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

This work proposes a new fuzzy logic based high resolution (HR), total variation diminishing (TVD) scheme in finite volume frameworks to compute an approximate solution of the shallow water equations (SWEs). Fuzzy logic enhances the execution of classical numerical algorithms. To test the effectiveness and accuracy of the proposed scheme, the dam-break problem is considered. A comparison of the numerical results by implementing some classical flux limiting methods is provided. The proposed scheme is able to capture both smooth and discontinuous profiles, leading to better oscillation-free results.

High Resolution TVD Scheme Based on Fuzzy Modifiers for Shallow-Water Equations.

This paper reviews some of the established high‐resolution (HR) schemes for computing solutions to hyperbolic systems by means of a finite volume (FV) framework. The objective of this research is to assess the ability of various HR flux limiting techniques to capture intense shocks and discontinuities. A crucial design factor in developing these methods is selecting a suitable flux‐limiter, so a review of different flux‐limiters for standard explicit FV frameworks has also been included. A brief discussion of the HR total variation diminishing (TVD) numerical schemes, starting from some historical introduction, going on to exploring the design principles behind the HR, second‐order, oscillation free schemes, turning towards high order nonlinear techniques and ending with an overview of some of the most significant developments and applications in the last decades has been covered. Several one‐dimensional and two‐dimensional benchmark examples with severe and challenging wave configurations have been utilized to compare the various numerical approaches. These nonlinear test cases have been selected such that the computational setups are as basic as possible.

A comparative study of high‐resolution methods for nonlinear hyperbolic problems

This paper expresses the concepts, methods, and applications of mathematical models in agriculture. We propose and analyze an ecosystem consisting of two types of preys and their predators; here the prey-I like sugarcane crops that take a long time to grow and prey-II like vegetables, which have a short life, are grown with sugarcane crops and predators that harm both prey-I and prey-II. The various equilibria of the system are obtained, and the stability conditions are analyzed. Furthermore, a comprehensive analysis of the optimal control strategy is also performed. The optimal control model includes the use of three control variables, such as pesticide application rate, biomass application rate, and control of Cassava mosaic virus in the system. Finally, we apply Pontryagin’s maximum principle to find the optimal control. Furthermore, analytical results are verified by numerical simulations.

A prey–predator model approach to increase the production of crops: Mathematical modeling and qualitative analysis

In the proposed model, a prey-predator system with migration in both species along with the disease in prey population is considered. To study the dynamics among healthy predator, susceptible prey and infected prey, Holling type II functional response is incorporated. The model has been examined using local stability analysis around the existing equilibrium points and by considering migration in susceptible prey as the control parameter, bifurcation conditions are discussed in brief. Basic reproduction number is determined which affirms the disease to be endemic or not and global stability analysis has been implemented using geometric approach. At the end, numerical simulation to support theoretical results is exhibited.

Vivek Kumar

Papers

High Resolution TVD Scheme Based on Fuzzy Modifiers for Shallow-Water Equations.

A comparative study of high‐resolution methods for nonlinear hyperbolic problems

A prey–predator model approach to increase the production of crops: Mathematical modeling and qualitative analysis

Dynamics of Predator–Prey System with Migrating Species and Disease in Prey Population

Learning Partial Differential Equations from Noisy Data using Neural Networks