Search results for: Fredholm integral equations

Authors: Leila Motamed-Jahromi, Mohsen Hatami, Alireza Keshavarz

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This research aims at obtaining the equations of pulse propagation in nonlinear plasmonic waveguides created with As₂S₃ chalcogenide materials. Via utilizing Helmholtz equation and first-order perturbation theory, two components of electric field are determined within frequency domain. Afterwards, the equations are formulated in time domain. The obtained equations include two coupled differential equations that considers nonlinear dispersion.

Keywords: nonlinear optics, plasmonic waveguide, chalcogenide, propagation equation

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Authors: Imen Soltani, Takoua Soltani, Taoufik Aguili

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Metamaterials crossover classic physical boundaries and gives rise to new phenomena and applications in the domain of beam steering and shaping. Where electromagnetic near and far field manipulations were achieved in an accurate manner. In this sense, 3D imaging is one of the beneficiaries and in particular Denis Gabor’s invention: holography. But, the major difficulty here is the lack of a suitable recording medium. So some enhancements were essential, where the 2D version of bulk metamaterials have been introduced the so-called metasurface. This new class of interfaces simplifies the problem of recording medium with the capability of tuning the phase, amplitude, and polarization at a given frequency. In order to achieve an intelligible wavefront control, the electromagnetic properties of the metasurface should be optimized by means of solving Maxwell’s equations. In this context, integral methods are emerging as an important method to study electromagnetic from microwave to optical frequencies. The method of moment presents an accurate solution to reduce the problem of dimensions by writing its boundary conditions in the form of integral equations. But solving this kind of equations tends to be more complicated and time-consuming as the structural complexity increases. Here, the use of equivalent circuit’s method exhibits the most scalable experience to develop an integral method formulation. In fact, for allaying the resolution of Maxwell’s equations, the method of Generalised Equivalent Circuit was proposed to convey the resolution from the domain of integral equations to the domain of equivalent circuits. In point of fact, this technique consists in creating an electric image of the studied structure using discontinuity plan paradigm and taken into account its environment. So that, the electromagnetic state of the discontinuity plan is described by generalised test functions which are modelled by virtual sources not storing energy. The environmental effects are included by the use of an impedance or admittance operator. Here, we propose a tunable metasurface composed of graphene-based elements which combine the advantages of reflectarrays concept and graphene as a pillar constituent element at Terahertz frequencies. The metasurface’s building block consists of a thin gold film, a dielectric spacer SiO₂ and graphene patch antenna. Our electromagnetic analysis is based on the method of moment combined with generalised equivalent circuit (MoM-GEC). We begin by restricting our attention to study the effects of varying graphene’s chemical potential on the unit cell input impedance. So, it was found that the variation of complex conductivity of graphene allows controlling the phase and amplitude of the reflection coefficient at each element of the array. From the results obtained here, we were able to determine that the phase modulation is realized by adjusting graphene’s complex conductivity. This modulation is a viable solution compared to tunning the phase by varying the antenna length because it offers a full 2π reflection phase control.

Keywords: graphene, method of moment combined with generalised equivalent circuit, reconfigurable metasurface, reflectarray, terahertz domain

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Authors: Yildiray Keskin, Omer Acan, Murat Akkus

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In this paper, the solution of fractional diffusion equations is presented by means of the reduced differential transform method. Fractional partial differential equations have special importance in engineering and sciences. Application of reduced differential transform method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results show that the approach is easy to implement and accurate when applied to fractional diffusion equations. The method introduces a promising tool for solving many fractional partial differential equations.

Keywords: fractional diffusion equations, Caputo fractional derivative, reduced differential transform method, partial

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Authors: Shai Sarussi

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Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. A characterization of the property of immediate neighbors in an Alexandroff topological space is given, in terms of closed and open subsets of appropriate subspaces. Moreover, two special subspaces of W are introduced, and a way in which their closed and open subsets induce W is presented.

