ABSTRACT. The talk deals with the numerical computation of hypersingular integrals of the type \[ \hint_0^{\ +\infty}\frac{f(x)}{(x-t)^{p+1}}\uu(x)dx,\] where the integral is understood in the Hadamard finite part sense, $p$ is a positive integer, $\uu(x)=e^{-x/2}x^\ga$ is a Laguerre weight and $t>0$. Many of the exiting methods make use of the decomposition

\begin{eqnarray*}&&\int_0^{+\infty}\frac{f(x)-\sum_{k=0}^p \frac{f^{(k)}(t)}{k!}(x-t)^{k} }{(x-t)^{p+1}}\uu(x)dx +\sum_{k=0}^p \frac{f^{(k)}(t)}{k!}\hint_0^{\ +\infty}\frac{\uu(x)}{(x-t)^{p+1-k}}dx, \end{eqnarray*}

\noindent requiring the samples of the function's derivatives. Here we propose a product integration rule based on a suitable Lagrange process, with the advantage of avoiding the derivatives computations. We determine conditions under which the rule is stable and convergent in suitable weighted uniform spaces. Finally, we show the performance of the procedure by proposing some numerical tests.

\textbf{Keywords:} Hypersingular integrals, Lagrange interpolation, Orthogonal polynomials, Product integration rules

ABSTRACT. The main message of the mismatch theorem in the Taylor's dissertation is that incurable breakdown in the Lanczos algorithm occurs only when a minimal realization of the transfer function given by the input of the algorithm has been found. In this talk we present how this result can be proved by means of the Gauss quadrature.

ABSTRACT. The approximation of functions defined on $(0,\infty)$ and with exponential monotonicity at the endpoints of the domain has not received attention in the literature.

Recently, the authors have indroduced the weight

\[w(x)=x^\gamma \mathrm{e}^{-x^{-\alpha}-x^\beta}\,,\qquad x>0\,,\]

and studied the corresponding sequence of orthonormal polynomials. This has allowed to develop the theory of polynomial approximation with the weight $w$ and constructive processes in numerical analysis.

In this talk we will show some approximation results using Lagrange interpolation.

ABSTRACT. We propose a Nystr\"om-type method to approximate the solution of integral equations of the form

\[ f(x)-\mu \int_0^{+\infty}k(x,y)f(y)w(y)\,\mathrm{d}y =g(x)\,,\quad x\in (0,+\infty), \]

where $\mu$ is a real parameter,

\[ w(y)=\mathrm{e}^{-y^{-\alpha}-y^\beta}\,, \quad \alpha>0\,, \ \beta>1\,, \]

the given functions $k$ and $g$ can grow exponentially with respect to their arguments, when they approach to $0^+$ and/or $+\infty$ \cite{MastroianniMilovanovicNotarangeloNew}.

Since the solution of this kind of equations can increase exponentially for $x\rightarrow 0^+$, the methods based on the weighted polynomial approximation with Laguerre-type weights are not suitable in this case. So, a first difficulty is to choose proper function spaces where these equations can be studied. To this aim, we introduce another exponential weight $u$ and new function spaces $C_u$ with weighted uniform metric. We prove that the proposed method is stable and convergent in this metric, using our recent results on polynomial approximation with the weight $u$ \cite{MastroianniNotarangelo13,MastroianniNotarangeloSzabados13,MastroianniNotarangelo14} and Gaussian rules with the weight $w$ \cite{MastroianniMilovanovicNotarangelo14}.

Finally, we give a priori error estimates and show some numerical examples, including a comparison with other Nystr\"om methods.

ABSTRACT. Procedures based on moments are developed for computing the three-term recurrence relation for orthogonal polynomials relative to the Binet, generalized Binet, squared Binet, and related subrange weight functions. Monotonicity properties for the zeros of the respective orthogonal polynomials are also established.

ABSTRACT. The aim of this paper is to modify generating functions for the Milne-Thomson polynomials, the Poisson-Charlier polynomials and the Hermite polynomials. We investigate and study some properties of these generating functions and their functional equations associated with some special analytic functions. We also give some remark and observation on orthogonality properties of these polynomials especially, it is well-known that the Poisson-Charlier polynomials which are orthogonal with respect to the Poisson distribution. Moreover, by using these generating functions and the $p$-adic integral method, we derive various identities and relations including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Poisson-Charlier polynomials, the Milne-Thomson polynomials, the Hermite polynomials and also the combinatorial sums.

