Showing papers on "Summation equation published in 2009"

On the stability of functional equations

Abstract: We give some theorems on the stability of the equation of homomorphism, of Lobacevski’s equation, of almost Jensen’s equation, of Jensen’s equation, of Pexider’s equation, of linear equations, of Schroder’s equation, of Sincov’s equation, of modified equations of homomorphism from a group (not necessarily commutative) into a $${\mathbb{Q}}$$ -topological sequentially complete vector space or into a Banach space, of the quadratic equation, of the equation of a generalized involution, of the equation of idempotency and of the translation equation. We prove that the different definitions of stability are equivalent for the majority of these equations. The boundedness stability and the stability of differential equations and the anomalies of stability are considered and open problems are formulated too.

Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result

Abstract: We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved by B. Perthame and G. Barles for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

Progress in solving a noncommutative quantum field theory in four dimensions

Abstract: We study the noncommutative \phi^4_4-quantum field theory at the self-duality point. This model is renormalisable to all orders as shown in earlier work of us and does not have a Landau ghost problem. Using the Ward identity of Disertori, Gurau, Magnen and Rivasseau, we obtain from the Schwinger-Dyson equation a non-linear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations are the starting point for a perturbative solution. In this way, the renormalised correlation functions are directly obtained, without Feynman graph computation and further renormalisation steps

Regularity, symmetry, and uniqueness of some integral type quasilinear equations

Abstract: We study integral equations corresponding to some quasilinear equations with nonlinearities of Lane–Emden and Hartree type. Regularity, symmetry, and uniqueness of these equations are considered. We obtain the uniqueness of the ground state of H 1 critical Hartree equation and extend the moving plane method of integral equation in [W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. LIX (2006) 0330–0343; W. Chen, C. Li, B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations 30 (1–3) (2005) 59–65] to some integral equations corresponding to the p -Laplace equation. We use ideas from the potential theories for the p -Laplace equations and Hessian equations.

Stability of an Additive-Cubic-Quartic Functional Equation

Abstract: In this paper, we consider the additive-cubic-quartic functional equation and prove the generalized Hyers-Ulam stability of the additive-cubic-quartic functional equation in Banach spaces.

Gaussian beams summation for the wave equation in a convex domain

Abstract: We consider the scalar wave equation in a bounded convex domain of Rn. The boundary condition is of Dirichlet or Neumann type and the initial conditions have a compact support in the considered domain. We construct a family of approximate high frequency solutions by a Gaussian beams summation. We give a rigorous justification of the asymptotics in the sense of an energy estimate and show that the error can be reduced to any arbitrary power of ε, which is the high frequency parameter.

On existence of integrable solutions of a functional integral equation under Carathéodory conditions

Abstract: We study the solvability of a functional integral equation in the space of Lebesgue integrable functions on an unbounded interval. Using the conjunction of the technique of measures of weak noncompactness with the classical Schauder fixed point principle we show that the equation in question is solvable in the mentioned function space. Our existence result is obtained under the assumption that functions involved in the investigated functional integral equation satisfy Caratheodory conditions. Moreover, that result generalizes several ones obtained earlier in many research papers and monographs.

Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

Abstract: The authors present a method of numerical approximation of the fixed point of an operator, specifically the integral one associated with a nonlinear Fredholm integral equation, that uses strongly the properties of a classical Schauder basis in the Banach space .

3D integral equation modeling with a linear dependence on dimensions

Abstract: We have improved the integral equation method for modeling 3D electromagnetic fields by using the separability of its inherent 3×3 dyadic Green’s tensors. Conventional integral equation approaches exhibit a quadratic dependence on model size, at least for the vertical dimension. In contrast, our approach has a linear dependence on all three dimensions. We tested our method on an example of induction logging in deviated boreholes.

The capacitance of the circular parallel plate capacitor obtained by solving the Love integral equation using an analytic expansion of the kernel

Abstract: The capacitance of the circular parallel plate capacitor is calculated by expanding the solution to the Love integral equation into a Fourier cosine series. Previously, this kind of expansion has been carried out numerically, resulting in accuracy problems at small plate separations. We show that this bottleneck can be alleviated, by calculating all expansion integrals analytically in terms of the Sine and Cosine integrals. Hence, we can, in the approximation of the kernel, use considerably larger matrices, resulting in improved numerical accuracy for the capacitance. In order to improve the accuracy at the smallest separations, we develop a heuristic extrapolation scheme that takes into account the convergence properties of the algorithm. Our results are compared with other numerical results from the literature and with the Kirchhoff result. Error estimates are presented, from which we conclude that our results is a substantial improvement compared with earlier numerical results.

Solutions of the three-dimensional sine-Gordon equation

Abstract: We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions.

A Calderón Multiplicative Preconditioner for the PMCHWT integral equation

Abstract: The interaction matrices originating from the Method of Moments (MoM) discretization of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation pertinent to the analysis from (piecewise) homogeneous penetrable objects become ill-conditioned when either part or the entire scattering surface requires a dense discretization. This behavior is similar to that of the Electric Field Integral Equation (EFIE) for the scattering from perfectly conducting surfaces and finds its origin in the presence of compact and hypersingular components in the integral equation kernel.

