Extension and application of residue theorem

Summation of Series by the Residue Theorem

The article analyzes scientific sources and diagnostic experiment results in order to substantiate the relevance of the problem of forming the readiness of technical university students’ for professional self-organization and self-education, and proves the expediency of interpreting such readiness as an educational result formulated in the form of competence. The relationship between the concepts of ability and competence based on their structural similarity is presented. The substantiation of the use of game technologies in teaching higher mathematics to students of a technical university in order to form their ability to professional self-organization and self-education is given. Examples of business games aimed at developing the desire to master mathematical knowledge and methods of action necessary for professional activity, independent search for information, mastering professional competencies, qualities such as flexibility, rationality, quick thinking, as well as students' awareness of the role of scientific organization of educational work in improving its effectiveness; mastering the simplest methods of scientific organization of work in the process of educational research and self-education are offered. The content possibilities of traditional technical university disciplines such as "Higher Mathematics", "Linear Algebra", "Mathematical Analysis" for the development of business and intellectual games are argued. Game scenarios are made in the concept of activity and competence-oriented scientific approaches.

A closed-form solution of beam on viscoelastic subgrade subjected to moving loads

The focus of this thesis is on some contemporary problems in the field of difference equations and discrete dynamical systems. The problems that I worked on range global attractivity results to all types of bifurcations for systems of difference equations in the plane. The major goal was to investigate the impact of nonlinear perturbation and the introduction of quadratic terms on linear fractional difference equations such as the Beverton-Holt as well as the Sigmoid Beverton-Holt Model with delay that describes the growth or decay of single species. The first Manuscript was on the study of the following equation: [Mathematical equations can not be displayed here, refer to PDF] Which was an open problem suggested by Dr. Kulenović. It is a a perturbation of the linear fractional difference equation: [Mathematical equations can not be displayed here, refer to PDF] The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter p can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the di erence equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium. In my second Manuscript, I considered the difference equation: [Mathematical equations can not be displayed here, refer to PDF] where C;D and F are positive numbers and the initial conditions x-1 and x0 are non-negative numbers. Equation (2) which is also a non-linear perturbation of the Beverton Holt model, belongs to the category of difference equations with a unique positive equilibrium that exhibit the Naimark-Sacker bifurcation. The investigation of the dynamics of such equation is very challenging as it depends on more than one parameter. However I give a method for proving that its dynamics undergoes the Naimark-Sacker bifurcation. Moreover I compute the direction of the Neimark-Sacker bifurcation for this difference equation and provide the asymptotic approximation of the invariant closed curve. Furthermore I give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium. The following theorem is the major result that I relied on to prove global asymptotic stability of the equilibria in my first two Manuscripts: Theorem

Some applications of fractional calculus involving summation of infinite series

The dynamic response of axisymmetric poroelastic bone subjected to transient pressure along the curved surface of the cylinder is studied. In the analysis, exact solutions that satisfy the boundary conditions are obtained by applying Laplace transformations to the equations of motion and fluid flow. The Laplace inversion is obtained from the residue theorem. The stress component σrr* (dimensionless) is plotted with T (dimensionless time) for various η* (dimensionless decaying parameter). It is seen that the stresses behave alike for different η*. In the region 1<T<1.5, for η*=2 a slight oscillation is present. The stress component σ* (dimensionless) is plotted for the same parameters. Similar behavior is exhibited with the crossing of curves for η*=2 and 3. The comparison between the elastic and poroelastic behavior of stress σrr* for η=1 is carried out. It is seen that there is rapid oscillation for the elastic case, while it is absent in the poroelastic case. For larger time there is no significant effect of stresses. It appears that the elastic constants of material which are considered without porosity may not exist in nature.

The problem of summation of Fourier series in finite form is formulated in the weak sense, which allows one to consider this problem uniformly both for classically convergent and for divergent series. For series with polynomial Fourier coefficients \(a_n, b_n \in \mathbb{R}[n]\), it is proved that the sum of a Fourier series can be represented as a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients \(a_n, b_n \in \mathbb{R}(n)\), it is shown that the sum of such a series is always a solution of a linear differential equation with constant coefficients whose right-hand side is a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. Thus, the issue of summing a Fourier series with rational coefficients is reduced to the classical problem of the theory of integration in elementary functions.

