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 paper singularly perturbed reaction difiusion equations of elliptic and parabolic types have been discussed using wavelet optimized flnite difierence (WOFD) method based on an interpolating wavelet transform using cubic spline on dyadic points as discussed in (1). Adaptive feature is performed automatically by thresholding the wavelet coe-cients. WOFD (2) works by using adaptive wavelet to generate an irregular grid which is then exploited for the flnite difierence method. Numerical examples are presented for elliptic and parabolic problems and comparisons have been made using cubic spline and WOFD. The proposed adaptive method is very efiective for studying singular perturbation problems in term of adaptive grid generation and CPU time.

/pdf/solving-singularly-perturbed-reaction-diffusion-problems-2akjw9th1x.pdf

Solving singularly perturbed reaction diffusion problems using wavelet optimized finite difference and cubic spline adaptive wavelet scheme

This paper deals with a delayed modified Holling-Tanner predator-prey model with refuge. The proposed model highlights the impact of delay and refuge on the dynamics of the system wherein analysis of the model in terms of local stability is performed. Both theoretical and experimental works point out that delay and refuge play an important role in the stability of the model and also it has been observed that due to delay, bifurcation occurred which results in considering delay as a bifurcation parameter. For some specific values of delay, Hopf bifurcation is investigated for the proposed model and direction of Hopf bifurcation with the stability of bifurcated periodic solutions by using normal form theory and central manifold reduction is also included in the domain of this study. At the end, few numerical simulations based on hypothetical set of parameters for the support of theoretical formulation are also carried out.

Bifurcation Analysis of a Delayed Modified Holling-Tanner Predator-Prey Model with Refuge

The aim of this paper is to solve numerically a class of problems on conservation laws, modelled by hyperbolic partial differential equations. In this paper, primary focus is over the idea of fuzzy logic-based operators for the simulation of problems related to hyperbolic conservation laws. Present approach considers a novel computational procedure which relies on using some operators from fuzzy logic to reconstruct several higher-order numerical methods known as the flux-limited methods. Further optimization of the flux limiters is discussed. The approach ensures better convergence of the approximation and preserves the basic properties of the solution of the problem under consideration. The new limiters are further applied to several real-life problems like the advection problem to demonstrate that the optimized schemes ensure better results. Simulation results are included wherever required.

Numerical Simulation of Hyperbolic Conservation Laws Using High Resolution Schemes with the Indulgence of Fuzzy Logic.

In this paper, a stage structured eco-epidemiological model with linear functional response is proposed and studied. The stages for both prey and predator have been considered. Infection occurs in the prey population only. The proposed mathematical model consists of five nonlinear ordinary differential equations to describe the interaction among juvenile prey, adult prey, infected prey, juvenile predator and adult predator populations. The model is analyzed by using linear stability analysis to obtain the conditions for which our model exhibits stability around the possible equilibrium points.

/pdf/dynamical-behavior-of-a-stage-structured-eco-epidemiological-2tfv7wkn94.pdf

Vivek Kumar

Papers

Solving singularly perturbed reaction diffusion problems using wavelet optimized finite difference and cubic spline adaptive wavelet scheme

Bifurcation Analysis of a Delayed Modified Holling-Tanner Predator-Prey Model with Refuge

Numerical Simulation of Hyperbolic Conservation Laws Using High Resolution Schemes with the Indulgence of Fuzzy Logic.

Dynamical behavior of a stage structured eco-epidemiological model

A New Reconstruction of Numerical Fluxes for Conservation Laws using Fuzzy Operators