Eleni G. Anastasselou

On the simultaneous determination of zeros of analytic or sectionally analytic functions

It is shown how the total-step and single-step iterative methods, as well as their improvements, for the simultaneous determination of simple zeros of polynomials can be used (with one slight modification) for the determination of simple zeros of analytic functions (inside or outside a simple smooth closed contour in the complex plane) or sectionally analytic functions (outside their arcs of discontinuity). Numerical results, obtained by the single-step method, are also presented.ZusammenfassungEs wird gezeigt, wie die Totalschritt- und Einschritt-Iterations-Verfahren für die gleichzeitige Bestimmung von einfachen Nullstellen von Polynomen sowie ihre Verbesserungen (mit einer kleinen Modifikation) für die Bestimmung von einfachen Nullstellen analytischer Funktionen (im inneren oder äußeren einer einfachen glatten abgeschlossenen Kontur in der komplexen Ebene) oder stückweise analytischer Funktionen (im äußeren ihrer Unstetigkeitsbögen) benutzt werden können. Numerische Ergebnisse, die mit der Einschrittmethode erhalten wurden, werden auch präsentiert.

A generalization of the Siewert–Burniston method for the determination of zeros of analytic functions

The Siewert–Burniston method for the derivation of closed‐form formulas for the zeros of sectionally analytic functions with a discontinuity interval along the real axis (based on the Riemann–Hilbert boundary value problem in complex analysis) is generalized to apply to the determination of the zeros of analytic functions (without discontinuity intervals) inside or outside simple smooth contours. An application of this method to the closed‐form solution of the transcendental equation zez =beb, appearing in the theory of neutron moderation in nuclear reactors, is also made.

A new method for obtaining exact analytical formulae for the roots of transcendental functions

A new method is proposed for the derivation of closed-form formulae for the zeros and poles of sectionally analytic functions in the complex plane. This method makes use of the solution of the simple discontinuity problem in the theory of analytic functions and requires the evaluation of real integrals only (for functions with discontinuity intervals along the real axis). Many transcendental equations of mathematical physics can be successfully solved by the present approach. An application to such an equation, the molecular field equation in the theory of ferromagnetism, is made and the corresponding analytical formulae are reported together with numerical results.

A modification of the delves-lyness method for locating the zeros of analytic functions

Abstract A new formula is proposed for the calculation of the integrals involved in the application of the Delves-Lyness method for locating the zeros of analytic functions inside an arbitrary simple smooth contour. This formula does not contain the derivative of the analytic function whose zeros are sought and, moreover, is free from multivaluedness problems for the integrand.

Derivation of the equation of caustics in cartesian coordinates with maple

Abstract The well-known equation of caustics about crack tips in experimental fracture mechanics is usually expressed in the form of two equations for the Cartesian coordinates (x,y) with a parameter (an appropriate polar angle playing the rǒle of this parameter). Here we derive the equivalent unique equation (but without a parameter) also in Cartesian coordinates with the help of the computer algebra system Maple V by using the available Grobner-bases algorithm. The obtained Cartesian equation is of the sixth degree and it can be solved in closed form with respect to y yielding an explicit result y =y(x). The same equation is also checked in two special cases, where it gives the same results as the equivalent pair of parametric equations. Analogous, more general results can also be derived.

Application of the Green and the Rayleigh-Green reciprocal identities to path-independent integrals in two- and three-dimensional elasticity

SummaryAn elementary but quite general method for the construction of path-independent integrals in plane and three-dimensional elasticity is suggested. This approach consists simply in using the classical Green formula in its reciprocal form for harmonic functions and, further, the more general Rayleigh-Green formula also in its reciprocal form, but for biharmonic functions. A large number of harmonic and biharmonic functions appears in a natural way in the theory of elasticity. Therefore, the construction of path-independent integrals (or, probably better, surface-independent integrals in the three-dimensional case) becomes really a trivial task. An application to the determination of stress intensity factors at crack tips is considered in detail and only the sum of the principal stress components is used in the path-independent integral. Further applications of the method are easily possible.

Gröbner bases in truss problems with maple

Abstract The well-known method of Grobner bases in computer algebra can be used in elementary truss problems, where the appearance of sine and cosine terms because of the angles of the bars of the truss to the Ox-axis (beyond the magnitudes of the forces in these bars) make the problem nonlinear. A simple truss is used for an illustration of the method during the determination of two angles in such a way that the forces in two bars have values given in advance. Similarly, it is observed, again by using Grobner bases, that the force equations at each joint of a loaded truss and the equation for the reactions of the truss are compatible. The whole approach seems to be one more application of Grobner bases related to their applications in nonlinear computational geometry. The computer algebra system Maple V has been used in the present application of Grobner bases.

On the location of straight discontinuity intervals of arbitrary sectionally analytic functions by using complex path-independent integrals

Abstract The classical method of locating zeros and poles of analytic and meromorphic functions in the complex plane via the evaluation of complex path-independent integrals on closed contours surrounding the sought zeros and poles is generalized to become applicable (for the first time) to the location of straight discontinuity intervals of completely general sectionally analytic functions possessing such an interval. Such is the case e.g. in crack problems in plane elasticity or in airfoils in plane fluid dynamics. This generalization makes use of appropriate or arbitrary quadrature rules and, in this way, it leads to a completely numerical method. The method is illustrated in the case of a simple sectionally analytic function, where the classical Chebyshev and Legendre polynomials are used and numerical results are presented. Possible generalizations of the present results are also reported in brief and are strongly expected in future.

A new, simple approach to the derivation of exact analytical formulae for the zeros of analytic functions

A very simple method for the construction of the polynomial whose zeros coincide with the zeros of an analytic function inside and along a simple closed contour in the complex plane, based on an appropriate application of the Cauchy theorem in complex analysis, is proposed. The present method was motivated by the classical Burniston-Siewert method, based on the theory of the Riemann-Hilbert boundary-value problem for the construction of the aforementioned polynomial, but, although essentially equivalent to the Burniston-Siewert method, it is much simpler.

A new approach to the derivation of exact analytical formulae for the zeros of sectionally analytic functions

Abstract A new method for the analytical determination of zeros of sectionally analytic functions in the cut complex plane (or of poles of sectionally meromorphic functions), based only on the first Plemelj formula, is proposed. This method, like the analogous earlier methods of Burniston and Siewert and of the senior author, leads to the determination of these zeros (or poles) as the roots of a polynomial. The coefficients of this polynomial contain regular integrals along the discontinuity interval, which can be computed by appropriate (usually Gaussian) quadrature rules. The advantage of the present approach over the existing ones is its simplicity as a concept and ease during the application to particular sectionally analytic (or meromorphic) functions.