Programmed to retain active responsivity to environmental stimuli, diverse types of synthetic gels have been attracting interests regarding various applications, such as versatile elastic biodevices. In a different approach, when the gels are made of tissue-derived biopolymers, they can act as an artificial extracellular matrix (ECM) for use as soft implants in regenerative medicine. To explore the physical properties of hydrogels in terms of statistical thermodynamics, the mean-field Flory-Huggins-Rehner theory has long been used with various analytical and numerical modifications. Here we suggest a novel mathematical model on the volume phase transition of a biological hybrid gel that is sensitive to ambient temperature. To mimic acellular soft tissues, the ECM-like hydrogel is modeled as a network of biopolymer chains, such as type I collagen and gelatin, which are covalently crosslinked and swollen in aqueous solvents. Within the network, thermoresponsive synthetic polymer chains are doped by chemical conjugation. Based on the Flory-Huggins-Rehner framework, our model phenomenologically illustrates a well-characterized volume phase behavior of engineered tissue mimics as a function of temperature by formulating the ternary mixing free energy of the polymer-solvent system and by generalizing the elastic free energy term. With this formalism, the decoupling of the Flory-Huggins interaction parameter between the thermoresponsive polymer and ECM biopolymer enables deriving a simple analytical equation for the volume phase transition as a function of the structural and compositional parameters. We show that the doping ratio of the thermoresponsive polymers affects the phase transition temperature of the ECM-like gels.

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.

Nickel alloys are widely used in the production of gas turbine parts. The alloys show resistance to mechanical and chemical degradation under severe long-term stress and high temperatures. One of the major mechanical properties of the alloys is the high-temperature rupture strength, which is measured after a specimen is heated to a certain temperature and held for a certain time considering deformation. Determining the influence of certain elements on the properties of an alloy is a complex scientific and engineering problem that affects the time and cost of developing new materials. Simulation is a great chance to cut costs. In this paper, we predict a high-temperature strength based on the composition of refractory elements in alloys using a deep learning artificial neural network. We build the model based on prior knowledge of the composition of the alloys, information on the role of alloying elements, type of crystallization, test temperature and time, and the tensile strength. Successful simulation results show the applicability of this method in practice.

We study the dependence of the eigenvalues of time-harmonic Maxwell's equations in a cavity upon variation of its shape. The analysis concerns all eigenvalues both simple and multiple. We provide analyticity results for the dependence of the elementary symmetric functions of the eigenvalues splitting a multiple eigenvalue, as well as a Rellich-Nagy-type result describing the corresponding bifurcation phenomenon. We also address an isoperimetric problem and characterize the critical cavities for the symmetric functions of the eigenvalues subject to isovolumetric or isoperimetric domain perturbations and prove that balls are critical. We include known formulas for the eigenpairs in a ball and calculate the first one.

Based on the least square method, we proposed a new algorithm to obtain the solution of the second kind regular and weakly singular Volterra-Fredholm integral equations in reproducing kernel spaces. The stability and uniform convergence of the algorithm are investigated in details. Numerical experiments verify the theoretical findings. Meanwhile this method is also applicable to the nonlinear Volterra integral equations. Test problems which have non-smooth solutions are also considered and our proposed method is efficient as some recent Krylov subspace methods such as LSQR and LSMR.

In this work, the Lie symmetry theory is used to study the propagation of waves in an elastic string with electric currents in a static magnetic field. Both linear and nonlin- ear cases of the governing equations of string motion are analyzed. The classification problem of finding the principal admitted Lie groups of symmetries is solved. Some invariant analytical solutions are constructed. The physics of invariant solutions is interpreted when it is possible.

In this article, an approximate analytical solution of an integro-differential system of equations is constructed, which describes the process of intense boiling of a superheated liquid. The kinetic and balance equations for the bubble-size distribution function and liquid temperature are solved analytically using the Laplace transform and saddle-point methods with allowance for an arbitrary dependence of the bubble growth rate on temperature. The rate of bubble appearance therewith is considered in accordance with the Dering-Volmer and Frenkel-Zeldovich-Kagan nucleation theories. It is shown that the initial distribution function decreases with increasing the dimensionless size of bubbles and shifts to their greater values with time.

Using monotonicity methods, the Lagrange multiplier rule and some variational arguments, we consider a type of localization results pertaining to the existence of critical points to action functionals on a closed ball. A variant of the Schechter critical point theorem on a ball in Hilbert and Banach spaces is obtained. Applications to nonlinear Dirichlet problem and to partial difference equations are given in the final part of this paper.

In this paper, an inverse time-fractional Schr$\ddot{o}$dinger problem of potential-free field is studied. This problem is ill-posed, i.e, the solution (if it exists) does not depend continuously on the data. Based on an a priori bound condition, the optimal error bound analysis is given. Moreover, a modified kernel method is introduced. The convergence error estimate obtained by this method under the priori regularization parameter selection rule is optimal, and the convergence error estimate obtained under the posteriori regularization parameter selection rule is order-optimal. Finally, some numerical examples are given to illustrate the effectiveness and stability of this method.

