In this paper, using the Lyapunov-like functions, the practical stability with respect to part of the variables of fractional-order nonlinear systems depending on a small parameter is studied.

We investigate mixed type boundary-transmission problems of the generalized thermo-electro-magneto elasticity (GTEME) theory for complex elastic anisotropic layered structures containing interfacial cracks. This type of problems are described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions for six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function). We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type mixed boundary-transmission problems in appropriate Sobolev spaces. We analyze smoothness properties of solutions near the edges of interfacial cracks and near the curves where different type boundary conditions collide.

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviours. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.

The enhanced thermal properties being the prominent objective behind the usage of nanofluids. Hence it is necessary to study the base/nanofluid physical properties at various conditions. This article projects the experimental results of thermal conductivity and viscosity of two different nanofluids. The ratio was considered as 60:40 and 40:40 by volume in water and ethylene glycol respectively. The preparation of nanofluids was started by scattering SiO2 nano-particles in EG and “water” (W) blended in “60:40” (60EGW) and “40:60” (40EGW) ratio by volume. The ratio was considered as 60:40 and 40:40 by volume in water and ethylene glycol respectively. The regression analysis was conducted with available data and correlations were formulated for thermal properties. The nanofluids were used in evaluating “viscosity” and “thermal conductivity” experimentally. From the results, the SiO2 particles have achieved enhancement of 34% and 32% in thermal conductivity with the two base-fluids. Similarly, enhancement of 102% and 62% were reported in viscosity. Hence, it can be observed that SiO2 nanofluids in 40EGW nanofluid are a better heat transfer fluid when compared to SiO2 in 60EGW nanofluid.

In this paper, we consider a system of initial boundary value problems for parabolic equations, as a generalized version of the “φ-η-θ model’‘ of grain boundary motion, proposed by Kobayashi [16]. The system is a coupled system of: an Allen–Cahn type equation as in (1.1) with a given temperature source; and a phase-field model of grain boundary motion, known as “Kobayashi–Warren–Carter type model”. The focus of the study is on a special kind of solution, called energy-dissipative solution, which is to reproduce the energy-dissipation of the governing energy in time. Under suitable assumptions, two Main Theorems, concerned with: the existence of energy-dissipative solution; and the large-time behavior; will be demonstrated as the results of this paper.

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation involving the fractional p-Laplace operator ( − Δ)ps. More precisely, we consider $$ M\left(^{2N}}}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s} u=\lambda f(x)|u|^{q-2}u+K(x)|u|^{p_{s}^{*}-2}u,\quad &{\rm in}\ \Omega,\\ u=0, \quad\quad &{\rm in}\ ^{N}\setminus \Omega, \\ $$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary ∂Ω, M(t)=a + btm − 1 with m > 1, a > 0, b > 0, dimension N > sp, $ p_{s}^{*}={N-ps}$ is the fractional critical Sobolev exponent, and the parameters λ > 0, 0 < s < 1 < q < p < ∞. Applying Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity of nonnegative solutions.

Bertoin, Roynette et Yor \cite{bertion} described new connections between the class $\Bd$ of L\’evy-Laplace exponents $\Psi$ (also called the class (sub)critical branching mechanism) and the class of Bernstein functions ($\BF$) which are internal, i.e. those Bernstein functions $\phi$ s.t. $\Psi \circ \phi$ remains a Bernstein function for every $\Psi$. We complete their work and illustrate how the class f internal function is rich from the stochastic point of view. It is well known that every $\phi \in \BF$ corresponds univocally to: (i) a subordinator ${(X_t)}_{t\geq 0}$ (or equivalently to transition semigroups ${\big(\pr(X_t\in dx)\big)}_{t\geq 0}$; (ii) a L\’evy measure $\mu$ (which controls the jumps of the subordinator). It is also known that, on $\oi$, the measure $\pr(X_t \in dx)/t$ converges vaguely to $\dd \delta_0(dx)+ \mu(dx)$ as $t\to 0$, where $\dd$ is the drift term, but rare are the situations where we can compare the transition semigroups with the L\’evy measure. Our extensive investigations on the composition of L\’evy-Laplace exponents $\Psi$ with Bernstein functions show, for instance, this remarkable facts: $\phi$ is internal is equivalent to: (a) $\phi^2 \in \BF$ or to (b) $t\mu(dx) - \pr(X_t\in dx)$ is a positive measure on $\oi$. We also provide conditions on $\mu$ insuring that $\phi$ is internal. We also show L\’evy-Laplace exponents are closely connected to the class of Thorin Bernstein function and provide conditions on $\mu$ insuring that $\phi$ is internal.

In this letter, two time delay dynamic models, TDD-NCP model and Fudan-CCDC model, are introduced to track the data of COVID-19. The TDD-NCP model is developed recently by Cheng's group group in Fudan and SUFE. The TDD-NCP model introduced the time delay process into the differential equations to describe the latent period of the epidemic. The Fudan-CDCC model is established when Wenbin Chen suggested to determine the kernel functions in the TDD-NCP model by the public data from CDCC. By the public data of the cumulative confirmed cases in different regions in China and different countries, these models can clearly illustrate that the containment of the epidemic highly depends on early and effective isolations.