ABSTRACT. This talk deals with a quadrature method for approximating the solutions of singular integro-differential equations of Prandtl's type \[\sigma(y)u(y)-\frac 1\pi\int_{-1}^1\frac{u'(x)}{x-y}dx+\frac 1\pi\int_{-1}^1 k(x,y)u(x)dx=g(y), \quad |y|\leq 1, \] where the unknown solution $u$ satisfies the additional conditions \[ u(-1)=u(1)=0 \] and $\sigma, k, g$ are given functions.

Several authors have studied this type of integro-differential equations and related numerical methods (see, for example, \cite[Ch. 3, Section I]{14}, \cite[Section 3]{13}, \cite[Section 9.53]{19}, \cite{3},\cite{4}).

We prove that the proposed method is stable and convergent. We give error estimates in weighted spaces of continuous functions equipped with uniform norms. Moreover we show some numerical tests that confirm the theoretical estimates.

\textbf{Keywords:} Cauchy singular integral equations, Singular integro-differential equations, Gaussian quadrature rules, Orthogonal polynomials

ABSTRACT. We consider second kind integral equations of Mellin convolution type having the form \begin{equation}\label{eq} f(y)+\int_0^1 k(x,y)f(x)dx+\int_0^1h(x,y)f(x)dx=g(y), \quad y\in (0,1], \end{equation} where $f(y)$ is the unknown, $h(x,y)$ and $g(y)$ are given sufficiently smooth functions and \begin{equation}\label{tildek} k(x,y)= \frac 1 x \tilde k \left(\frac y x\right) \end{equation} is a Mellin kernel with $\tilde k$ a given function on $[0,+\infty)$ satisfying proper assumptions. \newline The development of numerical methods for the solution of such kind of integral equations has a strong practical motivation due to the wide range of applications, particularly in engineering and physics.\newline The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral ope\-rator the chief theoretical barrier.\newline In this talk we address the concern over the stability of a numerical procedure for the solution of (\ref{eq}) in the case where the kernel $k(x,y)$ in (\ref{tildek}) satisfies the following condition \begin{equation}\label{tildek-cond} \int_0^{+\infty} t^{-1+\sigma}|\tilde k(t)|dt<+\infty, \quad \mathrm{for \, some} \quad \sigma>0. \end{equation} Under this assumption the Mellin integral operator \begin{equation}\label{K-def}(\mathcal{K} F)(y)=\int_0^1 \frac 1 x \tilde k \left(\frac y x\right)F(x)dx,\end{equation} is not necessarily bounded with respect to the uniform norm. We study the integral equation in a sui\-ta\-ble weighted space of continuous functions. Then, we consider an equivalent Mellin integral equation whose unknown is at least a continuous function. Finally, in order to approximate its solution, we apply a modified Nystr\"om method. \newline Since the definition of the integral operators associated to the new equation involves a Jacobi weight, the proposed method uses a Gauss-Jacobi quadrature formula for their discretization. Unfortunately, due to the fixed singularity of the Mellin kernel at the point $x=y=0$, such quadrature rule becomes inefficient for the approximation of the Mellin operator when $y$ is very close to $0$. Therefore, it becomes necessary to modify it in order to achieve stability and convergence results. This approach let us to reach our goal of proving theoretically the stability and the convergence of the proposed method. Furthermore, we are able to provide an error estimate in weighted uniform norm and to prove the well-conditioning of the involved linear systems which is crucial for the computation of the approximate solution. \newline %Here, we propose a Nystr\"om method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical examples illustrate the efficiency of the proposed procedures.

ABSTRACT. I aim to present a computational method for numerically solving a general form of time-fractional differential equation with boundary conditions. In this method, main problem is converted to a new problem with homogeneous conditions and then an equivalent integro-differential equation by proposing a technique. Next, the shifted Jacobi polynomials are implemented to approximate all the known and unknown functions in the equivalent integro-differential equation. Finally, a system of nonlinear algebraic equations is achieved by utilizing the collocation method which it is solved by Newton's iterative method. The benefits of this method are faster convergence and avoidance of a singular system.

ABSTRACT. The estimation of the parameter of a biased coin from the result of a few tosses is a classical problem in Probability. In this talk, we study this problem, showing an alternative procedure to the classical Maximum Likelihood method. This procedure exhibits a striking resemblance with the solution of a well--known problem in Approximation Theory, namely, the minimization of the Lebesgue function in polynomial interpolation. We use minimax approximation techniques to solve our problem, and the asymptotics of the solutions (optimal estimators) as the number of tosses tends to infinity is shown.

