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Über dieses Buch

This book contains original research papers presented at the International Conference on Mathematical Modelling, Applied Analysis and Computation, held at JECRC University, Jaipur, India, on 6-8 July, 2018. Organized into 20 chapters, the book focuses on theoretical and applied aspects of various types of mathematical modelling such as equations of various types, fuzzy mathematical models, automata, Petri nets and bond graphs for systems of dynamic nature and the usage of numerical techniques in handling modern problems of science, engineering and finance. It covers the applications of mathematical modelling in physics, chemistry, biology, mechanical engineering, civil engineering, computer science, social science and finance. A wide variety of dynamical systems like deterministic, stochastic, continuous, discrete or hybrid, with respect to time, are discussed in the book. It provides the mathematical modelling of various problems arising in science and engineering, and also new efficient numerical approaches for solving linear and nonlinear problems and rigorous mathematical theories, which can be used to analyze a different kind of mathematical models.

The conference was aimed at fostering cooperation among students and researchers in areas of applied analysis, engineering and computation with the deliberations to inculcate new research ideas in their relevant fields. This volume will provide a comprehensive introduction to recent theories and applications of mathematical modelling and numerical simulation, which will be a valuable resource for graduate students and researchers of mathematical modelling and industrial mathematics.



Certain Banach-Space Operators Acting on the Semicircular Elements Induced by Orthogonal Projections

The main purposes of this paper are (i) to construct-and-study (weighted-)semicircular elements induced by mutually orthogonal \(\left| \mathbb {Z}\right| \)-many projections, and the Banach \(*\)-probability space \(\mathbb {L}_{Q}\) generated by these operators, (ii) to establish \(*\)-isomorphisms on \(\mathbb {L}_{Q}\) from shifting processes on the set \(\mathbb {Z}\) of integers, (iii) to consider the \(*\)-isomorphisms of (ii) as Banach-space operators acting on \(\mathbb {L}_{Q}\) (by regarding the Banach \(*\)-algebra \(\mathbb {L}_{Q}\) as a Banach space), and (iv) to compare the free-distributional data affected by the operators of (iii) from the original data.
Ilwoo Cho

Explicit Expressions Related to Degenerate Cauchy Numbers and Their Generating Function

In the paper, by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind, the authors establish an explicit expression for degenerate Cauchy numbers and find explicit, meaningful, and significant expressions for coefficients in a family of nonlinear differential equations for the generating function of degenerate Cauchy numbers.
Feng Qi, Ai-Qi Liu, Dongkyu Lim

Statistical Deferred Riesz Summability Mean and Associated Approximation Theorems for Trigonometric Functions

The notion of deferred weighted statistical convergence was introduced by Srivastava et al. (Math Methods Appl Sci 41:671–683, 2018) [20]. In the present investigation, we have used (presumably new) the notion of approximation via statistical deferred weighted (Riesz) summability mean for trigonometrical periodic functions defined over a Banach space \(C_{2\pi }(\mathbb {R})\) and accordingly established a new approximation theorem (Korovkin-type). Furthermore, we have introduced the idea of the rate of statistical deferred weighted summability and also established another result for the same set of functions by using the modulus of continuity. Finally, We have also considered a number of fascinating special cases and examples in relevance to our results and definitions provided in this paper.
M. Patro, S. K. Paikray, B. B. Jena, Hemen Dutta

On Pointwise Convergence of a Family of Nonlinear Integral Operators

Let \(\Lambda \) be a non-empty index set consisting of \(\sigma \) indices and \( \sigma _{0}\) is allowed to be either accumulation point of \(\Lambda \) or infinity. We assume that the function \(K_{\sigma }\), \(K_{\sigma }: \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}\), has finite Lebesgue integral value on \(\mathbb {R}\) for all values of its second variable and for any \(\sigma \in \Lambda \) and satisfies some conditions. The main purpose of this work is to investigate the conditions under which Fatou type pointwise convergence is obtained for the operators in the following setting:
$$\begin{aligned} \left( T_{\sigma }f\right) \left( x\right) =\overset{\infty }{\underset{ -\infty }{\mathop {\displaystyle \int }}}K_{\sigma }\left( t,\underset{k=1}{\overset{\infty }{\mathop {\displaystyle \sum }}}P_{k,\sigma }f\left( x+\alpha _{k,\sigma }t\right) \right) dt,\ x\in \mathbb {R}, \end{aligned}$$
where \(P_{k,\sigma }\) and \(\alpha _{k,\sigma }\) are real numbers satisfying certain conditions, at \(p-\mu \)-Lebesgue point of function f. The obtained results are used for presenting some theorems for the rate of convergences.
Gumrah Uysal, Hemen Dutta

Existence and Ulam’s Type Stability of Integro Differential Equation with Non-instantaneous Impulses and Periodic Boundary Condition on Time Scales

