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2020 | Buch

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Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory

ICRAPAM 2018, New Delhi, India, October 23–25

herausgegeben von: Prof. Naokant Deo, Prof. Vijay Gupta, Prof. Ana Maria Acu, Prof. P. N. Agrawal

Verlag: Springer Singapore

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book collects original research papers and survey articles presented at the International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM), held at Delhi Technological University, India, on 23–25 October 2018. Divided into two volumes, it discusses major topics in mathematical analysis and its applications, and demonstrates the versatility and inherent beauty of analysis. It also shows the use of analytical techniques to solve problems and, wherever possible, derive their numerical solutions. This volume addresses major topics, such as multi-objective optimization problems, impulsive differential equations, mathematical modelling, fuzzy mathematics, graph theory, and coding theory. It is a valuable resource to students as well as researchers in mathematical sciences.

Inhaltsverzeichnis

Frontmatter

Exact Solution for Mixed Integral Equations by Method of Bernoulli Polynomials

Abstract

In this article, a new method has been developed for solving the mixed second kind Volterra–Fredholm integral equations numerically. A method is introduced in this paper is known as the Bernoulli matrix method. It is applied for solving mixed VFIE’s integral equations. The one property of this method is that it reduces the degree of the problem for solving a structure of algebraic equations. Our proposed method is introduced and it is applied to convert the integral equation into the algebraic equation using of Bernoulli matrix equation. Finally, there are some numerical results that have been given for illustrating the efficiency and exactness of this method.

Mithilesh Singh, Nidhi Handa, Shivani Singhal

Turing Patterns in a Cross Diffusive System

Abstract

In this paper we investigate the role of cross diffusion in pattern formation for a tritrophic food chain model. In the formulated model the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley Martin functional response. We have proved that the stationary uniform solution of the system is stable in the presence of diffusion and absence of cross diffusion but unstable in the presence of cross diffusion. Moreover we carry out numerical simulations to understand the Turing pattern formation for various self and cross diffusivity coefficients of the top level predator.

Nishith Mohan, Nitu Kumari

On Multi-objective Optimization Problems and Vector Variational-Like Inequalities

Abstract

This paper deals with nonsmooth multi-objective optimization problems involving locally Lipschitz \(V-r\)-invexity using Michel–Penot subdifferential. We consider vector variational-like inequalities of Stampacchia and Minty type and establish some results, which give necessary and sufficient conditions for a feasible point to be Pareto optimal solution of the MOP. We also establish various results related to weak Pareto optimal solution of the MOP and corresponding weak versions of the vector variational-like inequalities.

Vivek Laha, Harsh Narayan Singh

Controllability of Semilinear Control Systems with Fixed Delay in State

Abstract

This work studies the controllability of a class of delay differential equations. Instead of \(C_0\)-semigroup associated with the mild solution of the system, we use the concept of fundamental solution. Approximate controllability of the system is shown using sequence method. Finally, an illustrative example has been provided.

Abdul Haq, N. Sukavanam

Computational Performance of Server Using the Mx/M/1 Queue Model

Abstract

In this paper, we present the algorithms for evaluating the most effective and efficient transient solution to M^X/M/1 queueing model. The analytical results are expressed in modified Bessel functions and also use generalized Q-function. Numerical illustration has been obtained and compared with other algorithms by their own programs and results.

Jitendra Kumar, Vikas Shinde

Quantum Codes from the Cyclic Codes Over

Abstract

In this article, for any odd prime p, we study the cyclic codes over the finite ring \(R=\mathbb {F}_{p}[v,w]/\langle v^{2}-1,w^{2}-1,vw-wv\rangle \) to obtain the quantum codes over \(\mathbb {F}_{p}\). We obtain the necessary and sufficient condition for cyclic codes which contain their duals and as an application, some new quantum codes are presented at the end of the article.

