We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem is of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data is bounded below away from zero in appropriate norms.

Melt superheated treatment can affect the solidification structure, but the direct evidence requires strict experiments. The molecular dynamics simulation method can break the limited experimental conditions, and provide advanced prediction for research. In this study, influences of superheated temperature (Ts) on the nucleation undercooling (ΔT) of metallic melts (Ti, Co, Mg, Al, Ni, Fe, Ag) were studied by the molecular dynamics simulation. The results show that the value of ΔT increases with the rise of Ts until the maximal ΔT approaches. In the curve of ΔT vs. Ts, there is an inflexion region where the nucleus cluster was broken. Above this inflexion region, the number of nucleus clusters decreases with the rise of Ts. Based on the simulated results, a model was proposed for describing the relation of ΔT and Ts, with which the maximal undercooling for metals can be predicted.

The paper deals with the analysis of stable thermo-solutal dendritic growth in the presence of intense convection. The n-fold symmetry of crystalline anisotropy as well as the two- and three-dimensional growth geometries are considered. The steady-state analytical solutions are found with allowance for the convective-type heat and mass exchange boundary conditions at the dendritic surface. A linear morphological stability analysis determining the marginal wavenumber is carried out. The new stability criterion is derived from the solvability theory and stability analysis. This selection criterion takes place in the regions of small undercooling. To describe a broader undercooling diapason, the obtained selection criterion, which describes the case of intense convection, is stitched together with the previously known selection criterion for the conductive-type boundary conditions. It is demonstrated that the stitched selection criterion well describes a broad diapason of experimental undercoolings.

In this paper, authors introduce the generalized Bessel-Maitland transform whose kernel is the generalized Bessel-Maitland function. New identities are obtained for special cases of the generalized Bessel-Maitland function. Using these relations, several identities are obtained for generalized Bessel-Maitland integral transform. It is shown that some special cases of them are related with the Laplace transform and the Hankel transform. Also, some examples are given as representations of the outcomes presented here.

In this paper, we give a new method to show a monotonicity result for a function $f$ satisfying $(_{a}\nabla_{h}^{\nu}f)(t)\leq0$ ($(_{a}\nabla_{h,\ast}^{\nu}f)(t)\leq0$) with $\nu\in(0,1]$, which has never been solved in other papers. In addition, we give an example to illustrate one of our main results.

The present work is concerned with the well-posedness and efficient numerical algorithm for a terminal value problem with a generalized Caputo fractional derivative. We investigate the existence and uniqueness of the solution of the terminal value problem, and consider the continuous dependence of the solutions on the given data. To illustrate our theoretical results, we present a one step algorithm for solving the considered problems. Some numerical examples are shown to illustrate the theoretical results and the efficiency of the numerical method.

In this note, we show that the Hausdorff operator $H_{\Phi}$ is unbounded on a large family of Quasi-Banach spaces, unless $H_{\Phi}$ is a zero operator.

In this paper, we study the vaporization process of a polydisperse ensemble of liquid drops on the basis of a nonlinear set of balance and kinetics equations for the particle-radius distribution function and temperature in the gaseous phase. We found an exact parametric solution to this problem using a modified time variable and the Laplace integral transform method. The distribution function of vaporizing drops as well as its moments, the temperature dynamics in gas, and the unvaporized mass of drops are found. The initial particle-radius distribution shifts to smaller particle radii with increasing the vaporization time. As this takes place, the temperature difference between the drops and gas decreases with time. It is shown that the heat of vaporization and initial total number of particles in the system substantially influence the dynamics of a polydisperse ensemble of liquid drops.

We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a special solution to this problem.

In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: \begin{displaymath} \d u_{tt}-\Delta u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data.\\ For $\mu < \frac{N(q-1)}{2}$ and $\mu \in (0, \mu_*)$, where $\mu_*>0$ is depending on the nonlinearties’ powers and the space dimension ($\mu_*$ satisfies $(q-1)\left((N+2\mu_*-1)p-2\right) = 4$), we prove that the wave equation, in this case, behaves like the one without dissipation ($\mu =0$). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu}{(1+t)^\beta}u_t; \ \beta >1$ (\cite{LT3}), where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu}{1+t}u_t$ is important.

We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the $$-boundedness of the solution operator families with resolvent parameter $\lambda$ varying in a sector $$ for any $\gamma_0>0$ and $0<\varepsilon<\pi/2$, where $ =\{ \lambda\in\setminus\{0\} \mid \arg\lambda\leq\pi-\varepsilon, \ \lambda>\gamma_0 \}$. As applications, we obtain the maximal $L_p$-$L_q$ regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in $L_p$-$L_q$ setting. We make full use of the solution formula to take $\gamma_0>0$ arbitrarily, while in general domains we only know the $$-boundedness for $\gamma_0\gg1$ from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.

