Reduced coupled nonlinear partial differential equations like velocity and energy equations are numerically solved using the powerful and well-known mathematical software MATHEMATICA by built in NDSolve command. The influence of various important parameters on the velocity and temperature profile is summarised by graphs.

In this paper, we study the stability of the zero equilibrium and the occurrence of flip bifurcation on the following system of difference equations: \[x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\]\\ \[y_{n+1} =a_2\frac{z_n}{b_2+z_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}},\]\\ \[z_{n+1} =a_3\frac{x_n}{b_3+x_n} +c_3\frac{z_ne^{k_3-d_3z_n}}{1+e^{k_3-d_3z_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2,3$, are real constants and the initial values $x_0$, $y_0$ and $z_0$ are real numbers. We study the stability of this system in the special case when one of the eigenvalues is equal to -1 and the remaining eigenvalues have absolute value less than 1, using center manifold theory.

We present the mathematical analysis of the Isolation Random Forest Method (IRF Method) for anomaly detection, introduced in {\sc F.~T. Liu, K.~M. Ting, Z.-H. Zhou:}, {\it Isolation-based anomaly detection}, TKDD 6 (2012) 3:1–3:39. We prove that the IRF space can be endowed with a probability induced by the Isolation Tree algorithm (iTree). In this setting, the convergence of the IRF method is proved, using the Law of Large Numbers. A couple of counterexamples are presented to show that the method is inconclusive and no certificate of quality can be given, when using it as a means to detect anomalies. Hence, an alternative version of the method is proposed whose mathematical foundation is fully justified. Furthermore, a criterion for choosing the number of sampled trees needed to guarantee confidence intervals of the numerical results is presented. Finally, numerical experiments are presented to compare the performance of the classic method with the proposed one.

The set of hybrid numbers is a noncommutative number system unified and generalized the complex, dual and double(hyperbolic) numbers with the relation IH = − HI = ε + I. Two hybrid numbers P and Q are said to be similar if there exist a hybrid number X satisfying the equality X−1QX = P. And it is denoted by p ∼ q. In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations PX = XQ and QX − XP = C for P, Q, C ∈ 𝕂. KEYWORDS : Hybrid numbers, quaternions, coquaternions, dual and double numbers. MSC CLASSIFICATION : 15A66, 11R52, 15A18, 15A06, 16S50.

In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model’s ability to reproduce measured data on realistic problems.

A Coronavirus Disease 2019 (COVID-19) epidemiological model incorporating a boosted infection-acquired immunity and heterogeneity in infection-acquired immunity among recovered individuals is designed. The model is used to investigate whether incorporating these two processes can induce new epidemiological insights. Analytical findings reveal co-existence of multiple endemic equilibria on either regions divided by the fundamental threshold (control reproduction number). Numerical findings conducted to validate analytical results show that heterogeneity in infection-acquired immunity among recovered individuals can induce various bifurcation structures such as reversed backward bifurcation, forward bifurcation, backward bifurcation and reversed hysteresis effect. Moreover, numerical results show that reversed backward bifurcation is annihilated or switches to the usual forward bifurcation if infection-acquired immunity among recovered individuals with strong immunity is assumed to be everlasting. However, this is only possible if primary infection is more likely than reinfection. In case reinfection is more likely to occur than primary infection, reversed backward bifurcation structure switches to a backward bifurcation phenomenon. Further, longer duration of infection-acquired immunity does lead to COVID-19 decline over time but does not lead to flattening of the COVID-19 peak.

To understand dynamics of the COVID-19 disease realistically, a new SEIAPHR model has been proposed in this article where the infectious individuals have been categorized as symptomatic, asymptomatic and super-spreaders. The model has been investigated for existence of a unique solution. To measure the contagiousness of COVID-19, reproduction number R0 is also computed using next generation matrix method. It is shown that model is locally stable at disease free equilibrium point when R0 <1 and unstable for R0 >1. The model has been analyzed for global stability at both of the disease free and endemic equilibrium points. Sensitivity analysis is also included to examine the effect of parameters of the model on reproduction number R0. Couple of optimal control problems have been designed to study the effect of control strategies for disease control and eradication from the society. Numerical results show that the adopted control approaches are much effective in reducing new infections.

In this paper, the Dirichlet and Neumann boundary value problems for the steady state Stokes system of partial differential equations for a compressible viscous fluid with variable viscosity coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix, this problem is reduced to a system of direct segregated boundary-domain integral equations (BDIEs). The BDIEs in the two dimensional case have special properties in comparison with the three dimension because of the logarithmic term in the parametrix for the associated partial differential equations. Consequently, we need to set conditions on the function spaces or on the domain to ensure the invertibility of corresponding parametrix-based hydrodaynamic single layer and hypersingular potentials and hence the unique solvability of BDIEs. Equivalence of the BDIE systems to the Dirichlet and Neumann BVPs and the invertibility of the corresponding boundary-domain integral operators in appropriate Sobolev spaces are shown.

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.

By taking Sugeno-derivative into account, firstly, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive increasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard’s-Lindel\”of theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive increasing solution. Examples with graphs are provided to validate the results.

In this paper, a class of stochastic fractional differential equations (SFDEs) with impulses is considered. By virtue of the Monch’s fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen’s inequality, Holder inequality, Burkholder-Davis-Gundy inequality, Gronwall-Bellman inequality and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results.

In this paper, we further Meirong Zhang, et al.’s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied.

A document by Jorge Herrera de la Cruz. Click on the document to view its contents.

We are interested in problems of diffraction of an acoustic or electromagnetic wave by a perfectly conducting planar obstacle coated with thin multilayers of dielectric materials. The aim is to obtain boundary condition that replaces the effect of dielectric thin layers. This condition is constructed from an approximation of the Dirichlet-to-Neumann operator. In this paper, we analyze the construction and the approximation of this operator.

If an initial frame of vectors $\{e_i\}$ is related to a final frame of vectors $\{f_i\}$ by, in geometric algebra (GA) terms, a {\it rotor}, or in linear algebra terms, an {\it orthogonal transformation}, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or $4\times 4$ orthogonal matrix representing rotation and translation, given knowledge of initial and transformed points. In this paper we discuss methods in the literature for recovering such rotors and then outline a GA method which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of {\it characteristic multivectors} as discussed in the book by Hestenes \& Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create {\it fractional transformations} and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.

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.