On the numerical solution of the multidimensional singular integrals and integral equations, used in the theory of linear viscoelasticity

In the present report, we investigate the formulation, for the numerical evaluation of the multidimensional singular integrals and integral equations, used in the theory of linear viscoelasticity. Some simple formulas are given for the numerical solution of the general case of the multidimensional singular integrals. Moreover a numerical technique is also established for the numerical solution of some special cases of the multidimensional singular integrals like the two - and three - dimensional singular integrals. An application is given to the determination of the fracture behaviour of a thick, hollow circular cylinder of viscoelastic material restrained by an enclosing thin elastic ring and subjected to a uniform pressure

Hypersingular Integral Equations in Banach Spaces by the Quadrature Method

Abstract A new numerical method is introduced and investigated for the hypersingular integral equations defined in Banach spaces. The hypersingular integral equations belong to a wider class of singular integral equations having much more stronger singularities.The proposed approximation method is an extension beyond the quadrature method. Beyond the above, an error estimates theory is proposed and investigated for the hypersingular integral equations by proving some new theorems. Finally, the inequalities valid between the exact solutions of the hypersingular integral equations and the corresponding approximate solutions, are proposed and proved. 2010 Mathematics Subject Classification : 65L10, 65R20. Key Word and Phrases Hypersingular Integral Equations, Singularity, Quadrature Method, Error Estimates, Banach Spaces, Finite-Part Singular Integral Equations. Introduction The hypersingular integral equations consist to a wider class of singular integral equations. In particular the kernel of such integral equations has a stronger singularity as compared to the finitepart singular integral equations. Consequently, there is very big interest for the numerical evaluation of the hypersingular integral equations, as closed form solutions are not possible to be determined. J. Hadamard [1], [2] was the first scientist who introduced the concept of finite -part integrals, and L. Schwartz [3] studied very basic properties of them. Some years later, H.R. Kutt [4] proposed some algorithms for the numerical evaluation of the finite-part singular integrals and studied the difference between a finite -part integral and a "generalized principal value integral". On the contrary, M.A. Golberg [5] investigated the convergence of several numerical methods for the solution of finite-part integrals. He proposed a method, which was an extension beyond the Galerkin and collocation methods In addition, by E.G. Ladopoulos [9] -[15] were proposed several numerical methods for the solution of the finite-part singular integral equations of the first and the second kind. He further applied this type of singular integral equations to the solution of very important problems of elasticity, fracture mechanics and aerodynamics. Also, E.G. Ladopoulos, V.A. Zisis and D. Kravvaritis [16], By the current research are introduced and investigated the hypersingular integral equations, which have stronger singularity in comparison to the finite-part singular integral equations. The hypersingular integral equations belong therefore to a wider class of integral equations with kernels of very strong singularities. A numerical method is proposed for the solution of the hypersingular integral equations, defined in Banach spaces. The proposed approximation method is an extension beyond the quadrature method

Singular Integral Operators Method in Unsteady Spillway Flows

Abstract A classical open channel hydraulics problem is the determination of the free-surface profile of an unsteady flow over a spillway flo w. Thus, by using the Singular Integral Operators Method (S.I.O.M.) then the above problem can be solved by applying numerical evaluation. When a flo w rate Q is known, then the velocit ies and the elevations are computed on the free surface o f the spillway flow. For the nu merical evaluation of the singular integral equations are used both constant and linear elements. An applicat ion is finally given to the determination o f the free-surface profile of a special spillway and comparing the numerical results with corresponding results by the Boundary Integral Equation Method (B.I.E.M.)

Planar Airfoils in Two-dimensional Aerodynamics by Non-linear Multidimensional Singular Integral Equations

Abstract Planar airfoils are studied for the unsteady flow motion in two-dimensional aerodynamics. Such problems are reduced to the solution of a non-linear multidimensional singular integral equation, when the form of the source and vortex strength distribution is dependent on the history of these distributions on the NACA airfoil surface. A turbulent boundary layer model is also proposed, based on the formulation of the unsteady behavior of the momentum integral equation. Finally, an application is given to the determination of the velocity and pressure coefficient field around an aircraft by assuming constant vortex distribution. Key Word and Phrases Non-linear Aerodynamics, Two-dimensional NACA airfoil, Non-linear Multidimensional Singular Integral Equations, Constant Vortex Distribution, Aircraft, Velocity & Pressure Coefficient Field. Introduction Over the last years a continuously increasing interest has been given to the non-linear singular integral equations by which are solved very important problems of aerodynamics and fluid mechanics, especially these referred to unsteady flows. The computational methods which are used for the numerical evaluation of the non-linear singular integral equations consist of the latest high technology to the solution of general problems of solid and fluid mechanics. For this reason such computational methods are continuously improved. The aerodynamic characteristics of the NACA airfoils are too important for the design of the new generation aircrafts, with very high speeds. This new technology aerodynamic problems are therefore reduced to the solution of non-linear singular integral equations, which are used for the determination of the velocity and pressure coefficient field around the NACA airfoils. Hence, special attention should be concentrated to such computational methods used for the solution of the above mentioned aerodynamic and fluid mechanics problems of unsteady flows. The Over the last years, several other scientists made extensive calculations by using unsteady turbulent boundary layer methods. Among them we shall mention