Middle Convolution and Heun's Equation

Abstract: Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of ini- tial conditions of the sixth Painleve equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.

On a functional volterra-fredholm integral equation, via picard operators

Abstract: In this paper we present some results relative to existence, uniqueness, integral in- equalities and data dependence for the solutions of the functional Volterra-Fredholm integral equation with deviating argument in a Banach space:

Traveling wave solutions to a reaction-diffusion equation

Abstract: In this paper, we restrict our attention to traveling wave solutions of a reaction-diffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.

On existence and asymptotic behaviour of solutions of a fractional integral equation

Abstract: We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions, defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.

Integral Equations and the First Passage Time of Brownian Motions

Abstract: The first passage time problem for Brownian motions hitting a barrier has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the barrier itself have appeared. Most interestingly, Peskir(2002b) demonstrates that a master integral equation can be used to generate a countable number of new equations via differentiation or integration by parts. In this article, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of Volterra type integral equations of the first kind and prove uniqueness for a subclass. Furthermore, through the integral equations, we demonstrate how certain functional transforms of the boundary affect the density function. Finally, we demonstrate a fundamental connection between the Volterra integral equations and a class of Fredholm integral equations.

On a Constrained 2-D Navier-Stokes Equation

Abstract: The planar Navier-Stokes equation exhibits, in absence of external forces, a trivial asymptotics in time. Nevertheless the appearence of coherent structures suggests non-trivial intermediate asymptotics which should be explained in terms of the equation itself. Motivated by the separation of the different time scales observed in the dynamics of the Navier-Stokes equation, we study the well-posedness and asymptotic behaviour of a constrained equation which neglects the variation of the energy and moment of inertia.

Numerical Solution of Linear Fredholm Fuzzy Integral Equation of the Second Kind by Block-pulse Functions

Abstract: 3 Abstract: In this paper,we present a numerical method based on block-pulse funct ions(BPFs) for the s olut ion of a system of linear Fredholm integral equation of second kind with two variable. We w ill apply our method for an applied example,linear Fredholm fuzzy integral equation of the second kind(M.Friedman,M M ing,A.Kandel,Numerical solution of fuzzy differential and integral equation, fuzzy Sets and Systems 106 (1999)35-48 ).

Methods for solving an integral equation of the third kind with fixed singularities in the kernel

Abstract: We consider a linear integral equation of the third kind with fixed singularities in the kernel. For the approximate solution of this equation in the space of distributions, we suggest and justify a new generalized version of the collocation method.

A note on the Exp-function method and its application to nonlinear equations

Abstract: An improved Exp-function method is described and used for finding a unified solution of a nonlinear wave equation. The combined KdV–MKdV equation is selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free constants is obtained. This method can be applied also to many other equations.

Linear perturbations of a nonoscillatory second-order dynamic equation

Abstract: We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and improve recently given conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and a dominant solution of the unperturbed equation. The presented results are time scales analogues of a 2002 paper on the differential equations case by Trench. The difference equations case has not been treated yet, but is contained in our study together with other cases of dynamic equations.

Method of comparison equations and generalized Ermakov’s equation

Abstract: The connection between the method of comparison equations (generalized WKB method) and the Ermakov-Pinney equation is established. A perturbative scheme of solution of the generalized Ermakov-Pinney equation is developed and is applied to the construction of perturbative series for second-order differential equations with and without turning points.

On the theoretical justification of Pocklington's equation

Abstract: Pocklington's model consists in a one-dimensional integral equation relating the current at the surface of a straight finite wire to the tangential trace of an incident electromagnetic field. It is a simplification of the more usual single layer potential equation posed on a two-dimensional surface. We are interested in estimating the error between the solution of the exact integral equation and the solution of Pocklington's model. We address this problem for the model case of acoustics in a smooth geometry using results of asymptotic analysis.

Solution of Fredholm-Hammerstein Integral Equations with Walsh-Hybrid Functions

Abstract: In this paper a numerical method for finding the solution of FredholmHammerstein integral equations is proposed. At first the properties of the Walsh-hybrid functions, which combination of block-pulse functions and Walsh functions are proposed. The properties of the hybrid functions with the operational matrix of integration together Newton-Cotes nodes are then utilized to reduce the solution of integral equation to the solution of algebraic equations. The method is computationally attractive and application are demonstrated through illustrative examples.

A modification for solving Fredholm‐Hammerstein integral equation by using wavelet basis

Abstract: – The purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations., – L2[0, 1] was considered as solution space and the solution was projected to the subspaces of L2[0, 1] with finite dimension so that basis elements of these subspaces were orthonormal., – In this process, solution of Fredholm‐Hammerstein integral equation is found by solving the generated system of nonlinear equations., – Comparing the method with others shows that this system has less computation. In fact, decreasing of computations result from the modification.