Keywords: integral domains, Alexandroff topology, immediate neighbors, valuation domains

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Authors: Wedad Albalawi

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The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques, and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then, dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is an arbitrary nonempty closed subset of the real numbers. Then, the dynamic inequalities on time scales have received a lot of attention in the literature and has become a major field in pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on Hardy and Coposon inequalities, using Steklov operator on time scale in double integrals to obtain special cases of time-scale inequalities of Hardy and Copson on high dimensions. The advantage of this study is that it uses the one-dimensional classical Hardy inequality to obtain higher dimensional on time scale versions that will be applied in the solution of the Cauchy problem for the wave equation. In addition, the obtained inequalities have various applications involving discontinuous domains such as bug populations, phytoremediation of metals, wound healing, maximization problems. The proof can be done by introducing restriction on the operator in several cases. The concepts in time scale version such as time scales calculus will be used that allows to unify and extend many problems from the theories of differential and of difference equations. In addition, using chain rule, and some properties of multiple integrals on time scales, some theorems of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of hardy, inequality of coposon, steklov operator

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Authors: Liliana O. Martínez, Juan E González, Manuel Ramírez-Aranda, Ana Cervantes-Herrera

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A serious game category mobile application called Math Dominoes is presented. The main objective of this applications is to strengthen the teaching-learning process of solving algebraic equations and is based on the board game "Double 6" dominoes. Math Dominoes allows the practice of solving first, second-, and third-degree algebraic equations. This application is aimed to students who seek to strengthen their skills in solving algebraic equations in a dynamic, interactive, and fun way, to reduce the risk of failure in subsequent courses that require mastery of this algebraic tool.

Keywords: algebra, equations, dominoes, serious games

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Authors: Arcady Ponosov., Ramazan Kadiev

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The work studies the global moment stability of solutions of systems of nonlinear differential Itô equations with delays. A modified regularization method (W-method) for the analysis of various types of stability of such systems, based on the choice of the auxiliaryequations and applications of the theory of positive invertible matrices, is proposed and justified. Development of this method for deterministic functional differential equations is due to N.V. Azbelev and his students. Sufficient conditions for the moment stability of solutions in terms of the coefficients for sufficiently general as well as specific classes of Itô equations are given.

Keywords: asymptotic stability, delay equations, operator methods, stochastic noise

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Authors: Kamel Al-Khaled

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Reaction-diffusion equations are commonly used in population biology to model the spread of biological species. In this paper, we propose a fractional reaction-diffusion equation, where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. Based on the symbolic computation system Mathematica, Adomian decomposition method, developed for fractional differential equations, is directly extended to derive explicit and numerical solutions of space fractional reaction-diffusion equations. The fractional derivative is described in the Caputo sense. Finally, the recent appearance of fractional reaction-diffusion equations as models in some fields such as cell biology, chemistry, physics, and finance, makes it necessary to apply the results reported here to some numerical examples.

Keywords: fractional partial differential equations, reaction-diffusion equations, adomian decomposition, biological species

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Authors: Lila D. Sigdel, Chin J. Leo, Samanthika Liyanapathirana, Pan Hu, Minghao Lu

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Integral bridges are a class of bridges with integral or semi-integral abutments, designed without expansion joints in the bridge deck of the superstructure. Integral bridges are economical alternatives to conventional jointed bridges with lower maintenance costs and greater durability, thereby improving social and economic stability for the community. Integral bridges have also been proven to be effective in lowering the overall construction cost compared to the conventional type of bridges. However, there is significant uncertainty related to the design and analysis of integral bridges in response to cyclic thermal movements induced due to deck expansion and contraction. The cyclic thermal movements of the abutments increase the lateral earth pressures on the abutment and its foundation, leading to soil settlement and heaving of the backfill soil. Thus, the primary objective of this paper is to investigate the soil-abutment interaction under the cyclic translational movement of the abutment. Results from five experiments conducted to simulate different magnitudes of cyclic translational movements of abutments induced by thermal changes are presented, focusing on lateral earth pressure development at the abutment-soil interface. Test results show that the cycle number and magnitude of cyclic translational movements have significant effects on the escalation of lateral earth pressures. Experimentally observed earth pressure distributions behind the integral abutment were compared with the current design approaches, which shows that the most of the practices has under predicted the lateral earth pressure.