On the other hand, the problem is also framed in the Game Theory by means of a two--player game, for which the Nash--equilibrium is established and the corresponding pair of optimal strategies is studied and completely solved for the particular case of $n=2$ tosses.

Some numerical experiments are also displayed and further investigations are posed.

This is a joint work with D. Benko (Univ. South Alabama, Mobile, USA), D. Coroian and P. Dragnev (Indiana Purdue Univ., Fort Wayne, USA).

ABSTRACT. Generalized stochastic processes (GSPs) arise as solutions of stochastic partial differential equations (SPDEs) with singularities and represent a good theoretical framework to capture their singular behavior, e.g. if the process possesses infinite variance. The most famous generalized stochastic process is the Gaussian white noise process. We present two theoretical frameworks that provide also the possibility to undertake numerical approximations of generalized stochastic processes:

• Wiener-Ito polynomial chaos expansions,
• Colombeau-type regularizations.    

Both theories have the advantage to deal with nonlinear functions of GSPs and therefore they are precious for solving nonlinear SPDEs.

Considering generalized stochastic processes as elements of a topological inductive space constructed as the extension of the space of random variables with finite second moments, one can implement the theory of orthogonal polynomials and use a series expansion of GSPs via the Hermite polynomial basis. In this manner the process is uniquely determined by its Fourier coefficients. Truncating the series expansion provides an appropriate approximation of the process. Using this method any SPDE can be transformed into a lower triangular infinite system of PDEs that can be solved recursively. Summing up the obtained coefficients and proving the convergence of the obtained series one arrives to the solution of the initial SPDE.

The other method involves Colombeau algebras of generalized functions. These are equivalence classes of nets of stochastic processes with smooth sample paths which possess a moderate growth rate and they differ only by a negligible process i.e. a process that is rapidly decreasing to zero. Using this method the sample paths of all input data in a SPDE are now smoothed out by the help of a regularization parameter until they become smoothly differentiable and the SPDE is then solved pathwisely via these smooth sample paths. The final solution corresponds to the Stratonovich-integral solution of the original SPDE. 

In this talk we provide a comparison of the two methods and reflect on some recent advances and their applications to solving SPDEs.

ABSTRACT. Recenlty, the Lyndon words and their numbers have been investigated by researchers using various methods. Contrary to other studies, in this paper, we use the methods associated with a family of zeta functions interpolating a family of higher-order Apostol-type numbers and polynomials. The main purpose of this paper is not only to define power series including the numbers of Lyndon words and binomial coefficients, but also to construct new computation algorithms in order to simulate these series with numerical analysis and plots. With these algorithms, we provide novel computational methods to the area of the combinatorics on words. Moreover, in order to reduce algorithmic complexity of these algorithms, our other aim is to present an approximation to these series by rational functions of the Apostol-type numbers. Finally, we give some remarks, observations and comments on these polynomials and the numbers of Lyndon words.

ABSTRACT. A recently reported unconditionally-positive finite difference (UPFD) [1] and the standard explicit finite difference (EFD) schemes are compared to the analytical solution of the advection-diffusion reaction equation which describes the exponential traveling wave. It is found that although the unconditional positivity assures stability of the UPFD scheme regardless of the size of the discretization steps taken, this scheme is less accurate than the standard explicit finite difference scheme. This is because the UPFD scheme contains additional truncation-error terms in the approximations of the first and second derivatives with respect to x, which are evaluated at different moments in time. While these terms tend to zero as the mesh is refined, the UPFD scheme nevertheless remains less accurate than its standard explicit finite difference counterpart.

ABSTRACT. For constant and periodic boundary conditions, the one-dimensional advection-diffusion equation with constant coefficients is solved by the explicit finite difference method in a semi-infinite medium. It is shown how far the periodicity of the oscillating boundary carries on until diminishing to below appreciable levels a specified distance away, which depends on the oscillation characteristics of the source. Results have been tested against an analytical solution reported for a special case [1]. The explicit finite difference method is shown to be effective for solving the advection-diffusion equation with constant coefficients in semi-infinite media with arbitrary initial and boundary conditions.

ABSTRACT. We give a description of the p-regularity theory applications to nonlinear singular optimization problems and present numerical method for solving such problems. A construction of the p-factor operator has been used for the formulation of the pth-order necessary and sufficient optimality conditions.