Habibul Islam, Om Prakash, Ram Krishna Verma

Effect of Sterile Insect Technique on Dynamics of Stage-Structured Model Under Immigration

Abstract

A Stage-Structured Model has been proposed to study the effect of discharge of irradiated male insects and the immigration of wild insects using the Sterile Insect Technique (SIT). The release of sterile male insects will replace natural insects from the environment. The Logistic Growth Rate has been considered for the Larvae population. We have studied the dynamics of the model in three cases. In the first case, the model has been considered without the discharge of sterile male and immigration has not been taken into account. In the second case, the release of sterile male insect has been considered but immigration has not been taken into consideration. Lastly, in the third model, both immigration and discharge of sterile male have been taken into account. In all three cases, equilibrium points have been evaluated and stability analysis has been done. Thresholds for the sterile male population have been obtained which can be helpful in understanding and implementing the SIT.

Sumit Kaur Bhatia, Sudipa Chauhan, Priyanka Arora

Strict Practical Stability of Impulsive Differential System in Terms of Two Measures

Abstract

In this paper, an impulsive differential system is investigated to obtain sufficient conditions for strict practical stability. The investigations are carried out by perturbing Lyapunov function and by using the comparison principle. The stability properties are investigated in terms of two measures. Our results demonstrate that impulses do contribute to the system’s stability behaviour. An example is given to compliment on results.

Pallvi Mahajan, Sanjay Kumar Srivastava, Rakesh Dogra

Free Vibration Analysis of Rigidly Fixed Axisymmetric Viscothermoelastic Cylinder

Abstract

This paper represents the analysis of free vibrations of rigidly fixed, functionally graded generalized viscothermoelastic axisymmetric hollow cylinder which is considered undeformed at uniform temperature. The material of the cylinder is considered to be functionally graded according to the simple exponent law. The governing partial differential equations of motion and heat conduction have been transformed into ordinary differential equations due to time-harmonic analysis. The matrix Frobenius method of the series solution has been implemented to ordinary differential equations analytically to represent the solutions of displacement and temperature. The regular fixed boundary conditions are further solved by the use of numerical method of iteration technique with the help of MATLAB software tools. For numerical computations, we take polymethyl methacrylate material to represent natural frequencies, thermoelastic damping, frequency shift, temperature change and displacement. The behavior of frequencies, thermoelastic damping, temperature change and variation of displacement have been monitored (increase or decrease) with grading index (i.e. inhomogenous parameter).

Himani Mittal, D. K. Sharma

Study on a Free Boundary Problem Arising in Porous Media

Abstract

The present study discusses a free boundary problem arising from the steady two-dimensional seepage flow through a rectangular dam. The free boundary location, the potential velocity field, and the pressure field have been found using successive linearisation method (SLM) by solving a nonlinear partial differential equation arising as a governing equation for this problem. The SLM is a newly developed method, which is a very efficient and reliable method to handle nonlinear problems. The numerical and the graphical representation of the solution has been discussed using MATLAB under the certain valid assumption.

Bhumika G. Choksi, Twinkle R. Singh

Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population

Abstract

In this paper, we have formulated a prey–predator interaction model with native species as predator and exotic species as prey. The population is also effected by the habitat. It is assumed that the prey population can invade a new environment due to the unsuccessful exploitation of the exotic prey by the native predators. The existence of the steady-state solution and boundedness of the system is obtained. Further, the local and global stability analysis of the steady-state solution is evaluated. Finally, the sensitivity analysis of the system is done based on two parameters A, i.e. (loss of habitat due to exotic prey) and r, i.e. (growth rate of Habitat).

Namita Goel, Sudipa Chauhan, Sumit Kaur Bhatia

On Cliques and Clique Chromatic Numbers in Line, Lict and Lictact Graphs

Abstract

The line graph of a graph G denoted as L(G) has vertex set E(G) in which two vertices are adjacent if they correspond to adjacent edges in G. The lict graph and litact graph of G, denoted as \(L_c(G)\) and \(L_{ct}(G)\), respectively having vertex set \(E(G)\cup C(G)\) (here C(G) is the set of cut-vertices of G), two of these vertices will be adjacent in \(L_c(G)\) if they correspond to adjacent edges of G or one vertex is an edge e of G and other vertex is a cut-vertex c of G such that e is incident to c; and two vertices in \(L_{ct}(G)\) be adjacent if they are adjacent or incident elements of G. In this paper, we establish results on cliques and clique chromatic numbers in line, lict and litact graphs of any graph.

