No abstract

Propagation of spherical shock wave with azimuthal magnetic field and radiation heat flux in self-gravitating perfect gas is investigated. The azimuthal magnetic field and the initial density are assumed to vary according to power law. An exact similarity solution is reported when loss of energy due to radiation escape is notable and radiation pressure is non-zero. The entire energy of the shock wave is varying and increases with time. The effects of variation of the radiation pressure number, the initial density variation index, the Alfven-Mach number, the gravitational parameter and the adiabatic exponent are workout in detail. The shock strength increases with an increase in the initial density variation index. On the other hand, presence of magnetic field or an increment in the value of the radiation pressure number or the ratio of specific heats or gravitational parameter the shock strength decreases. It is obtained that increase in the radiation pressure number and gravitational parameter has same behavior on the flow variables. Also, it is observed that an increase in the value of gravitational parameter and the adiabatic exponent have same behavior on the fluid velocity, the material pressure, the radiation pressure, the mass and the radiation flux and azimuthal magnetic field.

We prove strong and Δ-convergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces using S-iteration process due to Agarwal et al. As uniformly convex hyperbolic spaces contain Banach spaces as well as CAT(0) spaces, our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces.

We introduce a new sequence of -integral operators. We show that it is a weighted approximation process in the polynomial space of continuous functions defined on unit interval. Weighted statistical approximation theorem, Korovkin type theorems for fuzzy continuous functions, and an estimate for the rate of convergence for these operators. IntroductionThe study of -Calculus is a generalization of any subjects, such as hyper geometric series, complex analysis, and particle physics. Currently it continues being an important subject of study. It has been shown that positive linear operators constructed by -numbers are quite effective as far as the rate of convergence is concerned and we can have some unexpected results, which are not observed for classical case. In the last decade, some new generalizations of wellknown positive linear operators, based on -integers, were introduced and studied by several authors. For example, -Meyer Konig and Zeller operators were studied by Trif [1], Dogru and Duman [2], Aral and Gupta [3], and so forth. In 20011, Aral and Gupta [4,5] introduced a -generalization of the classical Baskakov operators. In 2012, Sharma [6,7] introduced the -Durrmeyer type operators. Orkcu and Dogru [8] introduced Kantorovich type generalization ofSzasz-Mirakjan operators and discussed their -statistical approximation properties. In this paper motivated by Sharma we introduced a -analogue of the -Durrmeyer operators and we study better rate of convergence and statistical approximation properties.We mention some important definitions of -Calculus.Definition 1. For any fixed real numbers > 0 and ∈ , the -integers are defined byIn this way for real number one may writeDefinition 2. The -factorial is defined byDefinition 3. For any number ∈ (0, ), the -binomial coefficient is defined byAral and Gupta [4] introduced a -generalization of the classical Baskakov operators. For all ∈ [0, ∞), ∈ (0, 1), and each positive integer , the operators are defined as whereand they established some approximation results on it. Sharma [6] introduced the following -Durrmeyer type operators defined as follows: forwhereand they established some approximation results on it. In this paper motivated by Sharma [6,7,[10][11][12] we introduce a -analogue of the -Baskakov-Durrmeyer type operators defined as follows: for ∈ , ,where we set , ( ; ) = ( + −1 ) (Kasana et al.[13] obtained a sequence of modified Szâsz type operators for integrable function on [0, ∞) defined aswhere and belong to [0, ∞) and is fixed.In this paper, motivated by Kasana and Sharma, we introduce a -analogue of the -Baskakov-Durrmeyer type operators defined as follows: for ∈ , ,where and belong to , and is fixed. The aim of this paper is to study some approximation properties of a new generalization of operators based onintegers. We estimate moments for these operators. Also, we study statistical convergence and Korovkin type theorems for fuzzy continuous functions. Finally, we give better error estimations for operators (10) and (12). Estimation of MomentsWe ...

We establish the existence and uniqueness of solutions for a class of nonlinear Volterra integral and integro-differential equations using fixed-point theorems for a new variant of cyclic (ϕ, ψ, θ )-contractive mappings. Nontrivial examples are given to support the usability of our results. MSC: 47H10; 54H25