ABSTRACT. We are concerned with the solution to the general time-invariant matrix equation $AV(t)B=D$ and the time-varying matrix equation $A(t)V(t)B(t)=D(t)$ by means of gradient based neural network (GNN) model, called the GNNABD model. The resulting matrix generated by the GNNABD model is defined by the choice of the initial state and coincides with the general solution of the matrix equation $AVB=D$. Several particular appearances of this matrix equation and their applications in approximatting various inner and outer inverses are considered. Particularly, two particular cases of the general GNNABD model, globally convergent to the Moore-Penrose inverse and the Drazin inverse are defined and investigated theoretically and numerically. The influence of various nonlinear activation functions on several variants of the GNNABD model are investigated.

ABSTRACT. In this talk, the linear positive Stancu operator stated in [2] is considered. An extension of the Stancu operator is proposed by means of the technique that was used in [1]. Approximation results for the sequence of these extended operators are given in the space of continuous, real valued functions on [0,1]. For the rate of approximation, an estimate is obtained with the help of the modulus of smoothness. Some retaining properties of the new operator are also presented.

ABSTRACT. In this paper we analyse a class of conjugate gradient methods.

The conjugate gradient parameter $\beta_k$ is chosen in such a way that it is always nonnegative. Under the Wolfe * line search conditions, the sufficient descent always holds for each single method from this class. The global convergence of each method from this class is considered.

ABSTRACT. In applications, especially in engineering, often are encountered composite or layered structures, where the properties of individual layers can vary considerably from the properties of the surrounding material. Layers can be structural, thermal, electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. In this paper we investigate a mixed parabolic-hyperbolic initial-boundary value problem in two non-adjacent rectangles with nonlocal integral conjugation conditions. It was considered more examples of physical and engineering tasks which are reduced to transmission problems of similar type. For the model problem the existence and uniqueness of its weak solution in appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed.

ABSTRACT. One interesting class of parabolic problems model processes in heat-conduction media with concentrated capacity in which the heat capacity coefficient contains a Dirac delta function. Such problems are nonstandard and the classical tools of the theory of finite difference schemes are difficult to apply to their convergence analysis. In the present paper a finite-difference scheme, approximating the two-dimensional initial-boundary value problem for the heat equation with concentrated capacity and time dependent coefficients of the space derivatives, is derived. Abstract operator method is developed for analyzing this problem. Convergence in special discrete anisotropic Sobolev norms is proved.

ABSTRACT. Abstract: Paper present methodology of a possible interrogation of the vibro-impact nonlinear dynamics of two rolling bodies in series of the successive central collisions on curvilinear trace. Curvilinear trace consists of thee circle arches. Two rigid rolling bodies are with an axis of symmetry and one plane of symmetry and with different dimensions of circle cross section in the plane of symmetry. Each collision in series of the successive collision between rolling bodies is central collision. Between two successive collisions of two bodies are in rolling motion along corresponding branch of curvilinear tracing. For investigation of the kinetic parameters and discrete singular phenomena of vibro-impact dynamics of defined system, we use method of phase trajectory portraits, surface of system total mechanical energy and portraits of system constant total mechanical energy curves of rolling motion of each of bodies on curvilinear trace between two successive collisions from the series of the successive collisions. Also, theory of collision between rolling bodies is used for determination outgoing angular velocities after each collision necessary, as initial condition for each next phase trajectory branch between two successive collisions. For obtaining position of each rolling body on curvilinear trace at corresponding collision between two rolling bodies it is necessary to use approximations of the series of elliptic integrals and solving numerically series of the nonlinear transcendent equations. These numerical tasks are not simple, because need previously prognosis of the possible position of each of two bodies in position of collision along each of the circle arch as an branch of the complex curvilinear trace. Then it is necessary to summarize corresponding time intervals of motion of each of bodies up to corresponding position of collision. Next ii is to compose corresponding nonlinear transcendent equation by sums time intervals of each of rolling body along circle arches. For defined mechanical model, depending of radiuses of circle arches in curvilinear rolling trace, and radiuses of circle cross section in plane of body symmetry of each of rolling bodies, different combinations of phase trajectory portraits appeared. These phase trajectory portraits are with different positions of singular points and different forms of separatrix - homoclinic phase trajectories. We take into investigation one particular case with trigger of coupled three singular points and with an homoclinic phase trajectory in the form of number “eight”. Same conclusion of energy jumps between rolling bodies in series of successive collisions are presented.