Rashmi Jain, Anuj Kumar Jain

Friendship-Like Graphs and It’s Classiffication

Abstract

A distance-compatible set-labeling (dcsl) of a connected graph G is an injective set assignment \(f : V(G) \rightarrow 2^{X},\) X being a nonempty set, such that the corresponding induced function \(f^{\oplus } :V(G)\times V(G) \rightarrow 2^{X}\) given by \(f^{\oplus }(uv)= f(u)\oplus f(v)\) satisfies \(\mid f^{\oplus }(uv) \mid = k_{(u,v)}^{f}d_{G}(u,v) \) for every pair of distinct vertices \(u, v \in V(G),\) where \(d_{G}(u,v)\) denotes the usual path distance between u and v and \(k_{(u,v)}^{f}\) is a constant. A dcsl f of G is k-uniform if all the constant of proportionality with respect to f are equal to k, and if G admits such a dcsl then G is called a k-uniform dcsl graph. And it has been already proved that, for a finite graph G, k-uniform dcsl graph G with is a finite set X if and only if k-embedding of G into a hypercube \({\mathbf {H}}(X)\). In this paper, we introduce Friendship-like graphs and its classification and prove that it admits 2-uniform dcsl.

K. Nageswara Rao, P. Shaini, K. A. Germina

Chaotic Maps: Applications to Cryptography and Network Generation for the Graph Laplacian Quantum States

Abstract

In this article, we proposed a new chaotic map and is compared with existing chaotic maps such as Logistic map and Tent map. The value of maximal Lyapunov exponent of the proposed chaotic map goes beyond 1 and shows more chaotic behaviour than existing one-dimensional chaotic maps. This shows that proposed chaotic maps are more effective for cryptographic applications. Further, we are using one-dimensional chaotic maps to generate random time series data and define a method to create a network. Lyapunov exponent and entropy of the data are considered to measure the randomness or chaotic behaviour of the time series data. We study the relationship between concurrence (for the two-qubit quantum states) and Lyapunov exponent with respect to initial condition and parameter of the logistic map which is showing how chaos can lead to concurrence based on such Lyapunov exponents.

Anoopa Joshi, Atul Kumar

Consumer Behaviour Analysis for Purchasing a Passenger Car in Indian Context

Abstract

The automobile industry has faced a metamorphosis over the last three decades. There has been a shift from the scenario of a manufacturer’s market to a consumer’s market. The voice of the customer is put at priority even while deciding any specific model of a car. The bias created by the consumer towards one brand and the Indian mentality towards vehicular safety is a matter to investigate. This paper proposes a strategy for analyzing choices of consumers. Multi criteria decision-making tools, viz., TOPSIS, AHP and CROSSTAB ANALYSIS have been applied. Data of gender, age, profession and segment of the car collected in the survey is used as input parameters while making the purchase behaviour model. This analysis helps car manufacturers to incorporate the features that have been preferred by consumers and thus value would be created among them. Thereby the manufacturing firm would be able to improve its profit margins and strengthen its market shares. Simultaneously the consumer would be able to choose from a wide range of options.

Tanmay Agarwal, Nivedika Saroha, Girish Kumar

A New Hybrid Model Based on Triple Exponential Smoothing and Fuzzy Time Series for Forecasting Seasonal Time Series

Abstract

Triple exponential smoothing is one of the prominent linear models for seasonal time series forecasting. Fuzzy time series forecasting is originated as a new advent for forecasting the data which is imprecise and vague. In this work, we proposed a methodology using both triple exponential smoothing and fuzzy time series. It has the advantage of modelling aspects in linear and non-linear setup. Empirical results with real-world data sets show that the hybrid model is an efficient one based on forecasting accuracy than the component models used individually.

