Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation

In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Comput. Math. Appl. 54 (2007) 1018–1027]. It is found that the HPM is a special case of the HAM when =-1. However, the HPM solution is divergent for all x and t except t=0. It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results.     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation

There are very few results about analytic solutions of problems of optimal control with minimal L norm. In this paper, we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to find a control with minimal L norm that steers the system to the target.We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has been given by Bennighof and Boucher in Ref. 1.     

On the Class of Banach Spaces with James Constant $${\sqrt{}}$$ : Part II

The class of two-dimensional normed spaces with James constant \({\sqrt{2}}\) is studied. It is shown that, if the norm is \({\pi/2}\)-rotation invariant, then its James constant is \({\sqrt{2}}\) if and only if the norm is \({\pi/4}\)-rotation invariant. We also present a characterization of \({\pi/4}\)-rotation invariant norms using some properties of certain convex functions on the unit interval, which allow us to easily construct norms the James constant of which are \({\sqrt{2}}\). Moreover, two important examples are given, which show that neither absoluteness, symmetry nor \({\pi/2}\)-rotation invariance can be a characteristic property of the norms with James constant \({\sqrt{2}}\).     

Homotopy analysis solution of free convection flow on a horizontal impermeable surface embedded in a saturated porous medium

In this paper, the analytic solution of the buoyancy-driven flow over a horizontal impermeable flat plate embedded in a saturated porous medium is derived using the newly developed analytic method, namely the homotopy analysis method (HAM). The HAM results show great agreement comparing with numerical results. HAM contains an auxiliary parameter ? that provides a simple way of controlling and adjusting the convergence region. The resultant analytic solution is valid for all acceptable values of the temperature exponent parameter λ.     

Ramified coverings over a complete Kähler space

Let X and Y be reduced complex spaces with countable topology. Let be a locally semi-finite holomorphic map such that the analytic set is nowhere dense in X. If Y is complete Kähler, then we prove that X is also complete Kähler. Especially if is a (not necessarily finitely sheeted) ramified covering over a complete Kähler space Y, then X is also complete Kähler. Received: 2 August 2002     

On a problem connected with a theorem of Jarnik

Divide the plane into unit squares and plot a rectifiable closed jordan curve of lengthl. We show that the error created when approximating the number of unit squares contained inside this curve by the area enclosed by it is less than l, where is a constant satisfying

Analytic Solutions of Von Kármán Plate under Arbitrary Uniform Pressure — Equations in Differential Form

The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von Kármán [1]. This problem is reconsidered in this paper using an analytic approximation method, namely, the homotopy analysis method (HAM). Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of , where denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the Vincent's [2] and Chien's [3] methods) and the modified iteration method [4] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von Kármán's plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problems.     

Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential

We study the effect of spatially-dependent mass function over the solution of the Dirac equation with the Coulomb potential in the (3+1)-dimensions for any arbitrary spin-orbit κ state. In the framework of the spin and pseudospin symmetry concept, the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles are obtained by means of the Nikiforov-Uvarov method, in closed form. This physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the Dirac equation. The special cases , the constant mass and the non-relativistic limits are briefly investigated.     

Pseudo-steady-state solutions for solidification in a wedge

Polubarinova-Kochina's analytical differential equation methodis used to determine the pseudo-steady-state solution to problemsinvolving the freezing (solidification) of wedges of liquidwhich are initially at their fusion temperature. In particular,we consider four distinct problems for wedges which are: freezingwith the same constant boundary temperature, freezing with thesame constant boundary heat fluxes, freezing with distinct constantboundary temperatures and freezing with distinct constant fluxesat the boundaries. For the last two problems, a Heun's differentialequation with an unknown singularity is derived, which in bothcases admits a particularly elegant simple solution for thespecial case when the wedge angle is . The moving boundariesobtained are shown pictorially.     

Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order

On the basis of the exact solution of the linear Dirichlet problem , we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u _tt – u _xx = f(x, t, u, u _t).     

An inequality between thep- and (p, 1)-summing norm of finite rank operators fromC(K)-spaces

We show, for any operatorT from aC(K)-space into a Banach space with rank (T)≤n, the inequality , whereC≤4.671 is a numerical constant. The factor (1+logn)^1−1/p is asymptotically correct. This inequality extends a result of Jameson top ≠ 2. Several applications are given — one is a positive solution of a conjecture of Rosenthal and Szarek: For 1≤p<q<2,      

Calculus II: Analytic functors

Homotopy functors (for example, from spaces to spaces) are called analytic if, when evaluated on certain n-cubical diagrams, they satisfy certain connectivity estimates. Tools for verifying these estimates include certain generalizations of the triad connectivity theorem. Waldhausen's functor A is analytic. Analyticity has strong consequences, when combined with the concept derivative of a homotopy functor that was introduced in the previous article in this series. In particular, any analytic functor with derivative zero is, in a sense, locally constant.Research partially supported by NSF grant DMS-8806444 and a Sloan Fellowship.     

On coercive solvability of parabolic equations

In this paper we study the coercive solvability of abstract differential equations of parabolic type in the spaces of L. N. Slobodetskii W _p . It is established that the solution of equations with a constant operator A which generates an analytic semigroup belongs to the trace space E(, p, A). The results obtained are applied to the study of equations with a variable operator.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 409–419, April, 1972.     

On the first two eigenvalues of Sturm-Liouville operators

Among the Schrödinger operators with single-well potentials defined on with transition point at , the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric single-well potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.

     

Solution of a nonlinear singular integral equation with quadratic nonlinearity

Using methods of the theory of boundary-value problems for analytic functions, we prove a theorem on the existence of solutions of the equation

Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

The main results of this paper are as follows. (a) Let : M N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kähler structure andM a Sasakian structure. (b) Let be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature . If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.