The present manuscript is dedicated to the study of existence and stability of integro differential equation with periodic boundary condition and non-instantaneous impulses on time scales. Banach contraction theorem and non-linear functional analysis have been used to established these results. Moreover, to outline the utilization of these outcomes an example is given.
Vipin Kumar, Muslim Malik

Introduction to Class of Uniformly Fractional Differentiable Functions

In this paper, authors introduced new concept of uniformly fractional differentiable functions on an arbitrary interval I of R by using Caputo-type fractional derivative instead of the commonly used first-order derivative. Their interesting properties with few illustrations have been discussed in this paper.
Krunal B. Kachhia, Jyotindra C. Prajapati

Asymptotically Almost Automorphic Solution for Neutral Functional Integro Evolution Equations on Time Scales

The script is dedicated to look at the existence, uniqueness with stability consequence of asymptotically almost automorphic \((\mathcal {AAA})\) solution for integro neutral evolution equation on time scales by applying fixed point hypothesis. We give the time scale adaptation of \((\mathcal {AAA})\) functions. Toward the end, a precedent is given for the adequacy of the hypothetical outcomes.
Soniya Dhama, Syed Abbas

An Integral Relation Associated with a General Class of Polynomials and the Aleph Function

A new finite integral involving two general class of polynomials with the Aleph function has been obtained in the present paper. This integral is supposed to be one of the most universal integral evaluated until now and act as a key component from which we can obtain as its different special cases, integrals relating a large number of simpler special functions and polynomials. Some useful unique cases of the main outcome have also been discussed in the paper.
Monika Jain, Sapna Tyagi

On the New Fractional Operator and Application to Nonlinear Bloch System

In this chapter, we analyze the nonlinear Bloch system with a new fractional operator without singular kernel proposed by Michele Caputo and Mauro Fabrizio. The commensurate and non-commensurate order nonlinear Bloch system is considered. Special solutions using a numerical scheme based in Lagrange interpolations were obtained. We studied the uniqueness and existence of the solutions employing the fixed point theorem. Novel chaotic attractors with total order less than 3 are obtained.
J. F. Gómez-Aguilar, Behzad Ghanbari, Ebenezer Bonyah

Fractional Order Integration and Certain Integrals of Generalized Multiindex Bessel Function

We aim to introduce the generalized multiindex Bessel function \(J_{\left( \beta _{j}\right) _{m},\kappa ,b}^{\left( \alpha _{j}\right) _{m},\gamma ,c}\left[ z\right] \) and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function \(J_{\left( \beta _{j}\right) _{m},\kappa ,b}^{\left( \alpha _{j}\right) _{m},\gamma ,c}\left[ z\right] \). We prove that such integrals are expressed in terms of the Fox-Wright function \(_{p}\Psi _{q}(z)\). The results presented here are of general in nature and easily reducible to new and known results.
K. S. Nisar, S. D. Purohit, D. L. Suthar, Jagdev Singh

Fractional Variational Iteration Method for Time Fractional Fourth-Order Diffusion-Wave Equation

In the present article, Fractional variational iteration method (FVIM) is used to solve numerically time-fractional diffusion wave equation of order four. By using FVIM we obtain a sequence converging rapidly to the exact solution of the fourth order fractional diffusion wave equation. Two test problem are presented to prove the merit of the proposed technique. Plotted graph shows that the numerical solution acquired by employed technique is similar to the exact solution.
Amit Prakash, Manoj Kumar

Analytical Approach to Fractional Navier–Stokes Equations by Iterative Laplace Transform Method

In this paper, we have presented iterative Laplace transform scheme to examine fractional Navier–Stokes equations in cylindrical coordinates with initial conditions. The arbitrary ordered derivatives are described in terms of Caputo. By utilizing only the initial conditions, the analytical expressions are derived in the closed form. The results achieved with the aid of the proposed technique are graphically presented.
Rajendra K. Bairwa, Jagdev Singh

Biological Model of Dengue Spread with Non-Markovian Properties

A fatal and infectious called Dengue found in the tropical zone of the world is a mosquito-borne and caused by four viruses namely Den 1-Den 4. The transmission is achieved from one person to another via a bite of female adult Aedes mosquitoes. The dynamic of spread does not really follow the Markovian process therefore does have memory effect, thus can well be described by using nonlocal differential operators with non-singular and non-local kernel as these operators have a crossover from exponential decay law to power law as waiting time distribution. In this chapter, we reverted the classical model to fractional model by using the concept of recently established fractional differential operators known as the Caputo-Fabrizio derivative. To include into mathematical system the memory and the crossover effects. The new model was subjected to analysis of existence and uniqueness of the system solution to insure the well poseness of the modified system. Due to the complexity of the new system, a newly introduced numerical scheme was used to solve the system and some numerical simulations where performed to see the effect of the Mittag-Leffler law that brings the crossover effect.
Sonal Jain, Abdon Atangana

Approximate Solution of Higher Order Two Point Boundary Value Problems Using Uniform Haar Wavelet Collocation Method

