Related papers: Explicit positive solutions to $G$-heat equations …
It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we…
We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses,…
We construct a singular solution of a stationary nonlinear Schr\"{o}dinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity…
We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via…
We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces.…
We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on…
Algebraic Riccati equations are encountered in many applications of control and engineering problems, e.g., LQG problems and $H^\infty$ control theory. In this work, we study the properties of one type of discrete-time algebraic Riccati…
Let $\Omega$ be a smooth bounded domain in $\R^n$, $n\ge 5$. We consider the semilinear heat equation at the critical Sobolev exponent $$ u_t = \Delta u + u^{\frac{n+2}{n-2}} \inn \Omega\times (0,\infty), \quad u =0 \onn \pp\Omega\times…
An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it…
This paper reviews results about free convection near a vertical flat plate embedded in some saturated porous medium. We focus on a third order autonomous differential equation that gives a special class of solutions called similarity…
We show that the higher order linear differential equation possesses all solutions of infinite order under certain conditions by extending the work of authors about second order differential equation \cite{dsm2}.
In this paper we consider the overdetermined Cauchy problem for the heat equation. We prove that if the problem has a nontrivial nonnegative solution with a certain sequence of similar level sets, then the solution must be radially…
Let $\Omega$ be a bounded open interval, and let $p>1$ and $q\in\left(0,p-1\right) $. Let $m\in L^{p^{\prime}}\left(\Omega\right) $ and $0\leq c\in L^{\infty}\left(\Omega\right) $. We study existence of strictly positive solutions for…
We provide algorithms computing power series solutions of a large class of differential or $q$-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.
The specific heat-phonon spectrum inversion has played a significant role in solid physics. But for this inherently ill-posed problem, most of the known solutions are complex both in form and content, although they are rigorous and perfect.…
In this paper, the famous Klein-Gordon-Zakharov equations are firstly generalized, the new special types of Klein-Gordon-Zakharov equations with the positive fractional power terms (gKGZE) are presented. In order to derive the exact…
We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact…
We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are…
In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2{\alpha}.
In this paper we show how to find a closed form solution for third order difference operators in terms of solutions of second order operators. This work is an extension of previous results on finding closed form solutions of recurrence…