Keywords: Vibro-impact dynamics, Rolling bodies, Curvilinear trace, Successive central collisions, Outgoing angular velocities, Approximation, Elliptic integral, Nonlinear equations, Phase trajectory portrait, Singular points, Homoclinic trajectories.

Acknowledgements: Parts of this research were supported by the Ministry of Sciences and Technology of Republic of Serbia through Mathematical Institute SASA, Belgrade Grant ON174001 “Dynamics of hybrid systems with complex structures.”, Mechanics of materials and Faculty of Mechanical Engineering University of Niš.

References

[1] Hedrih (Stevanović) R. K., Nonlinear Dynamics of a Heavy Material Particle Along Circle which Rotates and Optimal Control, Chaotic Dynamics and Control of Systems and Processes in Mechanics (Eds: G. Rega, and F. Vestroni), p. 37-45. IUTAM Book, in Series Solid Mechanics and Its Applications, Editerd by G.M.L. Gladwell, Springer. 2005, XXVI, 504 p., Hardcover ISBN: 1-4020-3267-6. [2] Hedrih (Stevanović) R. K., A Trigger of Coupled Singularities, MECCANICA, Vol.39, No. 3, 2004., pp. 295-314. , DOI: 10.1023/B:MECC.0000022994.81090.5f, [3] Hedrih (Stevanović) R. K.,, Dynamics of Impacts and Collisions of the Rolling Balls, Dynamical Systems: Theoretical and Experimental Analysis, Springer Proceedings in Mathematics & Statistics, 2016, Volume Number: 182, Chapter 13, pp. 157-168. © Springer, ISBN 978-3-319-42407-1 ISBN 978-3-319-42408-8 (eBook), DOI 10.1007/978-3-319-42408-8 [4] Hedrih (Stevanović) R. K., Nonlinear Dynamics of a Gyro-rotor, and Sensitive Dependence on initial Conditions of a Heav Gyro-rotor Forced Vibration/Rotation Motion, Semi-Plenary Invited Lecture, Proceedings: COC 2000, Edited by F.L. Chernousko and A.I. Fradkov, IEEE, CSS, IUTAM, SPICS, St. Petersburg, Inst. for Problems of Mech. Eng. of RAS, 2000., Vol. 2 of 3, pp. 259-266. [5] Hedrih (Stevanović) K., (2009), Vibrations of a Heavy Mass Particle Moving along a Rough Line with Friction of Coulomb Type, ©Freund Publishing House Ltd., International Journal of Nonlinear Sciences & Numerical Simulation 10(11): 1705-1712, 2009. http://www.freundpublishing.com/International_Journal_Nonlinear_Sciences_Numerical%20Simulation/MathPrev.htm. [6] Hedrih (Stevanović) R. K.,, Vibro-impact dynamics of two rolling balls along curvilinear trace, Procedia Engineering, X International Conference on Structural Dynamics, EURODYN 2017, ScienceDirect, 217, pp. 1-6. 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. www.elsevier.com/locate/procedia (accepted, to appear) [7] Hedrih (Stevanović) R. K.,, (2016), Vibro-impact dynamics in systems with trigger of coupled three singular points: Collision of two rolling bodies, The 24th International Congress of Theoretical and Applied Mechanics (ICTAM 2016), Montreal, Canada, 21 - 26 August, 2016, Book of Papers, pp. 212 -213. IUTAM permanent site. ISBN: NR16-127/2016E-EPUB; Catalogue Number: 978-0-660-05459-9 [8] Hedrih (Stevanović) R. K., Non-linear dynamics of a heavy mass particle and rolling ball along curvilinear trace of series of circle arcs: Phase trajectory portraits, some analogies and vibro-impacts, MS-09 Nonlinear Dynamics in Engineering Systems,The 9th European Nonlinear Dynamics Conference (ENOC 2017) in Budapest, Hungary, 25-30 June, 2017 in the campus of Budapest University of Technology and Economics. www.congressline.hu/enoc2017; ID.342; ISBN 978-963-12-9168-1 http://www.congressline.hu/enoc2017/abstracts/342.pdf; [9] Mitrinović D. S., Djoković D. Ž., Special functions (Specijalne funkcije) , Gradjevinska knjiga, Beograd, 1964, p. 267.

ABSTRACT. Nonlinear structure of boundary layer flow models which are one of the mostly encountered physical models in nature attracts the interest of mathematicians to analyze the power and accuracy of the numerical methods. The aspire of this study is to present a semi-analytical solution to a boundary layer flow model by using a numerical method.