Molecular profiling of a rare rosette-forming glioneuronal tumor arising in the spinal cord

Rosette-forming glioneuronal tumor (RGNT) of the IV ventricle is a rare and recently recognized brain tumor entity. It is histologically composed by two distinct features: a glial component, resembling pilocytic astrocytoma, and a component forming neurocytic rosettes and/or perivascular rosettes. Herein, we describe a 33-year-old man with RGNT arising in the spinal cord. Following an immunohistochemistry validation, we further performed an extensive genomic analysis, using array-CGH (aCGH), whole exome and cancer-related hotspot sequencing, in order to better understand its underlying biology. We observed the loss of 1p and gain of 1q, as well as gain of the whole chromosomes 7, 9 and 16. Local amplifications in 9q34.2 and 19p13.3 (encompassing the gene SBNO2) were identified. Moreover, we observed focal gains/losses in several chromosomes. Additionally, on chromosome 7, we identified the presence of the KIAA1549:BRAF gene fusion, which was further validated by RT-PCR and FISH. Across all mutational analyses, we detected and validated the somatic mutations of the genes MLL2, CNNM3, PCDHGC4 and SCN1A. Our comprehensive molecular profiling of this RGNT suggests that MAPK pathway and methylome changes, driven by KIAA1549:BRAF fusion and MLL2 mutation, respectively, could be associated with the development of this rare tumor entity.Conselho Nacional de Desenvolvimento Científico e Tecnológico [475358/2011-2] to RMR (www.cnpq.br); Fundação de Amparo a Pesquisa do Estado de São Paulo [2012/19590-0] to RMR and [2011/08523-7 and 2012/08287-4] to LTB (www.fapesp.br); the Foundation for Science and Technology (FCT) [PTDC/SAU-ONC/115513/2009] to RMR; and the National Cancer Institute [P30CA046934] to MG

Further Developments of Non-linear Semigroups in Hilbert Spaces used in Oil & Gas Engineering

Abstract A new mathematical approach is investigated by using non-linear semigroups in order to prove the existence and uniqueness of solutions for the non-linear partial differential equation defined in Hilbert Spaces and derived from the general porous medium analysis. Such an equation is used in well test analysis in petroleum reservoir engineering for the determination of the properties of the reservoir materials. Hence, by the new method is estimated the size of the oil reserves after their exploration. Additionally, the existence and uniqueness of solutions for the non-linear porous medium equation is proved, by presenting some general boundary conditions. Finally, some properties of the solutions for the above non-linear partial differential equation are finally proved. Introduction Over the past years an increasing interest was realized on studying non-linear semigroups in general Banach spaces associated with the existence and uniqueness theory of partial differential equations arising in a big level of problems of mathematical physics and engineering. Hence, the study of the non-linear semigroups was derived directly from the examination of non-linear parabolic equations and from various non-linear boundary value problems. As a beginning the first work on semigroups was published by A.V

© Hindawi Publishing Corp. NONLINEAR UNSTEADY FLOW PROBLEMS BY MULTIDIMENSIONAL SINGULAR INTEGRAL REPRESENTATION ANALYSIS

A two-dimensional nonlinear aerodynamics representation analysis is proposed for the investigation of inviscid flowfields of unsteady airfoils. Such problems are reduced to the solution of a nonlinear multidimensional singular integral equation as the source and vortex strength distributions are dependent on the history of these distributions on the NACA airfoil surface. A turbulent boundary layer model is further investigated, based on the formulation of the unsteady behaviour of the momentum integral equation. An application is finally given to the determination of the velocity and pressure coefficient field around an aircraft by assuming linear vortex distribution. 2000 Mathematics Subject Classification: 65L10, 65R20