Keywords: integral bridge, cyclic thermal movement, lateral earth pressure, soil-structure interaction

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Authors: Artion Kashuri, Rozana Liko

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In this paper, the authors establish an integral identity using delta differentiable functions. By applying this identity, some new results via a general class of convex functions with respect to two nonnegative functions on a time scale are given. Also, for suitable choices of nonnegative functions, some special cases are deduced. Finally, in order to illustrate the efficiency of our main results, some applications to special means are obtained as well. We hope that current work using our idea and technique will attract the attention of researchers working in mathematical analysis, mathematical inequalities, numerical analysis, special functions, fractional calculus, quantum mechanics, quantum calculus, physics, probability and statistics, differential and difference equations, optimization theory, and other related fields in pure and applied sciences.

Keywords: convex functions, Hermite-Hadamard inequality, special means, time scale

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Authors: M. Naji, A. R. Khalim, M. Naji

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Integral Abutment Bridges (IAB) are defined as simple or multiple span bridges in which the bridge deck is cast monolithically with the abutment walls. This kind of bridges are becoming very popular due to different aspects such as good response under seismic loading, low initial costs, elimination of bearings and less maintenance. However, the main issue related to the analysis of this type of structures is dealing with soil-structure interaction of the abutment walls and the supporting piles. A two-dimensional, non-linear finite element (FE) model of an integral abutment bridge has been developed to study the effect of lateral time history displacement loading on the soil system.

Keywords: integral abutment bridge, soil structure interaction, finite element modeling, soil-pile interaction

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Authors: Natalia D. Vaysfel’d, Zinaida Y. Zhuravlova

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The loaded plane elastic semistrip, the lateral boundaries of which are fixed, is considered. The integral transformations are applied directly to Lame’s equations. It leads to one dimensional boundary value problem in the transformations’ domain which is formulated as a vector one. With the help of the matrix differential calculation’s apparatus and apparatus of Green matrix function the exact solution of a vector problem is constructed. After the satisfying the boundary condition at the semi strip’s edge the problem is reduced to the solving of the integral singular equation with regard of the unknown stress at the semis trip’s edge. The equation is solved with the orthogonal polynomials method that takes into consideration the real singularities of the solution at the ends of integration interval. The normal stress at the edge of the semis trip were calculated and analyzed.

Keywords: semi strip, Green's Matrix, fourier transformation, orthogonal polynomials method

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Authors: Naser Safdarian, Nader Jafarnia Dabanloo

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In this paper we used four features i.e. Q-wave integral, QRS complex integral, T-wave integral and total integral as extracted feature from normal and patient ECG signals to detection and localization of myocardial infarction (MI) in left ventricle of heart. In our research we focused on detection and localization of MI in standard ECG. We use the Q-wave integral and T-wave integral because this feature is important impression in detection of MI. We used some pattern recognition method such as Artificial Neural Network (ANN) to detect and localize the MI. Because these methods have good accuracy for classification of normal and abnormal signals. We used one type of Radial Basis Function (RBF) that called Probabilistic Neural Network (PNN) because of its nonlinearity property, and used other classifier such as k-Nearest Neighbors (KNN), Multilayer Perceptron (MLP) and Naive Bayes Classification. We used PhysioNet database as our training and test data. We reached over 80% for accuracy in test data for localization and over 95% for detection of MI. Main advantages of our method are simplicity and its good accuracy. Also we can improve accuracy of classification by adding more features in this method. A simple method based on using only four features which extracted from standard ECG is presented which has good accuracy in MI localization.

Keywords: ECG signal processing, myocardial infarction, features extraction, pattern recognition

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Authors: Khosrow Maleknejad, Yaser Rostami

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In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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Authors: Babitha Elizabeth Philip

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This paper aims to study in general about bridges without expansion joints. Integral Abutment Bridges (IAB) fall into this category of bridges. They are having a continuous deck and also the girders are integrated into the abutments. They are most cost effective system in terms of construction, maintenance, and longevity. The main advantage of IAB is that it is corrosion resistant since water is not allowed to pass through the structure. The other attractions of integral bridges are its simple and rapid construction, smooth and uninterrupted deck which provides a safe ride. Also damages to the abutments can be avoided to a great extent due to better load distribution at the bridge ends. Damages due to improper drainage are not seen in IAB because of its properly drained approach slabs thus eliminating the possibility of erosion of the abutment backfill and freeze and thaw damage resulting from saturated backfill.

