The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

A brief survey on numerical methods for solving singularly perturbed problems

Automated dynamic malware analysis is a common approach for detecting malicious software. However, many malware samples identify the presence of the analysis environment and evade detection by not performing any malicious activity. Recently, an approach to the automated detection of such evasive malware was proposed. In this approach, a malware sample is analyzed in multiple analysis environments, including a bare-metal environment, and its various behaviors are compared. Malware whose behavior deviates substantially is identified as evasive malware. However, a malware analyst still needs to re-analyze the identified evasive sample to understand the technique used for evasion. Different tools are available to help malware analysts in this process. However, these tools in practice require considerable manual input along with auxiliary information. This manual process is resource-intensive and not scalable. In this paper, we present MalGene, an automated technique for extracting analysis evasion signatures. MalGene leverages algorithms borrowed from bioinformatics to automatically locate evasive behavior in system call sequences. Data flow analysis and data mining techniques are used to identify call events and data comparison events used to perform the evasion. These events are used to construct a succinct evasion signature, which can be used by an analyst to quickly understand evasions. Finally, evasive malware samples are clustered based on their underlying evasive techniques. We evaluated our techniques on 2810 evasive samples. We were able to automatically extract their analysis evasion signatures and group them into 78 similar evasion techniques.

https://dl.acm.org/doi/pdf/10.1145/2810103.2813642

MalGene: Automatic Extraction of Malware Analysis Evasion Signature

Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

In this paper, we review many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade. In particular, it is a follow up article of the one published in 2005 [K.C. Patidar, On the use of non-standard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758]. It also includes those research contributions in this field that are very significant and published prior to the above article but were not included in the above paper simply because we did not have access to them when we wrote the above article. We also give a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades. All contributions are listed chronologically except that in some instances we have grouped certain works to show connectivity in those fields. While categorizing these research contributions, we considered a number of different application areas. Moreover, due to space limitations, firstly, we have not i...

Nonstandard finite difference methods: recent trends and further developments

In this work, a high-order compact finite-difference (HOCFD) scheme has been proposed to solve 1-dimensional (1D) and 2-dimensional (2D) elliptic and parabolic singularly-perturbed reaction-diffusion problems. A new kind of piecewise uniform mesh of Shishkin type (Miller et al. in Fitted Numerical Methods for Singular Perturbation Problems, 1996) has also been proposed and using this mesh the HOCFD scheme gives better results as compared to the results using the Shishkin mesh. Moreover, the stated method gives e-uniform convergence and improved orders of convergence which have also been provided in the results for some test problems.

High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

https://web.iitd.ac.in/~mmehra/publication/JWTA07.pdf

Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

In this paper, we study the strong convergence of SP iterative scheme with mixed errors for the accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces. We show that this iterative scheme is almost stable for both types of operators. Moreover, with the help of computer programs in C++, comparison between SP, Ishikawa and Noor iterative schemes with errors is also shown for both types of operators through examples.

Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces

We prove results concerning data dependence of Noor and SP iterative schemes using certain quasi-contractive operators in real Banach spaces. Our results reveal that by choosing an approximate quasi-contractive operator (for which it is possible to compute the fixed point); we can approximate the fixed point of the given operator. An example is also provided to explain the validity of our results. General Terms Computational Mathematics D Keywords SP iteration, Noor iteration, Quasi-Contractive Operators 1. INTRODUCTION Let E be a real Banach space and B be a nonempty closed, convex subset of E . Let T, S be two self operators on B . In a complete metric space, the Picard iterative process ^ ` n n 0 x f defined by 1 x Tx nn 1 n = 0, 1,… (1.1) has been employed to approximate the fixed points of mappings satisfying the inequality d Tx Ty d x y ( , ) ( , )dD 0 (1.2) for all

/pdf/data-dependence-of-noor-and-sp-iterative-schemes-when-1vyuj8enku.pdf

Data Dependence of Noor and SP Iterative Schemes when dealing with Quasi-Contractive Operators

The Jungck–Khan iterative scheme for a pair of nonself operators contains as a special case Jungck–Ishikawa and Jungck–Mann iterative schemes. In this paper, we establish improved results about convergence, stability, and data dependence for the Jungck–Khan iterative scheme.

https://journals.tubitak.gov.tr/math/issues/mat-16-40-3/mat-40-3-13-1503-1.pdf

Vivek Kumar

Papers

High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces

Data Dependence of Noor and SP Iterative Schemes when dealing with Quasi-Contractive Operators

Stability and data dependence results for the Jungck--Khan iterative scheme