The aim of this paper is to investigate, as precisely as possible, a boundary value problem involving a third order ordinary differential equation. Its solutions are the similarity solutions of a problem arising in the study of the phenomenon of high frequency excitation of liquid metal systems in an antisymmetric magnetic field within the framework of boundary layer approximation.

In this paper we give simple and low degree examples of one-parameter polynomial families of planar differential equations which present generic, codimension one, isolated, compact bifurcations. In contrast with some examples which appear in the usual text books each bifurcation occurs when the bifurcation parameter is zero. We study the total number of limit cycles that the examples present and we also make their phase portraits on the Poincaré sphere.

Existence results are established for the resonant problem $y^{\text{'}\text{'}}+{\lambda }_{m}ay=f(t,y)$ a.e. on $[0,1]$ with $y$ satisfying Dirichlet boundary conditions. The problem is singular since $f$ is a Carathéodory function, $a\in L_{\mathrm{l}oc}^1(0,1)$ with $a>0$ a.e. on $[0,1]$ and ${\int }_0^{1}x(1-x)a\left(x\right)\mathrm{d}x<\infty$.

We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with $$-Laplacian$$ . ((u)) = f(t, u, u), u(0) = A, u(T) = B, . $$where$$ is an increasing homeomorphism, $\left(ℝ\right)=ℝ$, $\left(0\right)=0$, $f$ satisfies the Carathéodory conditions on each set $[a,b]\times {ℝ}^2$ with $[a,b]\subset (0,T)$ and $f$ is not integrable on $[0,T]$ for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on $[0,T]$.

We consider linear differential equations of the form $${\left(p\left(t\right)x^{\text{'}}\right)}^{\text{'}}+\lambda q\left(t\right)x=0(p\left(t\right)>0,q\left(t\right)>0)\left(\mathrm{A}\right)$$ on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda$ for which () has principal solutions $x_0(t;\lambda )$ vanishing at $t=a$. This problem may well be called a singular eigenvalue problem, since requiring $x_0(t;\lambda )$ to be a principal solution can be considered as a boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\left\{{\lambda }_n\right\}$ of eigenvalues such...

A new class of singular fractional linear systems and electrical circuits is introduced. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and the Laplace transformation, the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional system if it contains at least one mesh consisting of branches only with an ideal supercapacitor and voltage sources or at least...

Using a method developed by the author for an analysis of singular integral inequalities a stability theorem for semilinear parabolic PDEs is proved.