The existence and uniqueness of the solution is proved for a nonlinear boundary value problem for ODE subject to an infinite condition \cite{1}, which describes the study of two-phase Stefan problems on the semi-infinite line [0,\infty). This result considerably extends the analysis of a recent work \cite{4}. A highly accurate analytic approximate solution of this problem is also provided via the Adomian decomposition method.

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.

A two-dimensional tumor-immune model with the time delay of the adaptive immune response is considered in this paper. The model accounts for the interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on the surface of a solid tumor with a necrotic kernel. The system has three equilibria. Both zero and maximum tumor volume equilibria are unstable, while the behavior of the positive equilibriums is closely related to the ratio of the immune killing rate to tumor volume growth rate. The positive equilibrium is more likely to be locally asymptotically stable when the ratio is smaller than a critical value, and unstable otherwise. We also derive conditions to guarantee the existence of Hopf bifurcation at the positive equilibrium. Applying the center manifold reduction and normal form method, we obtain explicit formulas to determine the properties of Hopf bifurcations. The global continuation of local Hopf bifurcation is studied based on the coincidence degree theory. The results reveal that long immune delay can lead to oscillation dynamics. We also carry out detailed numerical analysis for parameters to illustrate our qualitative analysis. Numerically, we find that a shorter immune response time leads to a longer patient survival time and the period and amplitude of a stable periodic solution increase with the immune response time. When CTLs recruitment rate and death rate vary, we observe how the ratio metioned above and the first bifurcation value change numerically, which yields further insights to the tumor-immune dynamics.

In this article, by using the notion of newly defined $q_{1}q_{2}$-derivatives and integrals some new Simpson’s type inequalities for co-ordinated convex functions are shown. The outcomes raised in this paper are extensions and generalizations of the comparable results in the literature on Simpson’s inequalities for co-ordinated convex functions.

In this paper, a fourth-order accurate conservative finite difference scheme for solving the coupled nonlinear Schr\”odinger (CNLS) equations is proposed. Conservation of the discrete energy and masses, priori estimates, existence and uniqueness of numerical solutions, convergence with second-order in time and fourth-order in space as well as stability of the present scheme are proved by discrete energy method. A convergent iterative method for the present scheme is developed. Numerical experiments are given to support the theoretical analysis.

A way to obtain approximate periodic solutions to nonlinear oscillators arising in a micro-electro-mechanical system (MEMS) is presented for the case of zero initial conditions and magnetostatic excitation. The frequency-amplitude relationship is derived by adopting He’s frequency formulation. The obtained analytical results are illustrated graphically. The proposed simple approach gives the fast insight into the dynamics of the singular oscillator and can be useful for design of MEMS devices.

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 exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficients in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.

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.

For the following wave equations with structural damping and nonlinear memory source terms \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t =\int_{0}^{t}(t-s)^{\gamma-1} \vert u (s)\vert^{p}\,\text{d}s, \] and \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t = \int_{0}^{t}(t-s)^{\gamma-1} \vert u_s (s)\vert^{p}\,\text{d}s, \] posed in $(x,t) \in \mathbb{R}^N \times [0,\infty) $, where $u=u(x,t)$ is real-value unknown function, $p>1$, $\alpha,\beta\in (0, 2]$, $\gamma\in (0,1)$, we prove the nonexistence of global solutions. Moreover, we give an upper bound estimate of the life span of solutions.

In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.

This paper is concerned with the existence and multiplicity of solutions for the following variable $s(\cdot)$-order fractional $p(\cdot)$-Kirchhoff type problem \begin{equation*} \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb R^{2N}}\frac{1}{p(x,y)}\displaystyle{\frac{|v(x)-v(y)|^{p(x,y)}}{|x-y|^{N+p(x,y)s(x,y)}}}dxdy\right)(-\Delta)^{s(\cdot)}_{p(\cdot)}v(x)+|v(x)|^{\overline{p}(x)-2}v(x) =\mu g(x,v)\ \ {\rm in}~\mathbb{R}^{N},\\ v\in W^{s(\cdot),p(\cdot)}(\mathbb{R}^{N}), \end{array}\right. \end{equation*} where $N>p(x,y)s(x,y)$ for any $(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$, $(-\Delta)^{s(\cdot)}_{p(\cdot)}$ is a variable $s(\cdot)$-order $p(\cdot)$-fractional Laplace operator with $s(\cdot):\mathbb R^{2N}\to(0,1)$ and $p(\cdot):\mathbb R^{2N}\to(1,\infty)$, $\overline{p}(x)=p(x,x)$ for $x\in\mathbb{R}^{N}$, and $M$ is a continuous Kirchhoff-type function, $g(x,v)$ is a Carath\’{e}odory function, $\mu>0$ is a parameter. We obtain that there are at least two distinct solutions for the above problem by applying the generalized abstract critical point theorem. Under the weaker conditions, we also show the existence of one solution and infinitely many solutions by using the mountain pass lemma and fountain theorem, respectively. In particular, the new compact embedding result of the space $ W^{s(\cdot),p(\cdot)}(\mathbb{R}^{N})$ into $L^{q(\cdot)}_{a(x)}(\mathbb{R}^{N})$ will be used to overcome the lack of compactness in $\mathbb{R}^N$. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.

In this work, an energy stable numerical scheme is proposed to solve the PFC model with a nonlocal Lagrange multiplier. The construction of the numerical scheme is based on invariant energy quadratization (IEQ) technique to transform a nonlinear system into a linear system, while the time variables is discretized by second order scheme. The stability term in the new scheme can balance the influence of nonlinear term. Moreover, we obtain the results of unconditional energy stability, uniqueness and uniform boundedness of numerical solution, and the numerical scheme is convergent with order of $\mathcal{O}(\delta t^2)$. Several numerical tests are conducted to confirm the theoretical results.

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.