A closed‐form deflection response of a beam rest is presented in this paper using the integral transform method. The theory of linear partial differential equations is used to represent the deflection of beam subjected to a moving harmonic line load in integration form. The solution is finally carried out using the inverse Fourier transform. To evaluate the integration analytically, poles of the integrand are identified with the help of algebraic equation theory. Residue theorem is then utilized to represent the integration as a contour integral in the complex plane. Closed‐form deflections and numerical results are provided for different combinations of load frequency and velocity. Copyright © 2002 John Wiley & Sons, Ltd.

The differential problem given by a parabolic equation describing the purely viscous flow generated by a constant or an oscillating motion of a boundary is the well-known Stokes’ problem. The one-dimensional equation is generally solved for unbounded or bounded domains; for the latter, either free slip (i.e., zero normal gradient) or no-slip (i.e., zero velocity) conditions are enforced on one boundary. Generally, the analytical strategy to solve these problems is based on finding the solutions of the Laplace-transformed (in time) equation and on inverting these solutions. In the present paper this problem is solved by making use of the residuals theorem; as it will be shown, this strategy allows achieving the solutions of First and Second Stokes’ problems in both infinite and finite depth. The extension to generally periodic boundaries with the presence of a periodic pressure gradient is also presented. This approach allows getting closed form solutions in the time domain in a rather fast and simple way. An ad hoc numerical algorithm, based on a finite difference approximation of the differential equation, has been developed to check the correctness of the analytical solutions.

Fractional Brownian Motions, Fractional Noises and Applications

People quit saying computers don't make mistakes after the Y2K scare that generated billions of dollars in hardware upgrades, and hence the dot.com bubble. Computer arithmetic with finite-length bit-strings has its limitations and creates concerns. In computational mathematics using computers, it is a pity to abandon rigor just at the point we have reduced the problem to computer algorithms. In fact we need not do so. By using pairs of machine numbers as upper and lower bounds of unknown numbers of interest, and by rounding upper bounds up and lower bounds down, we can modify ordinary floating-point machine arithmetic to produce mathematically rigorous results. This is called arithmetic with outward rounding, and it has been implemented and used for several decades. In particular it has been successfully used in non-trivial computer-aided proofs in mathematical analysis, such as the Kepler conjecture (a problem that was outstanding for more than 300 years) and many others. Practical applications also abound in chemical engineering, structural engineering, economics, aircraft control circuitry design, beam physics, global optimization, differential equations, etc. Rigor in computing depends on the integrity of order relations. Commonly used floating-point hardware can lose that integrity. A few examples are presented and it is shown how we can remedy the situation and regain rigor in computing by using outwardly rounded interval arithmetic.

This article is devoted to solving some of the problems of elementary mathematics with the help of a conformal invariant, which has its own advantages, applied in the theory of modern functions, as well as some important issues of mathematical analysis with the help of Komplex numbers, several results obtained with the help of the Eyler formula and their application in practice.

The cell is the basic structure and process unit that carries all the living characteristics of a living thing and has the ability to survive on its own under suitable conditions. The relationship of cell size with nutrient absorption and nutrient consumption in the cell membrane has been examined with the current model using the theory of differential equations in classical analysis. During these examinations, the cell considered was assumed to be spherical. In fact, the shapes of cells vary depending on their functional properties. Many have long appendages, cylindrical parts or branch-like structures. However, in this study, a simple global cell will be discussed, leaving all these complex situations aside. In the current model, the relationship between the change in the radius of the cell and the nutrient absorption and consumption in the cell membrane is detailed using classical differential equations. The answer to the question for which cell size is the consumption rate exactly balanced with the absorption rate was found in classical analysis. The current model consists of first-order differential equations. In this model, the dependent variables are the radius of the cell and the mass of the cell. The classical solutions of these models will be examined, the size of the cell and the cell membrane relationship will be examined, and details will be given with numerical examples. However, in order to consider this biological phenomenon from different perspectives and compare the results, the relevant event will be modeled using multiplicative analysis, one of the Non-Newtonian analyses. The new models will be solved using multiplicative analysis techniques, and the results will be compared with classical analysis. With this new model, it is planned to clarify the results obtained in the classical case, to reveal more clearly the relationship between the size of the cell and nutrient absorption and consumption in the cell membrane, and to obtain important results.

Dynamics and Control of Machines

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.