Volume 26, Issue 1

We investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.

The main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann–Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. The suggested schemes are applied to some problems and their practical usefulness is illustrated.

Motivated by the idea which has been introduced by Boulaaras and Guefaifia [S. Boulaaras and R. Guefaifia, Existence of positive weak solutions for a class of Kirchhoff elliptic systems with multiple parameters, Math. Methods Appl. Sci. 41 2018, 13, 5203–5210] and by Afrouzi and Shakeri [G. A. Afrouzi, S. Shakeri and N. T. Chung, Existence of positive solutions for variable exponent elliptic systems with multiple parameters, Afr. Mat. 26 2015, 1–2, 159–168] combined with some properties of Kirchhoff-type operators, we prove the existence of positive solutions for a new class of nonlocal p⁢(x){p(x)}-Kirchhoff parabolic systems by using the sub- and super-solutions concept.

In this article, we describe a function fitting method that has potential applications in machine learning and also prove relevant theorems. The described function fitting method is a convex minimization problem which can be solved using a gradient descent algorithm. We also provide qualitative analysis on fitness to data of this function fitting method. The function fitting problem is also shown to be a solution of a linear, weak partial differential equation (PDE). We describe a simple numerical solution using a gradient descent algorithm that converges uniformly to the actual solution. As the functional of the minimization problem is a quadratic form, there also exists a numerical method using linear algebra.

In this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space ℝn{\mathbb{R}^{n}} for any n≥2{n\geq 2}.

In this paper we will prove the Korovkin type theorem for modified Szász–Mirakyan operators via A -statistical convergence and the power summability method. Also we give the rate of the convergence related to the above summability methods, and in the last section, we give a kind of Voronovskaya type theorem for A -statistical convergence and Grüss–Voronovskaya type theorem.

In the class of analytic functions in the unit disc |z|<1{\lvert z\rvert<1} we prove some new sufficient conditions for functions to be univalent or to be close-to-convex in the unit disc. Also we extend Ozaki’s condition that ℜ⁢𝔢⁢{exp⁡(i⁢α)⁢f(p)⁢(z)}>0{\mathfrak{Re}\{\exp(i\alpha)f^{(p)}(z)\}>0} in |z|<1{\lvert z\rvert<1} implies that f⁢(z){f(z)} is at most p -valent in |z|<1{\lvert z\rvert<1}.

In this paper, we give some approximation properties by Stancu–Chlodowsky type λ-Bernstein operators in the polynomial weighted space and obtain the convergence properties of these operators by using Korovkin’s theorem. We also establish the direct result and the Voronovskaja type asymptotic formula.

This paper considers the following problem: for what value r , r<1{r<1} a function that is univalent in the unit disk |z|<1{|z|<1} and convex in the disk |z|<r{|z|<r} becomes starlike in |z|<1{|z|<1}. The number r is called the radius of convexity sufficient for starlikeness in the class of univalent functions. Several related problems are also considered.

In this paper, we introduce the class of asymptotically demicontractive multivalued mappings and establish a strong convergence theorem of the modified Mann iteration to a common fixed point of a finite family of asymptotically demicontractive multivalued mappings in a complete CAT⁢(0){\mathrm{CAT}(0)} space. We also give a numerical example of our iterative method to show its applicability.

In this paper, an algorithm for solving optimal control of linear time-varying systems with quadratic performance indices is presented. By using important matrices which are derived from Chebyshev wavelets properties, the original problem is converted to a quadratic programming one. This parameter optimization method is applied on both constrained and unconstrained control systems having linear state equations of integer and fractional orders. The computing time saved by this approach is much better than with other methods in which there is no need to calculate the optimal costs of systems by substituting the approximations of the state and control vectors and their values are default outputs of the quadprog solvers.

Delaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.