In this paper, we prove an optimal three-ball inequality for bi-Laplace equation in some open, connected set. The derivation of such estimate relies on a delicate Carleman estimate for the bi-Laplace equation and some Caccioppoli inequalities to estimate the lower-ters. Based on three -ball inequality, we then derive the vanishing order of solutions, which is a quantitative version of the strong unique continuation property.

The paper formulates the nonlinear problem of steady-state heat conduction at the constant electric potential difference on the surfaces of a plane dielectric layer with the temperature-dependent heat conduction coefficient and electrical resistivity. A fixed temperature value is set on one of the layer surfaces, and the convective heat exchange with the ambient medium occurs on the opposite surface. The formulation of the problem is transformed into integral ratios, which allows the calculation of the temperature distribution over the layer thickness, governed both by the monotonic and nonmonotonic function. The quantitative assay of the temperature state of a layer of a polymer dielectric made of amorphous polycarbonate is given as an example, as well as the analysis of nonuniformity of the absolute value of electric field intensity over the thickness of this layer.

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient non-linearity $|\nabla u(x,t)|^q$. The solution to this problem is understood in a weak sense. We show that for $1“3/2$ the existence of the only local-in-time weak solution of Cauchy’s problem.If $3/2”

A second-order time stepping scheme is developed for the binary fluid-surfactant phase field model coupled with hydrodynamics by using the scalar auxiliary variable approach and pressure correction method. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. By introducing one scalar auxiliary variable, the system is transformed into an equivalent form so that the nonlinear terms can be treated semi-explicitly. The scheme is linear and decoupled; thus, they can be solved eciently. We further prove that the semidiscretized scheme in time is unconditionally energy stable. Numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.

In this work, we present basic results and applications of Stepanov pseudo almost periodic functions with measure. Using only the continuity assumption, we prove a new composition result of $\mu$-pseudo almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions in Banach spaces with weak regular forcing terms. We prove the existence and uniqueness of $\mu$-pseudo almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations, respectively, provided that the nonlinear forcing terms are only Stepanov $ \mu $-pseudo almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Some examples illustrating our theoretical results are also presented.

We develop a one step matrix method in order to obtain approximate solutions of first order non-linear systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the exact solution. We apply the method to different well known examples and check its precision, in terms of local error, comparing it with the error produced by other methods. The advantage of the method over others widely used lies on the great simplicity of its implementation.

A document by BiYun Tang. Click on the document to view its contents.

This paper introduces advances in the use of Bessel functions (cylinder functions) in achieving optimal sharpness of shapes delineated by collections of holes (dark valley regions) and bright regions in different areas of a video frame. An image hole is an island of low voxel intensities surrounded by varying voxel intensity peaks. The basic approach is to identify dominant collections of lights (high voxel intensity peaks) and darks (low voxel intensity clusters) in video frames. A main finding in this paper is that surface objects (recorded in images) are sharper wherever there are preponderant contrasting differences between image lights and darks. These contrasting differences lead to the highly accurate detection of shape holes (dark valleys) that delineate surface shapes recorded in sequences of video frames). With the use of cylindrical Bessel functions, contrasting differences between peaks and valleys can be measured in terms of voxel intensities and voxel indices.

In this paper, we investigate the existence and uniqueness of solutions and Ulam's type stabilities including the well-known Ulam-Hyers stability and the newly extended Ulam-Hyers' conformable exponential stability for two classes of fractional differential equations with the conformable fractional derivative and the time delay. The Banach contraction principle, the technique of Picard operator, the Gronwall integral inequalities and generalized iterated integral inequality in the sense of conformable fractional integral are the main tools for deriving our main results. Finally, several illustrative examples will be presented to demonstrate our work.

This article presents an investigation on free convection flow and heat and mass transfer of fractional nanofluids formed by different base fluids with different shaped copper nanoparticles past an infinite vertical plate with radiation effect. The model which describes the system involving partial differential equations is converted into model of fractional nanofluids using Caputo time derivatives, which explains the impact of memory on behavior of nanofluids and are solved by employing technique of Laplace transform. The different base fluids - water $(H_2 O)$, $SA$:sodium alginate $(C_{6}H_{9}Na O_{7})$ and $EG$:ethylene glycol $(C_{2}H_{6}O_{2})$ and various shapes of nanoparticles - blade, brick, platelet and cylinder are considered for the study. The exact solutions are obtained for the temperature, concentration and velocity distributions and the respective Nusselt number, Sherwood number and skin-friction coefficient. The influence of non-dimensional parameters like fractional parameter, Schmidt number, radiation parameter, nanoparticle volume fraction, etc., which provides physical interpretations of temperature, concentration and velocity fields, Nusselt number, Sherwood number and skin-friction are discussed in detail with the aid of graphical representations. From the results it is found that fractional nanofluids formed by water based blade shaped nanoparticle exhibits more velocity and temperature distributions. Also, strengthen of fluid flow, temperature and concentration of nanofluids are inversely correlate with the order of fractional derivatives.