Keywords: continuous bridge, integral abutment bridge, joint bridge, life cycle cost, soil interaction

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Authors: Kamel Al-Khaled

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In this paper, numerical solutions for the nonlinear coupled Korteweg-de Vries, (abbreviated as KdV) equations are calculated by Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. First, discretizing time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a $\theta-$weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method.

Keywords: Nonlinear coupled KdV equations, Soliton solutions, Sinc-collocation method, Sinc functions

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Authors: Divij Gupta

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Pendular systems have a range of both mathematical and engineering applications, ranging from modelling the behaviour of a continuous mass-density rope to utilisation as Tuned Mass Dampers (TMD). Thus, it is of interest to study the differential equations governing the motion of such systems. Here we attempt to generalise these equations of motion for the plane compound pendulum with a finite number of N point masses. A Lagrangian approach is taken, and we attempt to find the generalised form for the Euler-Lagrange equations of motion for the i-th bob of the N -bob pendulum. The co-ordinates are parameterized as angular quantities to reduce the number of degrees of freedom from 2N to N to simplify the form of the equations. We analyse the form of these equations up to N = 4 to determine the general form of the equation. We also develop a MATLAB program to compute a solution to the system for a given input value of N and a given set of initial conditions.

Keywords: classical mechanics, differential equation, lagrangian analysis, pendulum

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Authors: Eda Gök

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Peridynamics is a new modeling concept of non-local interactions for solid structures. The formulations of Peridynamic (PD) theory are based on integral equations rather than differential equations. Through, undefined equations of associated problems are avoided. PD theory might be defined as continuum version of molecular dynamics. The medium is usually modeled with mass particles bonded together. Particles interact with each other directly across finite distances through central forces named as bonds. The main assumption of this theory is that the body is composed of material points which interact with other material points within a finite distance. Although, PD theory developed for discontinuities, it gives good results for structures which have no discontinuities. In this paper, displacement control of the isotropic plate under the effect of tensile and bending loading has been investigated by means of PD theory. A MATLAB code is generated to create PD bonds and corresponding surface correction factors. Using generated MATLAB code the geometry of the specimen is generated, and the code is implemented in Finite Element Software. The results obtained from non-local continuum theory are compared with the Finite Element Analysis results and analytical solution. The results show good agreement.

Keywords: non-local continuum mechanics, peridynamic theory, solid structures, tensile loading, flexural loading

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Authors: Fadi Awawdeh

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In this work, we highlight the key concepts in using semigroup theory as a methodology used to construct efficient formulas for solving inverse problems. The proposed method depends on some results concerning integral equations. The experimental results show the potential and limitations of the method and imply directions for future work.

Keywords: identification problems, semigroup theory, methods for inverse problems, scientific computing

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Authors: Samson Davoyan, Ashot Davoyan, Ani Khachatryan

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The indexes (Global Competitiveness Index, Economic Freedom Index, Human Development Index, etc.) developed by different international and non-government organizations in time and space express the quantitative and qualitative features of different fields of various reforms implemented in different countries. The main objective of our research is to develop new methodology that we will use to create integral index based on many indexes and that will include many areas of reforms. To achieve our aim we have used econometric methods (regression model for panel data method). The basis of our methodology is the development of the new integral index based on quantitative assessment of the change of two main parameters: the score of the countries by different indexes and the change of the ranks of countries for following two periods of time. As a result of the usage of methods for analyzes we have defined the indexes that are used to create the new integral index and the scales for each of them. Analyzing quantitatively and qualitatively analysis through the integral index for more than 100 countries for 2009-2014, we have defined comparative efficiency that helps to conclude in which directions countries have implemented reforms more effectively compared to others and in which direction reforms have implemented less efficiently.