An efficient collocation method is proposed for the numerical solution of second and fourth order two-point boundary value problems (B.V.P.) based on uniform Haar wavelet. We have converted higher order differential equations into a system of differential equations of lower order and then solve it by uniform Haar wavelet, which reduces the time and complexity of the system. The technique introduced here is easy to apply. The performance of the present method yield more accurate results on increasing the resolution level. To demonstrate the robustness and accuracy of the Haar wavelet collocation method, five problems have been solved and compared with the existing methods present in the literature [16].
Akmal Raza, Arshad Khan

Solving Multi-objective Fractional Transportation Problem

In classical fractional transportation programming problem, we want to optimize the objective function in the form of one or several ratios subject to some linear constraints. If in multi-objective transportation problem, objective function is in ration of two linear function under some linear restrictions, then the problem is called multi-objective linear fractional transportation problem (MOLFTP). In this paper we propose a new method to solve multi-objective linear fractional transportation problem which is extension of Nomani et al. (Int J Manag Sci Eng Manag (2016) [9]). Two numerical problems are presented to validate the proposed algorithm.
Vishwas Deep Joshi, Rachana Saini

On the Dark and Bright Solitons to the Negative-Order Breaking Soliton Model with (2+1)-Dimensional

This paper deal with the complex the dynamic of cnoidal waves via the negative-order breaking soliton model with (2+1)-dimensional. This model is arisen in the (2+1)-dimensional interaction of the Riemann wave propagated between y-axis and x-axis. The Improved bernoulli sub-equation function method is used in obtaining some complex and dark solutions with hyperbolic function structure. We present the interesting contour surfaces along with 2D and 3D graphics of the obtained analytical solutions in this study, plotted by using several computational programmes such as Matlap, Mathematica and so on. We finally present a comprehensive conclusion.
Haci Mehmet Baskonus

A Reliable Analytical Algorithm for Cubic Isothermal Auto-Catalytic Chemical System

In this work we apply an algorithm for the q-homotopy analysis transform method (q-HATM) to solve the Cubic Isothermal Auto-catalytic Chemical System (CIACS). This technique is a combination of the Laplace decomposition method and the homotopy analysis scheme. This method gives the solution in the form of a rapidly convergent series with h-curves are employed to determine the intervals of convergent. Averaged residual errors are used to determine the optimal values of h. We show the behavior of the solutions graphically. The q-HATM solutions are compared with Numerical results by Mathematica and with finite difference method and excellent agreement is found.
Khaled M. Saad, H. M. Srivastava, Devendra Kumar

Numerical Study of Effects of Adrenal Hormones on Lymphocytes

Lymphocytes play significant defensive role to keep the body healthy. However, there is substantial evidence that adrenal hormones such as epinephrine, norepinephrine, and cortisol generated by psychological stress suppress the activities of the immune system or alter the activation and mobilization several immune cells particularly lymphocytes during infections. Glucocorticoid receptors expressed by the immune cells makes binding those hormones possible. This work formulates a mathematical model to examine the impact of adrenal hormones on the immune system with respect to time evolution and spatial distribution cells in response to hormones concentration. The steady state of the model is studied and found to be uniformly and asymptotically stable subject to the secretion and decay rates of hormones. The numerical experiments using the free diffusion equations further investigates the dynamic behaviour of the “bound” lymphocytes secretion rate of the adrenal hormones induced by psychological stress.
Shikaa Samuel, Vinod Gill, Devendra Kumar, Yudhveer Singh

Mathematical Modelling of Poor Nutrition in the Human Life Cycle

Nutrition is very crucial in the survival of human race and more importantly the development of a child from the womb to adulthood. In some instances, the age of the individuals determines the kind of nutrients required. Therefore, the human cycle has something to do with the nutrients obtained. We formulate a mathematical model as a system of non-linear ordinary differential equations to investigate the effects of poor nutrition from conception to adulthood using the poor pregnant woman nutrient status. The steady states are studied and R0 of poor nutrition in the society are calculated. To keep the society healthy and free of malnutrition, malnourished pregnant females are encouraged to eat foods that contain all the nutrients needed for development. The model is supported with numerical simulation.
Ebenezer Bonyah, Kojo Ababio, Patience Pokuaa Gambrah

Characteristics of Homogeneous Heterogeneous Reaction on Flow of Walters’ B Liquid Under the Statistical Paradigm

In this article, significance of inclined MHD stagnant point flow of Walters B liquid because of stretched surface is investigated. Flow phenomenon is studied with Newtonian heating, homogeneous heterogeneous reactions, Joule heating and viscous dissipation. The nonlinear PDEs are converted to get nonlinear system of ODEs by invoking suitable transformations and solved by utilizing OHAM. Statistical methodology is used to check the significance and insignificance of the physical parameters via correlation coefficients and probable error. Characteristics of various sundry parameters on velocity, concentration and temperature fields are studied. Friction and Nusselt numbers are calculated and discuss in detail.
Anum Shafiq, T. N. Sindhu, Z. Hammouch
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