Volume 41, Issue 3

This paper is concerned with numerical methods for a coupled system of two partial differential equations (PDEs), modeling flow and transport of a contaminant in porous media. This coupled system, arising in modeling of flow and transport in heterogeneous porous media, includes two types of equations: an elliptic and a diffusion-convection equation. We focus on miscible flow in heterogeneous porous media. We use the mixed finite element method for the Darcy flow equation over triangles, and for the concentration equation, we use nonconforming finite volume methods in unstructured mesh. Finally, we show the existence and uniqueness of a solution of this coupled scheme and demonstrate the effectiveness of the methodology by a series of numerical examples.

In this paper, we obtain analytical solutions of an unsolved integral 𝐑S⁢(m,n){\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G -function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑S⁢(m,n){\mathbf{R}_{S}(m,n)} in the form of integrals ℧S*⁢(υ,b,c,λ,y){\mho_{S}^{*}(\upsilon,b,c,\lambda,y)}, ΞS⁢(υ,b,c,λ,y){\Xi_{S}(\upsilon,b,c,\lambda,y)}, ∇S⁡(υ,b,c,λ,y){\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧S⁢(υ,b,λ,y){\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G -functions. Moreover, as applications of Ramanujan’s integral 𝐑S⁢(m,n){\mathbf{R}_{S}(m,n)}, the three new infinite summation formulas associated with Meijer’s G -function are obtained.

In this article, we study certain results connected with a generalized Mittag-Leffler function. A generalized Mittag-Leffler function operator of Laplace and Sumudu conversions are investigated and some applications of the recognized results are also deduced as corollaries in this article. The outcomes of the present study are valuable in solving fractional order mathematical models in science, mathematics, finance and technology where the Mittag-Leffler function arises in a natural manner.

In this paper, we give some Minkowski–Clarkson’s type inequalities related to two finite sequences of real nonnegative numbers. In particular, we prove two inequalities which in some sense can be regarded as inverse Minkowski’s inequalities concerning the cases p≥2{p\geq 2} and 0<p≤1{0<p\leq 1}. Moreover, for 1<p<2{1<p<2} we prove another Minkowski–Clarkson’s type inequality.

In this paper, we will provide a complete study of the self-improving properties of the discrete Muckenhoupt class 𝒜p⁢(𝒞){\mathcal{A}^{p}(\mathcal{C})} of weights defined on ℤ+{\mathbb{Z}_{+}}. In addition, we will determine the range of the new constants which are related to the original constants via an algebraic equation. For illustration, we will give an example to prove that the results are sharp. The results will be obtained by employing a discrete version of an inequality due to Hardy–Littlewood and a new discrete Hardy-type inequality with negative powers.

In periodic homogenization problems, one considers a sequence (uη)η{(u^{\eta})_{\eta}} of solutions to periodic problems and derives a homogenized equation for an effective quantity u^{\hat{u}}. In many applications, u^{\hat{u}} is the weak limit of (uη)η{(u^{\eta})_{\eta}}, but in some applications u^{\hat{u}} must be defined differently. In the homogenization of Maxwell’s equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a curl-free field Y∖Σ¯→ℝ3{Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}}, where Y is the periodicity cell and Σ an inclusion, a vector in ℝ3{\mathbb{R}^{3}}. In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.