Browsing 3. Tεχνικές Αναφορές by Subject "Analytic functions"
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- ItemOpen AccessA new approach to the derivation of exact integral formulae for zeros of analytic functions
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA new method for the reduction of the problem of locating the zeros of an analytic function inside a simple closed contour to that of locating the zeros of a polynomial is proposed. The new method (exactly like the presently used classical relevant method) permits in this way the derivation of exact integral formulae for these zeros if they are no more than four. The present approach is based on the solution of a simple homogeneous Riemann–Hilbert boundary value problem. An application to a classical problem in physics concerning neutron moderation is also made and numerical results obtained by using the trapezoidal quadrature rule are presented.
- ItemOpen AccessA new method for the computation of the zeros of analytic functions
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Anastasselou, Eleni; Ιωακειμίδης, Νικόλαος; Αναστασέλου, ΕλένηA new method for the computation of the zeros of analytic functions (or the poles of meromorphic functions) inside or outside a closed contour C in the complex plane is proposed. This method is based on the Cauchy integral formula (in generalized forms) and leads to closed-form formulae for the zeros (or the poles) if they are no more than four. In general, for m zeros (or poles) these can be evaluated as the zeros of a polynomial of degree m. In all cases, complex contour integrals have to be evaluated numerically by using appropriate numerical integration rules. Several practical algorithms for the implementation of the method are proposed and the method of Abd-Elall, Delves and Reid is rederived by two different approaches as one of these algorithms. A numerical application to a transcendental equation appearing in the theory of neutron moderation is also made and numerical results of high accuracy are easily obtained.
- ItemOpen AccessA new quadrature method for locating the zeros of analytic functions with applications to engineering problems
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςA new method for the computation of real or complex zeros of analytic functions and/or poles of meromorphic functions outside a fundamental interval [a,b] of the real axis is proposed. This method is based on appropriately taking into account the error terms in the Gauss– and Lobatto–Chebyshev quadrature rules for ordinary integrals and it leads to a very simple non-iterative algorithm for the computation of these zeros. The results obtained by this algorithm with very few functional evaluations are of a very good accuracy and they can further be improved, if required, by local methods, which are generally inappropriate for the original localization of the zeros. The proposed method was tested in two engineering problems: a problem of neutron moderation in nuclear reactors and a problem of determining the critical buckling load of an elastic frame. The corresponding transcendental equations were solved by this method and numerical results for their zeros are presented. In all equations solved, numerical values for their zeros accurate to at least five significant digits were obtained by the present method with no more than thirteen functional evaluations.
- ItemOpen AccessA real closed-form integral formula for roots of nonlinear equations
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςAn analytical integral formula for a single simple root of a nonlinear equation in a finite interval is proposed. The derivation of this formula is based on Picard's method for the number of roots of nonlinear equations and on Green's theorem. The present formula is particularly useful for the location of roots of nonanalytic functions. A comparison between the present formula and a related classical formula based on the theory of analytic functions is also made.
- ItemOpen AccessApplication of complex path-independent integrals to locating circular holes and inclusions in classical plane elasticity
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςWe propose an elementary method, based on complex path-independent integrals and the classical complex potentials of Kolosov–Muskhelishvili, for the location of the position of the centre and the determination of the radius of circular holes and inclusions of a different material (either simply inserted or attached) in an infinite plane isotropic elastic medium. In practice, the method of pseudocaustics can be successfully used as the related experimental method. Generalizations of the present results follow trivially.
- ItemOpen AccessDirect solution of plane elasticity problems by using the Muskhelishvili functional equation and computer algebra software
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe classical Muskhelishvili functional equation for the solution of plane isotropic elasticity problems is revisited. The first complex potential φ(z) of Kolosov–Muskhelishvili is assumed to have the approximate form of a polynomial with unknown coefficients inside the elastic medium. Then the boundary conditions permit the expression of the boundary values for the second complex potential ψ(z) of Kolosov–Muskhelishvili in terms of the same coefficients. This potential, ψ(z), is then obtained by using the classical Cauchy integral formula in complex analysis. Finally, the unknown coefficients are determined by using an appropriate number of collocation points for ψ(z) outside the elastic medium. Both cases of the first and the second fundamental problems are considered. The present method was implemented and applied to the problem of an elliptical elastic medium by using the computer algebra system Mathematica and its modern and powerful language. Generalizations of the present results are also possible and suggested in brief. It is hoped that the present approach will be considered as an alternative to the classical methods of solution of plane isotropic elasticity problems like the finite and boundary element methods.
- ItemOpen AccessLocating inclusions of the same material in finite plane isotropic elastic media by using complex path-independent integrals
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe method of complex path-independent integrals on a closed contour is used for the location of an inclusion (of arbitrary but known shape) of the same material with the matrix and welded with the matrix in plane isotropic elasticity for a finite medium. Only the position of the inclusion and the external loading are not known in advance. The first complex potential of Kolosov–Muskhelishvili (or one of its two first derivatives) is used, together with optical methods for its evaluation, on the aforementioned contour. Beyond the location of the inclusion, a variety of generalizations of the proposed technique as well as a long discussion are also included.
- ItemOpen AccessOn the application of the generalized Plemelj formulas to crack problems
Γενικό Τμήμα (Τεχνικές Αναφορές)Ioakimidis, Nikolaos; Ιωακειμίδης, ΝικόλαοςThe singular integral equation for the simple plane elasticity problem of a straight crack under a known normal pressure distribution is derived by using the first generalized Plemelj formula (where a finite-part integral appears) on the basis of the complex-variable formulation of plane elasticity problems. The present results are believed to prove useful in a lot of more complicated elasticity problems, to which they can easily be applied, and to lead to the wide use of singular integral equations with finite-part integrals in elasticity problems.