In this work, we study a weakly singular Volterra integro differential equation with Caputo type fractional derivative. First, we derive a sufficient condition for the existence and uniqueness of the solution of this problem based on the maximum norm. It is observed that the condition depends on the domain of definition of the problem. Thereafter, we show that this condition will be independent of the domain of definition based on an equivalent weighted maximum norm. In addition, we have also provided a procedure to extend the existence and uniqueness of the solution in its domain of definition by partitioning it. We also derive a sufficient condition under which the model problem will provide an analytic solution. Next, we introduce a operator based parameterized method to generate an approximate solution of this problem. Convergence analysis of this approach is established here. Next, we have optimized this solution based on least square method. For this, residual minimization is used to obtain the optimal values of the auxiliary parameter. In addition, we have also provided an error bound based on this technique. Several numerical examples are produced to clarify the effective behavior of the convergence of the present method. Comparison of the standard method and optimized method based on residual minimization signify the better accuracy of modified one. In addition, we also consider an equivalent form of weakly singular integro differential equation of a Heat transfer problem to show the effectiveness of the present approach.

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.

A mixed boundary value problem for the L\’ame equation in a thin layer $\Omega^h:\cC\times[-h,h]$ around a surface $\cC$ with the Lipshitz boundary is investigated. The main goal is to find out what happens when the thickness of the layer tends to zero $h\to0$. To this end we reformulate BVP into an equivalent variational problem and prove that the energy functional has the $\Gamma$-limit being the energy functional on the mid-surface $\cC$. The corresponding BVP on $\cC$, considered as the $\Gamma$-limit of the initial BVP, is written in terms of G\”unter’s tangential derivatives on $\cC$ and represents a new form of the shell equation. It is shown that the Neumann boundary condition from the initial BVP on the upper and lower surfaces transforms into a right-hand side term of the basic equation of the limit BVP.

In this study, the viscosity of MgO-Water nanofluid in a different volume fraction of nanoparticles, temperatures, and shear rates has been predicted by Artificial Neural Networks (ANNs) and surface methods. In the ANN method, an algorithm is proposed to select the best neuron number for the hidden layer. In the fitting method, a surface is proposed for each volume fraction of nanoparticles, and finally, the results of ANN and surface fitting method have been compared. It can be observed that, increasing the volume fraction from 0.07% to 1.25% at temperatures of 25, 30, 40, 50, and 60 °C resulted in about two-fold increase in viscosity. Also, the best network has 24 neurons in the hidden layer. It can be seen that for a network with 24 neurons in the hidden layer has the best overall correlation, and this coefficient is 0.999035. The mean absolute value of errors in ANN and fitting method are 0.0118 and 0.0206, respectively.

In this paper, the Laplace transform combined with the local discontinuous Galerkin method is used for distributed-order time-fractional diffusion-wave equation. In this method,at first, we convert the equation to some time-independent problems by Laplace transform.Then we can solve these stationary equations by the local discontinuous Galerkin method to discretize diffusion operators at the same time. Then, by using a numerical inversion of the Laplace transform we can find the solutions of the original equation. One of the advantages of this procedure is its parallel implementation. Another advantage of this approach is that the number of stationary problems that should be solved is much less than that are needed in time-marching methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the method.

This study is concerned with the optimal time rates of weak solutions for the 2D magneto-micropolar equations with only micro-rotational dissipation and magnetic diffusion. Due to some new observations, we obtain the optimal time decay rates of weak solutions $\|\nabla u(t)\|_{L^2}+\|\nabla w(t)\|_{L^2}\le C(1+t)^{-2}$ and $ \|\nabla b(t)\|_{L^{p}}\le C(1+t)^{-\frac12-(1-\frac1p)}$ with $p\in[2,\infty)$.

We consider the following quasilinear Keller-Segel system \left\{ {l} u_t = \Delta u - \nabla (u \nabla v) + g(u), \quad &(x,t)\in \Omega \times [0,T_{max}) ,\\[6pt] 0= \Delta v - v + u, \qquad \quad &(x,t)\in \Omega \times [0,T_{max}), \\[6pt] \right. on a ball $\Omega \equiv B_R(0)\subset^n$, $n\geq 3$, $R>0$, under homogeneous Neumann boundary conditions and non negative initial data. The source term $g(u)$ is superlinear and of logistic type i.e. $g(u)=\lambda u - \mu u^k,\ k>1, \ \mu >0$, $\lambda \in $ and $T_{max}$ is the blow-up time.\\ The solution $(u,v)$ may or may not blow up in finite time. Under suitable conditions on data, we prove that the function $u$, which blows up in $L^{\infty} (\Omega)$-norm , blows up also in $L^p(\Omega)$-norm for some $p>1$. Moreover a lower bound of the lifespan (or blow-up time when it is finite) $T_{max}$ is derived. \\ In addition, if $\Omega \subset ^3$ a lower bound of $T_{max}$ is explicitly computable.

The present paper is focused on the analysis on the existence of positive solutions of a second order differential system with two delays \left \{{lcr} x_1''(t)-a_1(t)x_1(t)+m_1(t)f_1(t,x(t),x_\tau(t))=0,\ \ t>0,\\ x_2''(t)-a_2(t)x_2(t)+m_2(t)f_2(t,x(t),x_\tau(t))=0,\ \ t>0,\\ x_1(t)=0,\ \ -\tau_1\ \leq t \leq 0,\;\; x_1(t)=0,\\ x_2(t)=0,\ \ -\tau_2\ \leq t \leq 0,\;\; x_2(t)=0 \right. by using two well-known fixed point theorems.

By using some real analysis techniques, We study the structural characteristics of a multi-parameter Hilbert-type integral inequality with the hybrid kernel, and obtain some equivalent conditions for this inequality. We also consider the operator expression of the equivalent inequalities. The conclusions not only integrates some results of references, but also nds some new Hilbert-type integral inequalities with simple form by choosing suitable parameter values.

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives .