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About this book

This book discusses a variety of topics related to industrial and applied mathematics, focusing on wavelet theory, sampling theorems, inverse problems and their applications, partial differential equations as a model of real-world problems, computational linguistics, mathematical models and methods for meteorology, earth systems, environmental and medical science, and the oil industry. It features papers presented at the International Conference in Conjunction with 14th Biennial Conference of ISIAM, held at Guru Nanak Dev University, Amritsar, India, on 2–4 February 2018. The conference has emerged as an influential forum, bringing together prominent academic scientists, experts from industry, and researchers. The topics discussed include Schrodinger operators, quantum kinetic equations and their application, extensions of fractional integral transforms, electrical impedance tomography, diffuse optical tomography, Galerkin method by using wavelets, a Cauchy problem associated with Korteweg–de Vries equation, and entropy solution for scalar conservation laws. This book motivates and inspires young researchers in the fields of industrial and applied mathematics.

Table of Contents


Chapter 1. Certain Areas of Industrial and Applied Mathematics

This chapter is based on Presidential address at the International Conference and 14th Biennial Conference of Indian Society of Industrial and Applied Mathematics, GNDU, Amritsar, Feb 2–4, 2018.
Abul Hasan Siddiqi

Chapter 2. Schrödinger Operators with a Switching Effect

This paper summarizes the contents of a plenary talk given at the 14th Biennial Conference of Indian SIAM in Amritsar in February 2018. We discuss here the effect of an abrupt spectral change for some classes of Schrödinger operators depending on the value of the coupling constant, from below bounded and partly or fully discrete, to the continuous one covering the whole real axis. A prototype of such a behavior can be found in the Smilansky–Solomyak model devised to illustrate that an irreversible behavior is possible even if the heat bath to which the systems are coupled has a finite number of degrees of freedom and analyze several modifications of this model, with regular potentials or a magnetic field, as well as another system in which \(x^py^p\) potential is amended by a negative radially symmetric term. Finally, we also discuss resonance effects in such models.
Pavel Exner

Chapter 3. Distribution Theory by Riemann Integrals

It is the purpose of this article to outline a syllabus for a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably, these are needed for a deeper understanding of basic questions in signal analysis. Objects such as the Dirac delta and the Dirac comb should have a proper definition, and it should be possible to explain how one can reconstruct a band-limited function from its samples by means of simple series expansions. It should also be useful for graduate mathematics students who want to see how functional analysis can help to understand fairly practical problems, or teachers who want to offer a course related to the “Mathematical Foundations of Signal Processing” at their institutions. The course requires only an understanding of the basic terms from linear functional analysis, namely Banach spaces and their duals, bounded linear operators, and a simple version of \( w^{*} \)-convergence. As a matter of fact, we use a set of function spaces which is quite different from the collection of Lebesgue spaces \((L^p(\mathbb {R}_d),\Vert .\Vert _p)\) used normally. We thus avoid the use of the Lebesgue integration theory. Furthermore, we avoid topological vector spaces in the form of the Schwartz space. Although practically all the tools developed and presented can be realized in the context of LCA (locally compact abelian) groups, i.e., in the most general setting where a (commutative) Fourier transform makes sense, we restrict our attention in the current presentation to the Euclidean setting, where we have (generalized) functions over \(\mathbb {R}^d\). This allows us to make use of simple BUPUs (bounded, uniform partitions of unity), to apply dilation operators and occasionally to make use of concrete special functions such as the (Fourier invariant) standard Gaussian, given by \(g_0(t) = \exp (- \pi \vert t \vert ^{2})\). The problems of the overall current situation, with the separation of theoretical Fourier analysis as carried out by (pure) mathematicians and Applied Fourier analysis (as used in engineering applications) are getting bigger and bigger and therefore courses filling the gap are in strong need. This note provides an outline and may serve as a guideline. The first author has given similar courses over the past years at different schools (ETH Zürich, DTU Lyngby, TU Munich, and currently Charles University Prague) and so one can claim that the outline is not just another theoretical contribution to the field.
Hans G. Feichtinger, Mads S. Jakobsen

Chapter 4. Partial Differential Equations on Metric Graphs: A Survey of Results on Optimization, Control, and Stabilizability Problems with Special Focus on Shape and Topological Sensitivity Problems

We consider ordinary equations on metric graphs. In particular, we consider novelty and review earlier results in the context of shape and topology optimization for second-order equations on such metric graphs. For the sake of brevity, we concentrate on simple topologies, such as star graphs, in order to provide a simple representation of the concepts. In fact, we use the concept of Steklov–Poincaré operators in order to reduce complex graphs to star graphs. As for the differential operators, we also confine ourselves with constant coefficients. In that respect, the current article is the first one, where the results scattered in the literature are put in a unifying framework.
Günter Leugering

Chapter 5. Topological Analysis of a Weighted Human Behaviour Model Coupled on a Street and Place Network in the Context of Urban Terrorist Attacks

This article introduces a new model of weighted human behaviour in the context of urban terrorist attacks. In this context, one of the major challenges is to improve the protection of the population. In achieving this goal, it is important to better understand and anticipate both individual and collective human behaviour, and the dynamics of the displacements associated with these behaviours. Based on the recently published Panic-Control-Reflex (PCR) model, this new Coupled Weighted PCR model takes into account the role of spatial configurations on behavioural dynamics. It incorporates, via a bottleneck effect, the narrowness and the length of the streets, and thus the pressure and counter pressure of the crowd in dangerous and safe places. The numerical evacuation simulations highlight that, depending on their size, intermediate places or public squares modulate the dynamics and the speed of flow of the crowd as it evacuates to a safe place. This model features a user-friendly graphical representation, which allows planners to accurately decide where to organize host public events in a specific territorial context.
D. Provitolo, R. Lozi, E. Tric

Chapter 6. A New Model for Transient Flow in Gas Transportation Networks

We consider the flow of gas through networks of pipelines. A hierarchy of models for the gas flow is available. The most accurate model is the pde system given by the 1-d Euler equations. For large-scale optimization problems, simplifications of this model are necessary. Here we propose a new model that is derived for high-pressure flows that are close to stationary flows. For such flows, we can make the assumption of constant gas velocity. Under this assumption, we obtain a model that allows transient gas flow rates and pressures. The model is given by a pde system, but in contrast to the Euler equations, it consists of linear equations. Based upon this model, the fast computation of transient large-scale gas network states is possible.
Martin Gugat, Michael Herty

Chapter 7. Mixed-Integer Optimal Control for PDEs: Relaxation via Differential Inclusions and Applications to Gas Network Optimization

We show that mixed-integer control problems for evolution type partial differential equations can be regarded as operator differential inclusions. This yields a relaxation result including a characterization of the optimal value for mixed-integer optimal control problems with control constraints. The theory is related to partial outer convexification and sum-up rounding methods. The results are applied to optimal valve switching control for gas pipeline operations. A numerical example illustrates the approach.
Falk M. Hante

Chapter 8. Application of Solution of the Quantum Kinetic Equations for Information Technology and Renewable Energy Problem

In this paper, there is proved possibility application of the quantum kinetic equations toward the solution of the problems of information technology and renewable energy.
Mukhayo Rasulova

Chapter 9. Inverse Problems Involving PDEs with Applications to Imaging

In this chapter, we introduce the general idea of inverse problems particularly with applications to imaging. We use two well-known imaging modalities namely electrical impedance and diffuse optical tomography to introduce and describe inverse problems involving PDEs. We also discuss the mathematical difficulties and challenges for image reconstruction in practice. We describe both deterministic and statistical regularization techniques including Gauss–Newton method, Bayesian inversion, and sparsity approaches to provide a broad exposure to the field.
Taufiquar Khan

Chapter 10. Critical Growth Elliptic Problems with Choquard Type Nonlinearity: A Survey

This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involves the nonlinearity of the convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.
K. Sreenadh, T. Mukherjee

Chapter 11. Wavelet Galerkin Methods for Higher Order Partial Differential Equations

In this paper, we develop efficient and accurate wavelet Galerkin methods for higher order partial differential equations. Compactly supported Daubechies wavelets are used for spatial discretization, whereas stable finite difference methods are used for temporal discretization. The exact values of two-term connection coefficients are effectively used for the evaluation of integrals consisting of higher order derivatives. For the nonlinear elliptic partial differential equations, we have employed quasilinearization technique to obtain the nonlinear wavelet coefficients. Sparse GMRES solver is used to solve linear system of equations obtained after spatial and temporal discretization. Error analysis has been carried out to ensure the convergence of the proposed method. Finally, the method is successfully tested on few linear and nonlinear 1D and 2D PDEs.
B. V. Rathish Kumar, Gopal Priyadarshi

Chapter 12. Resilience and Dynamics of Coral Reefs Impacted by Chemically Rich Seaweeds and Unsustainable Fishing

Coral reefs are globally threatened by numerous natural and anthropogenic impacts. The proliferation of seaweeds in coral reefs is one of the most common and significant reasons for the decline of healthy corals. Some seaweeds release chemicals that are harmful to corals. The chemicals released by toxic seaweeds damage corals in areas of direct contact. While herbivorous reef fish play an important role in preventing the overgrowth of seaweeds on corals, unsustainable fishing of herbivores disrupts the ecological balance in coral reefs. This induces changes in the community structure from the dominant reef-building corals to one by seaweeds. We have considered a mathematical model of interactions between coral, toxic seaweeds, and herbivores to investigate the phase shifts from coral- to seaweed-dominated states. We investigate how seaweed toxicity and overfishing trigger negative effects on the ecological resilience of coral reefs through trophic cascades. It is observed that in the presence of seaweed toxicity and unsustainable fishing, the system can exhibit an irreversible dynamics through hysteresis cycles. Further, we employ Mawhin’s coincidence degree theory to investigate the existence of a unique positive almost periodic solution of the nonautonomous version of our model by incorporating synchronous or asynchronous seasonal variations in different parameters. The results from computer simulations have potential applications to control the overgrowth of seaweeds in coral reefs as well as to prevent coral bleaching.
Samares Pal, Joydeb Bhattacharyya

Chapter 13. Multigrid Methods for the Simulations of Surfactant Spreading on a Thin Liquid Film

A multigrid approach is proposed in this work for the simulations of surfactant spreading on a thin liquid film. The model equations for the descriptions of the surfactant dynamics are the coupled nonlinear partial differential equations in radial coordinate. The finite volume method on a uniform grid is used for the discretization of the governing equations in which the fluxes are discretized implicitly. The discretized system is solved using the nonlinear multigrid method such as the full approximation scheme. The obtained simulation results are discussed and validated with existing results.
Satyananda Panda, Aleksander Grm

Chapter 15. Treatment of Psoriasis by Interleukin-10 Through Impulsive Control Strategy: A Mathematical Study

Psoriasis is characterized by anomalous growth of keratinocytes (skin cells), which occurs due to abrupt signaling within immune cells and cytokines. The most significant immune cells, T cells go through differentiation with interaction of dendritic cells (DCs) to produce Type 1 T helper cell (\(\text {Th}_{1}\)) and Type 2 T helper cell (\(\text {Th}_{2}\)) subtypes. In psoriatic progression dynamics, the inflammation effect of \(\text {Th}_{1}\) mediated cytokines (pro-inflammatory) are responsible for the abnormal growth of keratinocytes. In this measure, the effect of anti-inflammatory cytokines secreted by \(\text {Th}_{2}\) subtype partially downregulate the growth of epidermal cell. In this research article, we have constructed a five-dimensional mathematical model involving T cells, dendritic cells, \(\text {Th}_{1}\), \(\text {Th}_{2}\), and keratinocyte cell populations for better understanding the development of psoriatic lesions. Moreover, we have evaluated the role of \(\text {Th}_{1}\), \(\text {Th}_{2}\), and interplay of various cytokine networks in Psoriasis through a set of nonlinear differential equations. Our analytical study reveals the preconditions for disease persistence and also validates the stability criteria of endemic equilibrium for the disease. Furthermore, we have used one-dimensional impulsive differential equation to examine the effects of different levels of biologic (\(\text{ Interleukin }\)-10) for different dosing intervals in keratinocytes cell population. We have also examined the qualitative behavior of keratinocyte by considering two different values of the parameter corresponding to the reduction of keratinocyte due to the impact of drug (IL-10). We have also found the perfect dosing intervals of biologic (\(\text{ Interleukin }\)-10) that could tolerate the keratinocytes at the desired level. Finally, our analytical and numerical computations reveal that the use of IL-10 through impulsive way is proven better treatment compared with other trivial therapeutic policies for psoriatic patients.
Amit Kumar Roy, Priti Kumar Roy

Chapter 16. On Fractional Partial Differential Equations of Diffusion Type with Integral Kernel

The main purpose of this work is to investigate the existence of solution of the fractional partial differential equations of diffusion type with integral kernel. The existence of solutions of the problem with Dirichlet boundary condition is established by using the Leray–Schauder fixed point theorem and Arzela–Ascoli theorem under suitable assumptions. Then, the result is generalized for Neumann boundary condition with the help of Green’s identity.
A. Akilandeeswari, K. Balachandran, N. Annapoorani

Chapter 17. Mathematical Study on Human Cells Interaction Dynamics for HIV-TB Co-infection

Co-infection of Tuberculosis (caused by Mycobacterium tuberculosis bacteria) and HIV (caused by Human Immunodeficiency virus) remains a global burden on public health system and poses particular diagnostic and therapeutic challenges. Due to co-infection, HIV speeds up the progression from latent to active TB and TB bacteria also accelerates the progress of HIV infection which ultimately leads to serious condition in individuals. In this research work, we formulate a six compartment mathematical model on the HIV-TB co-infection dynamics incorporating Macrophage (active and infected), T-cell (active and infected), Virus, and Bacteria population. Moreover, we explore the accelerating effect of both pathogens on each other in mathematical perceptive. Our analytical study reveals the conditions for the persistence of co-infection and also validates the stability criteria of equilibrium points for the disease. We also evaluate the disease-free condition using next generation method, expressed by the basic reproductive ratio (\(R_0\)). Moreover, our analytical and numerical simulations manifest the influence of certain key parameters on the threats posed by the impact of HIV-TB co-infection.
Suman Dolai, Amit Kumar Roy, Priti Kumar Roy

Chapter 18. Relative Controllability of Nonlinear Fractional Damped Delay Systems with Multiple Delays in Control

This paper is concerned with the relative controllability of fractional damped dynamical systems with multiple delays in control for finite-dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffer matrix function. An example is provided to illustrate the theory.
P. Suresh Kumar

Chapter 19. A Graphical User Interface-Based Fingerprint Recognition

Biometric authentication is a process of establishing an individual’s identity through measurable characteristics of their behavior, anatomy, or physiology. Fingerprint recognition is a biometric technology that has been extensively used in a various range of contexts from immigration control on airports, transactions in banks, applying for a driving license, a passport to Aadhar card in India, and personal computing. In recent emerging technologies, the usability aspects of system design have received less attention rather than technical aspects. The researches on fingerprint have shown many challenges for users like placing fingers to capture fingerprints, system feedback, and instructions to use fingerprint systems. This paper proposes a Graphical User Interface (GUI) system for studying various operations in recognizing fingerprints for biometric identification of individuals using an iterative, participative design approach. During this process, several different layouts have been identified. The fingerprint GUI provides facility to users to use by clicking on the buttons on the front-end interface of the system. The coding for the back-end interface functions is written in MATLAB. This study has been tested over DB1 of FVC2006 database. The dataset consists of 1800 images captured by electric field sensor at 250 dpi. The volunteers were asked to put their fingers naturally on the acquisition device and no constraints were enforced to guarantee a minimum quality in the images. The minutiae and texture features of fingerprints have been studied and the results show 100% matching of an individual from the collected database. Fingerprint recognition using GUI is reliable and easy to understand the operations and results more efficiently.
Rohit Khokher, Ram Chandra Singh

Chapter 20. Existence and Stability Results for Stochastic Fractional Delay Differential Equations with Gaussian Noise

In this paper, the existence and uniqueness of solutions of stochastic fractional delay differential equations is obtained by using Picard–Lindelöf successive approximation scheme. Further, the stability results are established using the Mittag-Leffler function. Examples are provided to illustrate the theory.
P. Umamaheswari, K. Balachandran, N. Annapoorani

Chapter 21. Asymptotic Stability of Implicit Fractional Volterra Integrodifferential Equations

In this paper, we discuss the stability of fractional Volterra integrodifferential equations using a method based on eigenvalue criterion. Lyapunov’s definition of stability is used and of the two methods, Lyapunov’s first or indirect method is used to prove the stability results. Some sufficient conditions ensuring asymptotic stability of the system involving implicit fractional derivative are established. Examples are provided to demonstrate the effectiveness of the method.
Kausika Chellamuthu
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