Keywords: development, rank, reforms, comparative, index, economic, corruption, social, program

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Authors: Bhavini Pandya

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This paper considers Hη: T2 → T2 the Perturbed Cerbelli-Giona map. That is a family of 2-dimensional nonlinear area-preserving transformations on the torus T2=[0,1]×[0,1]= ℝ2/ ℤ2. A single parameter η varies between 0 and 1, taking the transformation from a hyperbolic toral automorphism to the “Cerbelli-Giona” map, a system known to exhibit multifractal properties. Here we study the multifractal properties of the family of maps. We apply a box-counting method by defining a grid of boxes Bi(δ), where i is the index and δ is the size of the boxes, to quantify the distribution of stable and unstable manifolds of the map. When the parameter is in the range 0.51< η <0.58 and 0.68< η <1 the map is ergodic; i.e., the unstable and stable manifolds eventually cover the whole torus, although not in a uniform distribution. For accurate numerical results we require correspondingly accurate construction of the stable and unstable manifolds. Here we use the piecewise linearity of the map to achieve this, by computing the endpoints of line segments which define the global stable and unstable manifolds. This allows the generalized fractal dimension Dq, and spectrum of dimensions f(α), to be computed with accuracy. Finally, the intersection of the unstable and stable manifold of the map will be investigated, and compared with the distribution of periodic points of the system.

Keywords: feed forward, gradient descent, neural network, integral equation

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Authors: A. M. Sagir

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Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: block method, first order ordinary differential equations, hybrid, self-starting

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Authors: Jyh-Yang Wu, Sheng-Gwo Chen

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In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations.

Keywords: conservation laws, diffusion equations, Cahn-Hilliard equations, evolving surfaces

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Authors: Ernesto Pineda Leon, Dante Tolentino Lopez, Janis Zapata Lopez

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The current paper shows the application of the boundary element method for the analysis of plates under shear stress causing plasticity. In this case, the shear deformation of a plate is considered by means of the Reissner’s theory. The probability of failure of a Reissner’s plate due to a proposed index plastic behavior is calculated taken into account the uncertainty in mechanical and geometrical properties. The problem is developed in two dimensions. The classic plasticity’s theory is applied and a formulation for initial stresses that lead to the boundary integral equations due to plasticity is also used. For the plasticity calculation, the Von Misses criteria is used. To solve the non-linear equations an incremental method is employed. The results show a relatively small failure probability for the ranges of loads between 0.6 and 1.0. However, for values between 1.0 and 2.5, the probability of failure increases significantly. Consequently, for load bigger than 2.5 the plate failure is a safe event. The results are compared to those that were found in the literature and the agreement is good.

Keywords: boundary element method, failure, plasticity, probability

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Authors: F. Maass, P. Martin, J. Olivares

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The aim of this work is the application of the program GeoGebra in teaching the study of nonhomogeneous linear second order differential equations with constant coefficients. Different kind of functions or forces will be considered in the right hand side of the differential equations, in particular, the emphasis will be placed in the case of trigonometrical functions producing the resonance phenomena. In order to obtain this, the frequencies of the trigonometrical functions will be changed. Once the resonances appear, these have to be correlationated with the roots of the second order algebraic equation determined by the coefficients of the differential equation. In this way, the physics and engineering students will understand resonance effects and its consequences in the simplest way. A large variety of examples will be shown, using different kind of functions for the nonhomogeneous part of the differential equations.

Keywords: education, geogebra, ordinary differential equations, resonance

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Authors: Armin Ardekani, Mohammad Akbari

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We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields.

Keywords: computational mathematics, differential equations, engineering, series

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Authors: Kholod Mohammad Abualnaja

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A new algorithm is presented to find some new iterative methods for solving nonlinear equations F(x)=0 by using the variational iteration method. The efficiency of the considered method is illustrated by example. The results show that the proposed iteration technique, without linearization or small perturbation, is very effective and convenient.

Keywords: variational iteration method, nonlinear equations, Lagrange multiplier, algorithms

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Authors: Shai Sarussi

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Let S be an integral domain which is not a field, let F be its field of fractions, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R ∩ F = S and F R = A. A topological space whose set of open sets is closed under arbitrary intersections is called an Alexandroff space. Inspired by the well-known Zariski-Riemann space and the Zariski topology on the set of prime ideals of a commutative ring, we define a topology on the set of all S-nice subalgebras of A. Consequently, we get an interplay between Algebra and topology, that gives us a better understanding of the S-nice subalgebras of A. It is shown that every irreducible subset of S-nice subalgebras of A has a supremum; and a characterization of the irreducible components is given, in terms of maximal S-nice subalgebras of A.

