The existence of solutions for boundary value problem of fractional hybrid differential equations

In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equations$D_{0^{+}}^{\alpha }\left[\frac{x(t)}{f(t\text{,}x(t))}\right]+g(t\text{,}x(t))=0\text{,}0<t<1\text{,}$$x(0)=x(1)=0\text{,}$where $1<\alpha ?2$ is a real number, $D_{0^{+}}^{\alpha }$ is the Riemann–Liouville fractional derivative. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. As an application, examples are presented to illustrate the main results.     

Approximate analytical solution to the Falkner–Skan wedge flow with the permeable wall of uniform suction

Based on a new analytical method, namely homotopy analysis method (HAM), an approximate analytical solution to the Falkner–Skan wedge flow; with the boundary conditions of $f(0)=\gamma >0\text{,}{f}^{\prime }(0)=0\text{,}{f}^{\prime }(+\infty )=0$, i.e. the permeable wall mass transfer conditions of uniform suction, is given. The comparisons are also made between the results of the present work and numerical method by 4th-order Runge–Kutta method combined with Newton–Raphson technique. It is found that the results of the present work agree well with those by numerical method, which verifies the validity of the present work.     

Le nombre des diviseurs unitaires d'un entier dans les progressions arithmétiques

Let the functions $d_{k\text{,}l}^{*}(n)$ and $d_{k\text{,}l}(n)$ be number of unitary divisors (see below) and number of divisors n in arithmetic progressions $\{l+mk\}$; k and l are integers relatively prime such that $1?l?k$ and let, for $n?2$

$F(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}(n))\ln (\phi (k)\ln n)}{\ln n}\text{,}{F}^{*}(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}^{*}(n))\ln (\phi (k)\ln n)}{\ln n}\text{and}$

$D^{*}(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}(n)/d_{k\text{,}l}^{*}(n))\ln (\phi (k)\ln n)}{\ln n}\text{,}$

where $\phi (k)$ is Euler's totient. The function $F(n\text{;}k\text{,}l)$ has been studied in [A. Derbal, A. Smati, C. A. Acad. Sci. Paris, Ser. I 339 (2004) 87–90]. In this Note we study the functions $F^{*}(n\text{;}k\text{,}l)$ and $D^{*}(n\text{;}k\text{,}l)$. We give explicitly their maximal orders and we compute effectively the maximum of $F^{*}(n\text{;}k\text{,}l)$ for $k=1\text{,}2\text{,}3$ and that of $D^{*}(n\text{;}k\text{,}l)$ for $k=1\text{,}3\text{,}5\text{,}7\text{,}8\text{,}9\text{,}10\text{,}11\text{,}13$. To cite this article: A. Derbal, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

Transcendence degree one function fields over a finite field with many automorphisms

Let $\mathbb{K}$ be the algebraic closure of a finite field ${\mathbb{F}}_q$ of odd characteristic p. For a positive integer m prime to p, let $F=\mathbb{K}(x,y)$ be the transcendence degree 1 function field defined by $y^q+y=x^m+x^{?m}$. Let $t=x^{m(q?1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let K be the Galois closure of F with respect to H. By Stichtenoth [20], K has genus $\mathfrak{g}(K)=(qm?1)(q?1)$, p-rank (Hasse–Witt invariant) $\gamma (K)={(q?1)}^2$ and a $\mathbb{K}$-automorphism group of order at least $2q^{2}m(q?1)$. In this paper we prove that this subgroup is the full $\mathbb{K}$-automorphism group of K; more precisely ${\text{Aut}}_{\mathbb{K}}(K)=\mathrm{\Delta }?D$ where Δ is an elementary abelian p-group of order $q^2$ and D has an index 2 cyclic subgroup of order $m(q?1)$. In particular, $\sqrt{m}|{\text{Aut}}_{\mathbb{K}}(K)|>\mathfrak{g}{(K)}^{3/2}$, and if K is ordinary (i.e. $\mathfrak{g}(K)=\gamma (K)$) then $|{\text{Aut}}_{\mathbb{K}}(K)|>{\mathfrak{g}}^{3/2}$. On the other hand, if G is a solvable subgroup of the $\mathbb{K}$-automorphism group of an ordinary, transcendence degree 1 function field L of genus $\mathfrak{g}(L)\ge 2$ defined over $\mathbb{K}$, then $|{\text{Aut}}_{\mathbb{K}}(K)|\le 34{(\mathfrak{g}(L)+1)}^{3/2}<68\sqrt{2}\mathfrak{g}{(L)}^{3/2}$; see [15]. This shows that K hits this bound up to the constant $68\sqrt{2}$.Since ${\text{Aut}}_{\mathbb{K}}(K)$ has several subgroups, the fixed subfield $F^N$ of such a subgroup N may happen to have many automorphisms provided that the normalizer of N in ${\text{Aut}}_{\mathbb{K}}(K)$ is large enough. This possibility is worked out for subgroups of Δ.