This study provides the optimization of thermophysical properties of Cu/engine oil nanofluid. In this optimization, the objective functions were determined with the experimental data of viscosity and TC of nanofluid using RSM. Two equations for predicting thermal conductivity (TC) and viscosity data were presented which can accurately predict these properties. The NSGA-II method was used for multi-objective optimization (Mo-O) and Pareto’s front was introduced to study optimal viscosity and TC responses. According to the results, the highest TC and the lowest viscosity occurs when the temperature and solid volume fraction (SVF) of the nanoparticle are at their maximum values. Among the results, those with the highest TC and the lowest viscosity are referred to as optimal points.

This paper is concerned with a parabolic-elliptic Keller-Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it Dictyostelium discoideum} cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal-dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non-constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller-Segel system with signal-dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy.

Mittag-Leffler functions has many applications in various areas of Physical, biological ,applied, earth Sciences and Engineering. It is used in solving problems of fractional order differential, integral and difference equations. In this paper, we aim to define the m-parameter Mittag-Leffler function, which can be reduced to various already known extensions of Mittag-Leffler function. We then, discuss its various properties like recurrence relations, differentiation formula and integral representations. We also represent the new m-parameter Mittag-Leffler function in terms of some known special functions such as Generalized hypergeometric function, Mellin Barnes integral, Wright hypergeometric function and Fox H-function. We also discuss its various integral transforms like Euler-Beta, Whittaker, Laplace and Mellin transforms. Further, fractional differential and integral operators are considered to discuss few properties of m-parameter Mittag-Leffler function. Also, we use the m-parameter Mittag-Leffler function to define a generalization of Prabhakar integral and discuss its properties. Further, relation of m-parameter Mittag Leffler function with various other functions such as exponential, trigonometric, hypergeometric and algebraic functions is obtained and represented graphically using MATHEMATICA 12.

In this paper, we introduce two simple inertial algorithms for solving the split variational inclusion problem in Banach spaces. Under mild and standard assumptions we establish the weak and strong convergence of the proposed methods, respectively. As theoretical realization we study existence of solutions of the split common fixed point problem in Banach spaces. Several numerical examples in finite and infinite dimensional spaces compare and illustrate the performances of our schemes. Our work generalize and extend some recent relate results in the literature and also propose a simple and applicable method for solving split variational inclusions. .2in _2000 Mathematics Subject Classification:_ 65K10; 65K05; 47H10; 47L25. _Keywords:_ Split variational inclusion problem; inertial technique; Banach spaces.

Inspired by papers by M.A. Bokhari, A. Qadir, and H. Al-Attas [On Gauss-type quadrature rules, Numer. Funct. Anal. Optim. 31 (2010), 1120-1134] and by M.R. Rapaic, T.B. Sekara, and V. Govedarica [A novel class of fractionally orthogonal quasi-polynomials and new fractional quadrature formulas, Appl. Math. Comput. 245 (2014), 206-219], in this paper we investigate a few types of orthogonal polynomials on finite intervals and derive the corresponding quadrature formulas of Gaussian type for efficient numerical computation of the left and right fractional Riemann-Liouville integrals. Several numerical examples are included to demonstrate the numerical efficiency of the proposed procedure.

Current pagination concerns with exploration regarding thermal characteristics induced by Hybridization of  with MWCNT in a permeable tank filled with viscous fluid with magnetization. Roseland approximation is obliged to configure radiative heat flux aspects. Modeling and structuring of considered problem is established in partial differential setup. CVFEM is implemented to seek out solution of constructed differential layout. Data establishing properties of    with MWCNT’s is disclosed in tabular format. Impression of sundry variables in view of isotherm and stream pattern is divulged. Local convective thermal rate against involved variables in 3D snapshots is captured. Test about grid independence and comparison with conducted work to get assurance of present finding is explicated. It is deduced that heat transfer along walls enhances verses Rayleigh and Darcy parameters whereas delineates against mounting effect of magnetic field strength. 

The objective of the present paper is to investigate the dynamics of Hepatitis B-type virus (HBV) infection through mathematical model. Distinct to the existing mathematical models on HBV, the present model considers the various factors such as immune impairment, the maximum number of T-cells (total carrying capacity), logistic growth term. Besides, for more accuracy, the role of antiretroviral therapies are also involved in the analysis. In addition, time delays are inevitable during the activation of immune response and during the antiretroviral therapy. Considering these factors while formulating the mathematical model which helps to gain insights into the disease progression. With the derived model, the qualitative analysis such as stability analysis, bifurcation analysis and stabilization analysis can be performed to investigate the performance of the model over the period of time. The significance of the model parameters are revealed through Hopf-type bifurcation analysis and the global stability analysis of the proposed model. With the help of dataset values that are extracted from the literature the efficiency of the derived theoretical results are explored.