In this article, we first presented a new general identity concerning differentiable mappings defined on $m$-invex set via $k$-fractional integrals. By using the concept of generalized relative semi-$(r;m,h_{1},h_{2})$-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to trapezium-like integral inequalities via $k$-fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article. Applications to special means for trapezium-like integral inequalities via $k$-fractional integrals are provided as well.

This paper develops a numerical approach for solving coupled systems of nonlinear fractional order integro-differential equations(NFIDE). Shifted discrete Chebyshev polynomials (SDCPs) have been introduced and their attributes have been checked. Fractional operational matrices for the orthogonal polynomials are also acquired. A numerical algorithm supported by the discrete orthogonal polynomials and operational matrices are used to approximate solution of coupled systems of NFIDE. The operational matrices of fractional integration and product are applied for approximate the unknown functions directly. These approximations were put in the coupled systems of NFIDE. A comparison has been made between the absolute error of approximate solutions of SDCPs method with previous published. The gained numerical conclusions disclose that utilizing discrete Chebyshev polynomials are more efficient in comparison to the other methods.

In this paper, we prove some fixed point theorems under generalized $(\epsilon-\delta)$ type rational contractions in which the fixed point may be a point of discontinuity. Our results generalize and improve a host of well-known fixed point theorems existing in the literature. In addition to it we give a fixed point theorem for $(\epsilon-\delta)$ non-expansive mappings in metric spaces. Several examples are given to illustrate our results.

In this manuscript, numerical solutions of the equations in the form of $u_{t}=Au+B(u)$ have been sought for, where $A$ and $B$ are linear and nonlinear operators, respectively. The modified equal width (MEW) equation has been converted into two sub problems. Then, the sub problems were solved according to the Strang splitting scheme by applying the cubic B-spline collocation finite element method. Thus, more accurate results of the equation MEW have been obtained than those non-splitting users. In order to test the accuracy and efficiency of the present method; single soliton, interaction of two solitons and Maxwellian initial condition pulse problems have been considered. Moreover, the stability analysis of each sub problem has been investigated by von-Neumann analysis method.

The aim of this short paper is two-fold: (i) to construct a $\mathsf{TQ}$-localization functor on algebras over a spectral operad $\mathcal{O}$, in the case where no connectivity assumptions are made on the $\mathcal{O}$-algebras, and (ii) more generally, to establish the associated $\mathsf{TQ}$-local homotopy theory as a left Bousfield localization of the usual model structure on $\mathcal{O}$-algebras, which itself is almost never left proper, in general. In the resulting $\mathsf{TQ}$-local homotopy theory, the ''weak equivalences'' are the $\mathsf{TQ}$-homology equivalences, where ''$\mathsf{TQ}$-homology'' is short for topological Quillen homology, which is also weakly equivalent to stabilization of $\mathcal{O}$-algebras. More generally, we establish these results for $\mathsf{TQ}$-homology with coefficients in a spectral algebra $\mathcal{A} $. A key observation, that goes back to the work of Goerss-Hopkins on moduli problems, is that the usual left properness assumption may be replaced with a strong c ofibration condition in the desired subcell lifting arguments: Our main result is that the $\mathsf{TQ}$-local homotopy theory can be established (e.g., a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are $\mathsf{TQ}$-equivalences.

In this paper, by using operator matrix techniques, we present necessary and sufficient conditions for the existence of a solution to the system of equations $AXD+FX^{*}B=C,$ $GXF^{*}+FX^{*}G^{*}=H$ for adjointable operators between Hilbert $C^*$-modules, and derive an expression for the general solution to the system. We establish necessary and sufficient conditions for the existence of a solution to the system of adjointable operator equations $AXF=H_{1},$ $CXD=H_{2},$ $BXD=H_{3}$ over Hilbert $C^*$-modules. Some of the findings of this paper extend some known results in the literature.

Obtaining exact solutions of the evolution equation is one of the very important subjects in mathematics, science and technology. For this purpose, many different methods have been constructed and developed. In this article, a new technique which is called extended modified Exp(-$\Omega (\xi )$) function method is going to be studied for seeking new exact solutions of Burger-Fisher equation and Phi Four equation. The method is capable of deriving many number of solutions. With the aid of the method, various exact solutions including trigonometric, hyperbolic and rational solutions have been obtained and using a software the graphical representation of the solutions have been presented. In conclusion, we can say that the present method can also be used for the solutions of a wide range of problems.

In this work the estimate about modulus of smoothness of the derivatives of the functions is obtained in Orlicz spaces. The relations between the modulus of smoothness of the functions with $n$th partial and Vallée-Poussin sums of the Fourier series in Orlicz spaces are studied.

The present paper conduct to introduce the connection and duplication formulas associated with the Boas-Buck-Appell polynomials. Examples providing the analogues results for certain members related to the Boas-Buck-Appell polynomials are considered.

In this work, the 2D-Appell and the Bessel polynomials are combined to introduce the family of the 2D-Appell-Bessel polynomials. The generating function, quasi-monomial properties, series definition and determinant form of these polynomials are established. Examples of some members belonging to this family are considered. The graphs of some hybrid special polynomials are also drawn for suitable values of the indices.

In the present study, a Lumped Galerkin finite element method using quadratic B-splines has been applied to the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) and coupled modified Korteweg-de Vries (mKdV) equations. The numerical solutions of discretized equations using Lumped Galerkin finite element method have been obtained using the fourth order Runge-Kutta method which is widely used for the solution of ordinary differential equation system. The numerical solutions obtained for various space and time values have been compared with exact ones using the error norms $L_{2}$ and $L_{\infty}$. Lumped Galerkin finite element method is an effective one which can be applied to a wide range of nonlinear evolution equations.

The object of this paper is to solve the generalized radical cubic functional equation, and discuss the stability problem in quasi-$\beta$-Banach spaces and then the stability by using subadditive and subquadratic functions in ($\beta, p$)-Banach spaces for the generalized radical cubic functional equation. Also certain non-stability results are investigated via specific counterexamples. Our results are generalization of the main results which are established by Z. Alizadeh and A. G. Ghazanfari in 2016.

In his work, K. Alladi considered the partition function $pod(n)$, the number of partitions of an integer $n$ with odd parts distinct (the even parts are unrestricted). He obtained a series expansion for the product generating function of these partitions. Later Hirschhorn and Sellers obtained some internal congruences involving the infinite families and Ramanujan's type congruences for $pod(n)$. Let $B_{4, 5}(n)$ denote the number of $(4, 5)$-regular bipartitions of a positive integer $n$ with odd parts distinct. In this paper, we establish many infinite families of congruences modulo powers of $2$ for $B_{4, 5}(n)$.