𝒜 Let $𝒜$ be the class of analytic functions in the open unit disk . We define ${\Theta }^{\alpha ,\beta }:𝒜\to 𝒜$ by $\left({\Theta }^{\alpha ,\beta }f\right)$$\left(z\right):=\Gamma \left(2-\alpha \right)z^{\alpha }{D}_z^{\alpha }$$(\Gamma \left(2-\beta \right)z^{\beta }{D}_z^{\beta }f\left(z\right)),$$\left(\alpha ,\beta \ne 2,3,4\dots \right),$ where $D_z^{\gamma }f$ is the fractional derivative of $f$ of order $\gamma$. If $\alpha ,\beta \in \left[0,1\right]$, then a function $f$ in $𝒜$ is said to be in the class $\text{S}{\text{P}}_{\alpha ,\beta }$ if ${\Theta }^{\alpha ,\beta }f$ is a parabolic starlike function. In this paper, several properties and characteristics of the class $\text{S}{\text{P}}_{\alpha ,\beta }$ are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class-preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved.

Direct solution of a class of $n\text{th}$-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78).

We present new oscillation criteria for the differential equation of the form ${\left[r\left(t\right)U\left(t\right)\right]}^{\prime }+p\left(t\right)k_2\left(x\left(t\right),\hspace{0.17em}{x}^{\prime }\left(t\right)\right){\left|x\left(t\right)\right|}^{\nu }$$U\left(t\right)+q\left(t\right)\varphi \left(x(g_1\left(t\right)\right),x^{\prime }\left(g_2\left(t\right))\right)f\left(x\left(t\right)\right)=0$, where $U\left(t\right)=k_1\left(x\left(t\right),x^{\prime }\left(t\right)\right){\left|x^{\prime }\left(t\right)\right|}^{\alpha -1}{x}^{\prime }\left(t\right)$, $\alpha \le \beta ,\hspace{0.17em}\hspace{0.17em}\nu =\left(\beta -\alpha \right)/\left(\alpha +1\right)$. Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking $\beta =\alpha$, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.

A perturbing nonlinear Schrodinger equation is studied under general complex nonhomogeneities and complex initial conditions for zero boundary conditions. The perturbation method together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the symbolic solution algorithm is tested through computing the possible approximations under truncation procedures. The method of solution is illustrated through case studies and figures.

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function $x^{\prime \prime }(z)+c{x}^{\prime }(z)=x\left(az+bx(z)\right)$ in the complex field $ℂ$. By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function ${\lambda }^2{g}^{\prime \prime }(\lambda z)g^{\prime }(z)-\lambda g^{\prime }(\lambda z)g^{\prime \prime }(z)$ + $c{\left(g^{\prime }\left(z\right)\right)}^2\left(\lambda g^{\prime }\left(\lambda z\right)-a{g}^{\prime }(z)\right)$ = ${\left(g^{\prime }\left(z\right)\right)}^3\left(g\left({\lambda }^{2}z\right)-ag\left(\lambda z\right)\right)$, we get its local invertible analytic solutions.

The oscillation susceptibility of the ADMIRE aircraft along the path of longitudinal flight equilibriums is analyzed numerically in the general and in a simplified flight model. More precisely, the longitudinal flight equilibriums, the stability of these equilibriums, and the existence of bifurcations along the path of these equilibriums are researched in both models. Maneuvers and appropriate piloting tasks for the touch-down moment are simulated in both models. The computed results obtained in the models are compared in order to see if the movement concerning the landing phase computed in the simplified model is similar to that computed in the general model. The similarity we find is not a proof of the structural stability of the simplified system, what as far we know never been made, but can increase the confidence that the simplified system correctly describes the real phenomenon.

For the differential equation $u^{\prime }\left(t\right)=Au\left(t\right)+f\left(t\right)$, $t\ge 0$ on a Hilbert space $H$, we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

We provide an alternate representation to the result that the Lie algebra of generators of the system of $n$ differential equations, $(y^a{)}^{″}=0$, is isomorphic to the Lie algebra of the special linear group of order $\left(n+2\right)$, over the real numbers, $sl\left(n+2,ℝ\right)$. In this paper, we provide an alternate representation of the symmetry algebra by simple relabelling of indices. This provides one more proof of the result that the symmetry algebra of $(y^a{)}^{″}=0$ is $sl\left(n+2,ℝ\right)$.

Double integral operators which were considered by S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008), 591–597) are discussed. In order to show the analytic function $f\left(z\right)$ is starlike of order $\beta$ in the open unit disk $𝕌$, the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions for $f\left(z\right)$ to be starlike of order $\beta$ in $𝕌$ concerned with second-order differential inequalities and double integral operators.

The combined effects of a uniform vertical magnetic field and a nonuniform basic temperature profile on the onset of steady Marangoni convection in a horizontal layer of micropolar fluid are studied. The closed-form expression for the Marangoni number $M$ for the onset of convection, valid for polynomial-type basic temperature profiles upto a third order, is obtained by the use of the single-term Galerkin technique. The critical conditions for the onset of convection have been presented graphically.

A nonlinear thermoelastic system with memory is considered, which is derived from a physical model with vibration in temperature environment. By some skillful and technical arguments, results of existence, uniqueness, and uniform decay on this generalized system are obtained.

We study the global bifurcation of the differential inclusion of the form $-{\left(k{u}^{\prime }\right)}^{\prime }+g\left(\cdot ,u\right)\in \mu F\left(\cdot ,u\right),u^{\prime }\left(0\right)=0=u^{\prime }\left(1\right)$, where $F$ is a “set-valued representation” of a function with jump discontinuities along the line segment $\left[0,1\right]\times \left\{0\right\}$. The proof relies on a Sturm-Liouville version of Rabinowitz's bifurcation theorem and an approximation procedure.

We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of $H$-measures to investigate the zero diffusion-dispersion-smoothing limit.

We discuss the existence of solutions for a nonlinear antiperiodic boundary value problem of integrodifferential equations of fractional order $q\in \left(1,2\right]$. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to establish the results.

An analytical expression for the solution of the prey-predator problem by an adaptation of the homotopy analysis method (HAM) is presented. The HAM is treated as an algorithm for approximating the solution of the problem in a sequence of time intervals; that is, HAM is converted into a hybrid numeric-analytic method called the multistage HAM (MSHAM). Comparisons between the MSHAM solutions and the fourth-order Runge-Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for the nonlinear systems of ordinary differential equations.