Volume 2 (2018) Issue 2

In this report we present new sixth order iterative methods for solving non-linear equations. The derivation of these methods is purely based on variational iteration technique. To check the validity and efficiency we compare of methods with Newton’s method, Ostrowski’s method, Traub’s method and modified Halleys’s method by solving some test examples. Numerical results shows that our developed methods are more effective. Finally, we compare polynomigraphs of our developed methods with Newton’s method, Ostrowski’s method, Traub’s method and modified Halleys’s method.

In this work we study the global solution, uniqueness and asymptotic behaviour of the nonlinear equation
$$\begin{array}{r}{u}_{tt}–{\mathrm{\Delta }}_{p}u=\mathrm{\Delta }u–g\ast \mathrm{\Delta }u\end{array}$$
where ${\mathrm{\Delta }}_{p}u$ is the nonlinear $p$-Laplacian operator, $p\ge 2$ and $g\ast \mathrm{\Delta }u$ is a memory damping. The global solution is constructed by means of the Faedo-Galerkin approximations taking into account that the initial data is in appropriated set of stability created from the Nehari manifold and the asymptotic behavior is obtained by using a result of P. Martinez based on new inequality that generalizes the results of Haraux and Nakao.

This paper deals with the determination of a coefficient in the diffusion term of some degenerate /singular one-dimensional linear parabolic equation from final data observations. The mathematical model leads to a non convex minimization problem. To solve it, we propose a new approach based on a hybrid genetic algorithm (married genetic with descent method type gradient). Firstly, with the aim of showing that the minimization problem and the direct problem are well posed, we prove that the solution’s behavior changes continuously with respect to the initial conditions. Secondly, we chow that the minimization problem has at least one minimum. Finally, the gradient of the cost function is computed using the adjoint state method. Also we present some numerical experiments to show the performance of this approach.

In this paper, we find a solution of a new type of Langevin equation involving Hilfer fractional derivatives with impulsive effect. We formulate sufficient conditions for the existence and uniqueness of solutions. Moreover, we present Hyers-Ulam stability results.

Let $\mathcal{A}$ be the class of analytic and univalent functions in the open unit disc $\mathrm{\Delta }$ normalized such that $f(0)=0=f^{\mathrm{\prime }}(0)-1.$ In this paper, for $\psi \in \mathcal{A}$ of the form $\frac{z}{f(z)},f(z)=1+\sum _{n=1}^{\mathrm{\infty }}{a}_{{}_n}{z}^n$ and $0\le \alpha \le 1,$ we introduce and investigate interesting subclasses ${\mathcal{H}}_{\sigma }(\varphi ),S_{\sigma }(\alpha ,\varphi ),M_{\sigma }(\alpha ,\varphi ),$ ${\mathrm{\Im }}_{\alpha }(\alpha ,\varphi )$ and ${\beta }_{\alpha }(\lambda ,\varphi )(\lambda \ge 0)$ of analytic and bi-univalent Ma-Minda starlike and convex functions. Furthermore, we find estimates on the coefficients $\left|a_1\right|$ and $\left|a_2\right|$ for functions in these classess. Several related classes of functions are also considered.

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish both the existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.

It is proved that if the problem ${\mathrm{\nabla }}^{2}u=1$ in $D$, $u{|}_S=0$, $u_N=m:=|D|/|S|$ then $D$ is a ball. There were at least two different proofs published of this result. The proof given in this paper is novel and short.

In this paper, we use Riccati transformation technique to establish some new oscillation criteria for the second order nonlinear dynamic equation with damping on time scales $$(r(t)(x^{\mathrm{\Delta }}(t){)}^{\alpha }{)}^{\mathrm{\Delta }}-p(t)(x^{\mathrm{\Delta }}(t){)}^{\alpha }+q(t)f(x(t))=0.$$ Our results not only generalize some existing results, but also can be applied to the oscillation problems that are not covered in literature. Finally, we give some examples to illustrate our main results.

We define fractional transforms ${\mathcal{R}}_{\mu }$ and ${\mathcal{H}}_{\mu }$, $\mu >0$ on the space $\mathbb{R}\times {\mathbb{R}}^n$. First, we study these transforms on regular function spaces and we establish that these operators are topological isomorphisms and we give the inverse operators as integro differential operators. Next, we study the $L^p$-boundedness of these operators. Namely, we give necessary and sufficient condition on the parameter $\mu$ for which the transforms ${\mathcal{R}}_{\mu }$ and ${\mathcal{H}}_{\mu }$ are bounded on the weighted spaces $L^p([0,+\mathrm{\infty }[\times {\mathbb{R}}^n,r^{2a}dr\otimes dx)$ and we give their norms.

Let $S$ be a $C^1$-smooth closed connected surface in ${\mathbb{R}}^3$, the boundary of the domain $D$, $N=N_s$ be the unit outer normal to $S$ at the point $s$, $P$ be the normal section of $D$. A normal section is the intersection of $D$ and the plane containing $N$. It is proved that if all the normal sections for a fixed $N$ are discs, then $S$ is a sphere. The converse statement is trivial.