Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.

Extended convergence of a sixth order scheme for solving equations under ω–continuity conditions

The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

Existence and multiplicity results for a Steklov problem involving (p(x), q(x))-Laplacian operator

In this work, we are concerned with a generalized Steklov problem with (p(x), q(x))-Laplacian operator. Under some appropriate conditions on the data involved in the elliptic problem, we prove the existence of at least three solutions using Ricceri’s three critical points theorem.

The ideal of Lipschitz classical p-compact operators and its injective hull

We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

Pricing American Put Option using RBF-NN: New Simulation of Black-Scholes

The present work proposes an Artificial Neural Network framework for calculating the price and delta hedging of American put option. We consider a sequence of Radial Basis function Neural Network, where each network learns the difference of the price function according to the Gaussian basis function. Based on Black Scholes partial differential equation, we improve the superiority of Artificial Neural Network by comparing the performance and the results achieved used in classic Monte Carlo Least Square simulation with those obtained by Neural networks in one dimension. Thus, numerical result shows that the Artificial Neural Network solver can reduce the computing time significantly as well as the error training.

Nonlinear Random Differential Equations with n Sequential Fractional Derivatives

This article deals with a solvability for a problem of random fractional differential equations with n sequential derivatives and nonlocal conditions. The existence and uniqueness of solutions for the problem is obtained by using Banach contraction principle. New random data concepts for the considered problem are introduced and some related definitions are given. Also, some results related to the dependance on the introduced data are established for both random and deterministic cases.

Bu’s theorem in the positive situation

In this paper, we valorize the relationship between positive p−summing operators and positive strongly q−summing operators using (Contemp. Math. 328, 145 − 149 (2003)).

A new class of mixed fractional differential equations with integral boundary conditions

This paper deals with a new class of mixed fractional differential equations with integral boundary conditions. We show an important equivalence result between our problem and nonlinear integral Fredholm equation of the second kind. The existence and uniqueness of a positive solution are proved using Guo-Krasnoselskii’s fixed point theorem and Banach’s contraction mapping principle. Different types of Ulam-Hyers stability are discussed. Three examples are also given to show the applicability of our results.

Further generalized refinement of Young’s inequalities for τ -mesurable operators

In this paper, we prove that if a, b > 0 and 0 ≤ v ≤ 1. Then for all positive integer m(1) - For v ∈ v \in \left[ {0,{1 \over {{2^n}}}} \right], we have {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{v^m}{{\left( {\sqrt {{b^m}} - \root {{2^k}} \of {\left( {a{b^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m}.}(2) - For v ∈ v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right], we have {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{{\left( {1 - v} \right)}^m}{{\left( {\sqrt {{a^m}} - \root {{2^k}} \of {\left( {b{a^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m},} we also prove two similar inequalities for the cases v ∈ v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right] and v ∈ v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right]. These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.