In this paper, we study a class of doubly nonlinear parabolic equations (1.1). The nonlinearity B(u) brings great difficulties to the study of the problem. First we show that the problem has a unique solution. Then we prove that the process corresponding to the problem is norm-to-weak continuous. After that, by using Legendre transform, we obtain uniform estimates and asymptotic compactness properties that allow us to ensure the existence of pullback D-attractors for the associated process to the problem

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 main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: $$u_{tt}- \Delta u+ \mu(-\Delta)^{\sigma/2} u_t= |u_t|^p,\quad u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x),$$ with $\mu>0$, $n\geq1$, $\sigma \in (0,2]$ and $p>1$. In particular, we are going to prove the non-existence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.

The time-fractional Gardner Burger (TFGB) equation is an efficient model for studying nonlinear fluctuations of different types of wave profiles, such as the gravity solitary waves in the ocean, ion-acoustic wave (IAW) in a plasma environment, etc. Here, to build an example of the existence of the classical Gardner Burger (GB) equation, a multi-component plasma environment is considered and a classical GB equation is derived by employing reductive perturbation technique (RPT) from the basic governing equation. Further, the classical GB equation is converted into the TFGB equation by applying Agrawal’s approach, where the Riesz fractional derivative is adopted on the time-fractional term. A new approach using the improved Bernoulli sub-equation function method (IBSEFM) is carried out to solve the TFGB equation. Finally, some $2D$ and $3D$ graphs are plotted through which the physical structures of the solution are explored and the effect of the Burgers term and fractional order of the equation are determined.

In this paper, the critical condition to achieve rational solutions of the multi-component nonlinear Schr\”odinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions,an explicit formula of the $n$th-order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the $n$th-order rogue waves is summarized. This conjecture also holds for rogue waves of the multi-component complex modified Korteweg-de Vries equation. Finally, the semi-rational solutions of the Manakov system are discussed.

in this work, we introduce an efficient meshless technique for solving the two-dimensional variable-order time-fractional mobile/immobile advection-diffusion model with Dirichlet boundary conditions. The main advantage of this scheme is to obtain a global approximation for this problem which reduces such problems to a system of algebraic equations. To approximate the first and fractional variable-order against the time, we use the finite difference relations. The proposed method is based on the Moving Kriging (MK) interpolation shape functions. To discretization this model in space variables, we use the MK interpolation. Duo to the fact that the shape functions of MK have Kronecker’s delta property, boundary conditions are imposed directly and easily. To illustrate the capability of the proposed technique on regular and irregular domains, several examples are presented in different kinds of domains and with uniform and nonuniform nodes. Also, we use this scheme to simulating anomalous contaminant diffusion in underground reservoirs.

We consider a new generic reaction-diffusion system, given as the following form: ∂u/∂t - div(g(│(∇u_σ)│)∇u)=f(t,x,u,v,∇v), in Q_T ∂v/∂t - d_v Δv=p(t,x,u,v,∇u), in Q_T u(0,.)=u_0, v(0,.)=v_0, in Ω (1) ∂u/∂η=0, ∂v/∂η=0, in ∑_T. Where Ω=]0,1[?×]0,1[, Q_T =]0,T [? and T =]0,T [?, (T > 0), η is an outward normal to domain Ω and u_0, v_0 is the image to be processed, x ∈Ω, σ >0, ∇u_σ= u∗ ∇G_σ and G_σ= 1/√2πσ exp(-│x│^2/4σ). In this study we are going to proof that there is a global weak solution to the ptoblem (1), we truncate the system and show that it can be solved by using Schauder fixed point theorem in Banach spaces. Finally by making some estimations, we prove that the solution of the truncated system converge to the solution of the problem.

In this paper, the authors obtain some new blow up criteria for the smooth solution to the three dimensional magnetic B\’enard equation without thermal diffusion in terms of pressure. We prove that if $\pi\in L^2(0, T; L^{\f3r}(\R^3))$ with $0

Numerous experimental data on rapid solidification of eutectic systems exhibit the formation of metastable solid phases with the initial (nominal) chemical composition. This fact is explained by suppression of eutectic decomposition due to diffusionless (chemically partitionless) solidification beginning at a high but a finite growth velocity of crystals. A model considering the diffusionless growth is developed in the present work to analyze the atomic diffusion ahead of lamellar eutectic couples growing into supercooled liquid. A general solution of the model is presented from which two regimes are followed. The first presents diffusion-limited regime with the existence of eutectic decomposition if the solid-liquid interface velocity is smaller than characteristic diffusion speed in bulk liquid. The second shows suppression of eutectic decomposition under diffusionless transformation from liquid to one-phase solid if the solid-liquid interface velocity overcomes characteristic diffusion speed in bulk liquid.