Showing papers on "Green's theorem published in 2011"

On a perturbed quadratic fractional integral equation of Abel type

Abstract: We present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in C[0,1]. Our equation contains the famous Chandrasekhar integral equation as a special case. The concept of the measure of noncompactness related to monotonicity, introduced by J. Banas and L. Olszowy, and a fixed point theorem due to Darbo are the main tools in carrying out our proof. Moreover, we give an example for indicating the natural realizations of our abstract result presented in this paper.

A Fubini Theorem in Riesz Spaces for the Kurzweil-Henstock Integral

Abstract: A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the "extended" real plane.

Retrieval of Green's function and generalized optical theorem for the scattering of complete dyadic fields.

Abstract: Green’s function retrieval has been widely used in different research fields due to the fact that the Green’s function can be extracted by cross-correlating the records at two receivers. In this paper, the retrieval of the dyadic Green’s function is studied by investigating the representation theorem. The generalized optical theorem for the dyadic fields is derived based on the elastic dynamic interferometric equation. By addressing the cross-correlation recorded at two receivers, the important role of the generalized optical theorem and energy equipartition in retrieving the exact Green’s function is shown. The presented derivation also shows the Newton–Marchenko equation holdsif the condition of equipartition is not satisfied.

On Mean Value Theorem of Integrals

Abstract: This paper gives a new proof method for the first integral mean value theorem with ξ∈(a,b).Secondly,in the case of strengthened part of the traditional conditions,it proves also the second integral mean value theorem with ξ∈(a,b),the traditional proof method be changed and simplified,the new proof method is suitable for non-mathematics students.

A Theorem for Numerical Verification on Local Uniqueness of Solutions to Fixed-Point Equations

Abstract: We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Frechet differentiable operators The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation The conditions are formulated to apply readily to numerical verification methods We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations

Divergent Integrals in Elastostatics: General Considerations

Abstract: This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.

The Extension of Double Integral Mean Value Theorem

Abstract: In this paper,from geometric meaning and basic form of the double integral value theorem starting,a necessary condition for the establishment of the double integral mean value theorem was found.The continuity condition is weakened to the condition with intermediate value property in the double integral mean value theorem,the double integral mean value theorem was discussed,the generalized format of the double integral mean value theorem is given by intermediate value property.Further on the basis of function continuity of the double integral mean value theorem,monotonic of the function on two variables is increased,(monotonic increasing,monotonic decreasing),the other generalized format of the double integral mean value theorem is given.In the last special case of the double integral mean value theorem is given the extension of Integral mean value theorem.

Thermal Analysis: VCSELs on an SiOB

Abstract: This paper introduces a method combining a general electrothermal network π-model in system level and the associated mathematical technique, Green's theorem, in terms of the adopted materials and system geometries to build up an equivalent electrothermal circuit model (EETCM) for efficient thermal analysis and behavior prediction in a thermal system. Heat conduction and convection equations in integral forms are derived using the theorem and successfully applied for the thermal analysis of a 3-D optical stack, vertical-cavity surface-emitting lasers (VCSELs) on a silicon optical bench. The complex stack structure in conventional simulators can be greatly simplified using the method by well-predicting probable heat flow paths, and the simplification can eventually achieve the goal of CPU time saving without having complicated mesh designing or scaling. By comparing the data from the measurement, the finite-element simulation, and the method calculation shows that an excellent temperature match within ±~0.5 °C and 90% CPU time saving can be realized.

Generalizing the Converse to Pascal's Theorem via Hyperplane Arrangements and the Cayley-Bacharach Theorem

Abstract: Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in k triple points and if we color the remaining lines so that each triple point lies on a red and blue line then the points of intersection of the red and blue lines lying off the green line lie on a unique curve of degree k-1. We also use these ideas to extend a second generalization of the Braikenridge-Maclaurin Theorem, due to Mobius. Finally we use Terracini's Lemma and secant varieties to show that this process constructs a dense set of curves in the space of plane curves of degree d, for degrees d <= 5. The process cannot produce a dense set of curves in higher degrees. The exposition is embellished with several exercises designed to amuse the reader.

Valuation of Two-Factor Interest Rate Contingent Claims Using Green's Theorem

Abstract: Over the years a number of two-factor interest rate models have been proposed that have formed the basis for the valuation of interest rate contingent claims. This valuation equation often takes the form of a partial differential equation that is solved using the finite difference approach. In the case of two-factor models this has resulted in solving two second-order partial derivatives leading to boundary errors, as well as numerous first-order derivatives. In this article we demonstrate that using Green's theorem, second-order derivatives can be reduced to first-order derivatives that can be easily discretized; consequently, two-factor partial differential equations are easier to discretize than one-factor partial differential equations. We illustrate our approach by applying it to value contingent claims based on the two-factor CIR model. We provide numerical examples that illustrate that our approach shows excellent agreement with analytical prices and the popular Crank–Nicolson method.