The maximum norm analysis of a nonmatching grids method for a class of parabolic $p(x)$-Laplacien equation

Motivated by the work of Boulaaras and Haiour in [7], we provide a maximum norm analysis of Schwarz alternating method for parabolic p(x)-Laplacien equation, where an optimal error analysis each subdomain between the discrete Schwarz sequence and the continuous solution of the presented problem is established

Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in $R^{n}\backslash\{0\}$

We consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 < \alpha(t) < 1$, $0 \leq \beta \leq 1$, and $0 < q < 1$ is studied. Moreover, an example is provided to demonstrate the result.

Peculiarities of the Activities of the Physical Education Teacher and the Trainer

Teaching and upbringing in physical education lessons is a complex multifaceted process, the effectiveness of which depends not only on what the teacher himself knows and can do, but also on how he transfers knowledge and skills to students. Pedagogical activity is a continuous solution of pedagogical problems. A feature of the conditions for the activity of a physical education teacher is the need to show physical exercises and ensure students when they perform physical exercises, as well as move with students when exercising in the air, on hikes, etc. The indicator of the coach's efficiency is the successful achievement of the goal with the most rational use of forces and means. In other words, efficiency presupposes the correspondence of the structure and functioning of the trainer's psyche to the structure and dynamics of his activity.

On the spatially inhomogeneous particle coagulation-condensation model with singularity

The spatially inhomogeneous coagulation-condensation process is an interesting topic of study as the phenomenon’s mathematical aspects mostly undiscovered and has multitudinous empirical applications. In this present exposition, we exhibit the existence of a continuous solution for the corresponding model with the following \emph{singular} type coagulation kernel: \[K(x,y)~\le~\frac{\left( x + y\right)^\theta}{\left(xy\right)^\mu}, ~~\text{for} ~x, y \in (0,\infty), \text{where}~ \mu \in \left[0,\tfrac{1}{2}\right] \text{ and } ~\theta \in [0, 1].\] The above-mentioned form of the coagulation kernel includes several practical-oriented kernels. Finally, uniqueness of the solution is also investigated.

On the existence of solutions of the Beltrami equations with conditions on inverse dilatations

We have found one of possible conditions under which the degenerate Beltrami equation has a continuous solution of the Sobolev class. This solution is H\"{o}lder continuous in the ''weak'' (logarithmic) sense with the exponent power $\alpha=1/2.$ Moreover, it belongs to the class $W^{1, 2}_{\rm loc}.$ Under certain additional requirements, it can also be chosen as a homeomorphic solution. We give an appropriate example of the equation that satisfies all the conditions of the main result of the article, but does not have a homeomorphic Sobolev solution.

Exact Solution of the Schrödinger Equation for Composition of Coulomb and Oscillator Potentials

The spherically symmetric potential is considered whose dependence on the distance r is described by the smooth composition of Coulomb at r < r0 and oscillator at r > r0 potentials. The boundary distance r0 is determined by the parameters of these potentials. The exact continuous solution of the radial Schrödinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.

Convergence and Numerical Solution of a Model for Tumor Growth

In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the discretization of the parabolic–hyperbolic–parabolic–elliptic system by means of the explicit formulae of the GFDM. We provide a theoretical proof of the convergence of the spatial–temporal scheme to the continuous solution and we show several examples over regular and irregular distribution of points. This shows the feasibility of the method for solving this nonlinear model appearing in Biology and Medicine in complicated and realistic domains.

A non-local macroscopic model for traffic flow

In this work we propose a non-local Hamilton-Jacobi model for traffic flow and we prove the existence and uniqueness of the solution of this model. This model is justified as the limit of a rescaled microscopic model. We also propose a numerical scheme and we prove an estimate error between the continuous solution of this problem and the numerical one. Finally, we provide some numerical illustrations.