Keywords: Alexandroff topology, integral domains, Zariski-Riemann space, S-nice subalgebras

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Authors: Riadh Zorgati, Thomas Triboulet

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In quite diverse application areas such as astronomy, medical imaging, geophysics or nondestructive evaluation, many problems related to calibration, fitting or estimation of a large number of input parameters of a model from a small amount of output noisy data, can be cast as inverse problems. Due to noisy data corruption, insufficient data and model errors, most inverse problems are ill-posed in a Hadamard sense, i.e. existence, uniqueness and stability of the solution are not guaranteed. A wide class of inverse problems in physics relates to the Fredholm equation of the first kind. The ill-posedness of such inverse problem results, after discretization, in a very ill-conditioned linear system of equations, the condition number of the associated matrix can typically range from 109 to 1018. This condition number plays the role of an amplifier of uncertainties on data during inversion and then, renders the inverse problem difficult to handle numerically. Similar problems appear in other areas such as numerical optimization when using interior points algorithms for solving linear programs leads to face ill-conditioned systems of linear equations. Devising efficient solution approaches for such system of equations is therefore of great practical interest. Efficient iterative algorithms are proposed for solving a system of linear equations. The approach is based on a preconditioning of the initial matrix of the system with an approximation of a generalized inverse leading to a stochastic preconditioned matrix. This approach, valid for non-negative matrices, is first extended to hermitian, semi-definite positive matrices and then generalized to any complex rectangular matrices. The main results obtained are as follows: 1) We are able to build a generalized inverse of any complex rectangular matrix which satisfies the convergence condition requested in iterative algorithms for solving a system of linear equations. This completes the (short) list of generalized inverse having this property, after Kaczmarz and Cimmino matrices. Theoretical results on both the characterization of the type of generalized inverse obtained and the convergence are derived. 2) Thanks to its properties, this matrix can be efficiently used in different solving schemes as Richardson-Tanabe or preconditioned conjugate gradients. 3) By using Lp norms, we propose generalized Kaczmarz’s type matrices. We also show how Cimmino's matrix can be considered as a particular case consisting in choosing the Euclidian norm in an asymmetrical structure. 4) Regarding numerical results obtained on some pathological well-known test-cases (Hilbert, Nakasaka, …), some of the proposed algorithms are empirically shown to be more efficient on ill-conditioned problems and more robust to error propagation than the known classical techniques we have tested (Gauss, Moore-Penrose inverse, minimum residue, conjugate gradients, Kaczmarz, Cimmino). We end on a very early prospective application of our approach based on stochastic matrices aiming at computing some parameters (such as the extreme values, the mean, the variance, …) of the solution of a linear system prior to its resolution. Such an approach, if it were to be efficient, would be a source of information on the solution of a system of linear equations.

Keywords: conditioning, generalized inverse, linear system, norms, stochastic matrix

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Authors: Win Ko Ko, A. N. Temnov

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The problem of nonlinear oscillations of a two-layer liquid completely filling a limited volume is considered. Using two basic asymmetric harmonics excited in two mutually perpendicular planes, ordinary differential equations of nonlinear oscillations of the interface of a two-layer liquid are investigated. In this paper, hydrodynamic coefficients of linear and nonlinear problems in integral relations were determined. As a result, the instability regions of forced oscillations of a two-layered liquid in a cylindrical tank occurring in the plane of action of the disturbing force are constructed, as well as the dynamic instability regions of the parametric resonance for different ratios of densities of the upper and lower liquids depending on the amplitudes of liquids from the excitations frequencies. Steady-state regimes of fluid motion were found in the regions of dynamic instability of the initial oscillation form. The Bubnov-Galerkin method is used to construct instability regions for approximate solution of nonlinear differential equations.

Keywords: nonlinear oscillations, two-layered liquid, instability region, hydrodynamic coefficients, resonance frequency

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