A. J. Saleena, C. Jessy John

New Fuzzy Divergence Measure and Its Applications in Multi-criteria Decision-Making Using New Tool

Abstract

Fuzzy set theory is well suited for dealing with uncertainty and vagueness. In this research paper, we introduced new convex function, new fuzzy divergence measure and its generalization with the proof of its validity. Further, we established relations between new and well-known fuzzy divergence measures. Also, we discussed applications of new fuzzy divergence measure in multi-criteria decision-making using a new tool and its comparison with the TOPSIS method.

Ram Naresh Saraswat, Adeeba Umar

An SIRS Age-Structured Model for Vector-Borne Diseases with Infective Immigrants

Abstract

In this paper, we develop a SIRS age-structured model with infective immigrants. We consider a fraction of the juvenile immigrants and a fraction of the adult immigrants to be infective. We calculate the equilibrium points and then check the stability of these points. The reproduction number is calculated using the Next- Generation Method. Mathematical simulation for the model is also conducted using MATLAB. It is observed that an increase in the infective immigrants does not affect the total infective persons in the population. However, there is an increase in the infective population if the rate of immigration is increased. Also, the recovered population increases as the recovery rate increases. It is seen that as the mosquito population increases due to an increase in their birth rate, the infective human population also increases.

Nisha Budhwar, Sunita Daniel, Vivek Kumar

Numerical Study of Conformable Space and Time Fractional Fokker–Planck Equation via CFDT Method

Abstract

In this article, conformable fractional differential transform (CFDT) method has been successfully implemented to compute the numerical solution of space–time fractional Fokker–Planck equation with conformable fractional derivative. The computed results are compared with the existing results in the literature, and also depicted graphically for \(\alpha =\beta =1\). The accuracy of the computed results for different values of \(\alpha \) and \(\beta =1\) is measured in terms of \(L_2\) error norms. The findings show that the present results agreed well with the results by various well-known methods such as Adomian decomposition method (ADM), variational iteration method (VIM), fractional variational iteration method (FVIM) and fractional reduced differential transform method (FRDTM), and so forth. The proposed results converge to the exact solutions.

Brajesh Kumar Singh, Anil Kumar

Multispectral Bayer Color Image Encryption

Abstract

A first approach for Bayer color image encryption and decryption is proposed. The original image is downsampled into three basic color components using multispectral property, then each color component is encrypted by elliptic curve cryptography followed by generalized Arnold transformation. In the first stage of encryption, on each color component separate keys of ECC are employed, and the next stage of encryption considers keys from independent parameters of the coefficient matrix of Arnold transform. The two steps of encoding apply disjoint keys for each color component in both stages, which gives a higher level of security and robustness. Simulation analysis and experimental results are performed on several test images to show the strength of the proposed technique, and a comparison is established with other proposed models.

Binay Kumar Singh, Jagat Singh

Investigation of Prospective Elementary Teachers’ Opinions About Problem Concept

Abstract

This study aims to not only determine the views of prospective elementary teacher about Polya’s four-step problem-solving process used in mathematics teaching, the problem concept and characteristics of nonroutine problems and their solutions, but also to investigate the level of knowledge about some important points concerning these concepts that prospective elementary teacher need to sufficiently know. The research sample consists of 43 prospective elementary teachers from an education faculty of a state university in Turkey. The data were obtained from the answers the prospective elementary teachers wrote on the working papers that include various styles (multiple choice, true–false judgments selection, and open-ended) of questions. Both quantitative and qualitative assessment tools were employed in the analysis of the data.

Danyal Soybaş, Sevim Sevgi

Titel: Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory
herausgegeben von: Prof. Naokant Deo
Prof. Vijay Gupta
Prof. Ana Maria Acu
Prof. P. N. Agrawal
Verlag: Springer Singapore
Electronic ISBN: 978-981-15-1157-8
Print ISBN: 978-981-15-1156-1
DOI: https://doi.org/10.1007/978-981-15-1157-8