DSpace at Texas State University https://digital.library.txstate.edu:443 The University Scholarship digital repository system captures, stores, indexes, preserves, and distributes digital research material. 2022-09-30T04:45:39Z Stability and phase portraits of susceptible-infective-removed epidemic models with vertical transmissions and linear treatment rates https://digital.library.txstate.edu/handle/10877/16186 Stability and phase portraits of susceptible-infective-removed epidemic models with vertical transmissions and linear treatment rates Hoti, Marvin; Huo, Xi; Lan, Kunquan We study stability and phase portraits of susceptible-infective-removed (SIR) epidemic models with horizontal and vertical transmission rates and linear treatment rates by studying the reduced dynamical planar systems under the assumption that the total population keeps unchanged. We find out all the ranges of the parameters involved in the models for the infection-free equilibrium and the epidemic equilibrium to be positive. The novelty of this paper lies in the demonstration and justification of the parameter conditions under which the positive equilibria are stable focuses or nodes. These phase portraits provide more detailed descriptions of behaviors and extra biological understandings of the epidemic diseases than local or global stability of the models. Previous results only discussed the stability of the SIR models with horizontal or vertical transmission rates and without treatment rates. Our results involving vertical transmission and treatment rates will exhibit the effect of the vertical transmissions and the linear treatment rates on the epidemic models. 2017-12-14T00:00:00Z Approximation of the leading singular coefficient of an elliptic fourth-order equation https://digital.library.txstate.edu/handle/10877/16185 Approximation of the leading singular coefficient of an elliptic fourth-order equation Abdelwahed, Mohamed; Chorfi, Nejmeddine; Radulescu, Vicentiu The solution of the biharmonic equation with an homogeneous boundary conditions is decomposed into a regular part and a singular one. The later is written as a coefficient multiplied by the first singular function associated to the bilaplacian operator. In this paper, we consider the dual singular method for finding the value of the leading singular coefficient, and we use the mortar domain decomposition technique with the spectral discretization for its approximation. The numerical analysis leads to optimal error estimates. We present some numerical results which are in perfect coherence with the analysis developed in this paper. 2017-12-14T00:00:00Z Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent https://digital.library.txstate.edu/handle/10877/16184 Non-homogeneous problem for fractional Laplacian involving critical Sobolev exponent Cheng, Kun; Wang, Li <p>In this article, we study the existence of positive solutions for the nonhomogeneous fractional equation involving critical Sobolev exponent</p> <pre>(-∆)<sup>s</sup>u + λu = u<sup>p</sup> + μƒ(x), u > 0 in Ω,<br> u = 0, in ℝ<sup>N</sup> \ Ω,</pre> <p>where Ω ⊂ ℝ<sup>N</sup> is a smooth bounded domain, N ≥ 1, 0 < 2s < min{N, 2}, λ and μ > 0 are two parameters, p = N+2s/N-2s and ƒ ∈ C<sup>0,α</sup>(Ω̅), where α ∈ (0, 1). ƒ ≥ 0 and ƒ ≢ 0 in Ω. For some λ and N, by the barrier method and mountain pass lemma, we prove that there exists 0 < μ̅ ≔ μ̅, (s, μ, N) < +∞ such that there are exactly two positive solutions if μ ∈ (0, μ̅) and no positive solutions for μ > μ̅ . Moreover, if μ = μ̅, there is a unique solution (μ̅; u<sub>μ̅</sub>), which means that (μ̅/ u<sub>μ̅</sub>) is a turning point for the above problem. Furthermore, in case λ > 0 and N ≥ 6s if Ω is a ball in ℝ<sup>N</sup> and ƒ satisfies some additional conditions, then a uniqueness existence result is obtained for μ > 0 small enough.</p> 2017-12-11T00:00:00Z Stability of solutions for a heat equation with memory https://digital.library.txstate.edu/handle/10877/16183 Stability of solutions for a heat equation with memory Tatar, Nasser Eddine; Kerbal, Sebti; Al-Ghassani, Asma This article concerns the heat equation with a memory term in the form of a time-convolution of a kernel with the time-derivative of the state. This problem appears in oil recovery simulation in fractured rock reservoir. It models the fluid flow in a fissured media where the history of the flow must be taken into account. Most of the existing papers on related works treat only (in addition to the well-posedness which is by now well understood in various spaces) the convergence of solutions to the equilibrium state without establishing any decay rate. In the present work we shall improve and extend the existing results. In addition to weakening the conditions on the kernel leading to exponential decay, we extend the decay rate to a general one. 2017-12-11T00:00:00Z