Mathematical modeling of uncertain FIDEs is an extremely significant topic in electric circuits, signal processing, electromagnetics, and anomalous diffusion systems. Based on the RKA, a touching numerical approach is considering in this study to solve groups of FFIDEs with ABC fractional distributed order derivatives. The solution-based approach ‎lies in generating infinite orthogonal basis from kernel functions, where uncertain condition is fulfilled. Thereafter, an orthonormal basis is erected to figurate fuzzy ABC solutions with series shape in idioms of η-cut extrapolation in Hilbert space A(D) and B(D). In this orientation, fuzzy ABC fractional integral, fuzzy ABC fractional derivative, and fuzzy ABC FIDE are utilized and comprised. The competency and accuracy of the suggested approach are indicating by employing several experiments. From theoretical viewpoints, the acquired results signalize that the utilize approach has several merits in feasibility and opportunity for treating with many fractional ABC distributed order models. In the end, highlights and future suggested research work are eluded.

The dynamical behavior of the rational vectorial operator associated with a multidimensional iterative method in polynomial systems, gives us interesting information about the stability of the iterative scheme. The stability of fixed points, dynamic planes, bifurcation diagrams, etc. are known tools that act in this sense. In this manuscript, we introduce a new tool, which we call isonormal surface, to complement the information about the stability of the iterative method provided by the dynamic elements mentioned above. These dynamical instruments are used for analyze the stability of a parametric family of multidimensional iterative schemes in terms of the value of the parameter. Some numerical tests confirm the obtained dynamical results.

SVEIR epidemic model; age-dependence; basic reproduction number; local and global stability; uniform persistence

A computation analysis is performed to study double-diffusive natural convection in a right-angle trapezoidal cavity packed with a porous medium. The horizontal top and bottom boundaries are insulated and impermeable. The vertical left sidewall is kept at a constant heat flux and high concentration, whereas the temperature and concentration of the mixture at the inclined sidewall are held at lower temperatures and lower concentration. The dimensionless nonlinear system is solved by employing finite difference method along Successive Under Relaxation technique. The findings are compared and validated with the existing literature for the Darcy flow driven through a single buoyancy effect (difference in density is only due to temperature variations) in a porous square enclosure. The numerical results are expressed in the form of dimensionless temperature, concentration, streamlines, isotherms and iso-concentrations, local and average Nusselt and Sherwood numbers. It is demonstrated that the Rayleigh number and buoyancy parameter enhance both surface heat and concentration rates.

In this paper, we deal with a new approximation of a diffusive predator--prey model with Leslie--Gower term and social behavior for the prey subject to Neumann boundary conditions. A new approach for a predator-prey interaction in the presence of prey social behavior has been considered. Our main topic in this work is to study the influence of the prey's herd shape on the predator-prey interaction in the presence of Leslie--Gower term. First of all, we examine briefly the system without spatial diffusion. By analyzing the distribution of the eigenvalues associated with the constant equilibria, the local stability of the equilibrium points and the existence of Hopf bifurcation have been investigated. Then, the spatiotemporal dynamics introduced by self diffusion was determined, where the existence of the positive solution, Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been derived. Further, the effect of the prey's herd shape rate on the prey and predator equilibrium densities as well as on the Hopf bifurcating points has been discussed. Finally, by using the normal form theory on the center manifold, the direction and stability of the bifurcating periodic solutions have also been obtained. To illustrate the theoretical results, some graphical representations are given.

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 a class of higher-dimensional Hilbert-type inequalities on a fractal set $(\mathbb{R}_+^{\alpha n})^{k}$. The crucial step in establishing our results are higher-dimensional spherical coordinates on a fractal space. Further, we impose the corresponding conditions under which the constants appearing in the established Hilbert-type inequalities are the best possible. As an application, our results are compared with the previous results known from the literature.

The aim of this paper is to give an alternative technique for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long-term well-posedness result established by S. Israwi in [2] but for the original system, in which the partial operator ∇× is not involved.