How far one can go with the Exp-function method?

A criterion determining if an exact solution of a differential equation can be expressed in a form comprising a finite number of exponential functions is constructed in this paper. This criterion is based on the concept of ranks of Hankel matrixes constructed from sequences of coefficients produced by symbolic multiplicative operator techniques. The employment of this criterion also gives an answer on the structure of the solution. Several examples are used to illustrate this concept.     

Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

Starlikeness and subordination properties of a linear operator

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator L_k^α. For functions in this class H_k^α(p,λh) we estimate the coefficients. Furthermore, some subordination properties related to the operator L_k^α are also derived.     

New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations

In this letter, the Kaup–Kupershmidt, (2+1)$(2+1)$-dimensional Potential Kadomtsev–Petviashvili (shortly PKP) equations are presented and the Exp-function method is employed to compute an approximation to the solution of nonlinear differential equations governing the problem. It has been attempted to show the capabilities and wide-range applications of the Exp-function method. This method can be used as an alternative to obtain analytic and approximate solution of different types of differential equations applied in engineering mathematics.     

Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation

Some methods to look for exact solutions of nonlinear differential equations are discussed. It is shown that many popular methods are equivalent to each other. Several recent publications with “new” solitary wave solutions for the Kuramoto-Sivashinsky equation are analyzed. We demonstrate that all these solutions coincide with the known ones.      

Spectral collocation and radial basis function methods for one-dimensional interface problems

Differential equations with singular sources or discontinuous coefficients yield non-smooth or even discontinuous solutions. This problem is known as the interface problem. High-order numerical solutions suffer from the Gibbs phenomenon in that the accuracy deteriorates if the discontinuity is not properly treated. In this work, we use the spectral and radial basis function methods and present a least squares collocation method to solve the interface problem for one-dimensional elliptic equations. The domain is decomposed into multiple sub-domains; in each sub-domain, the collocation solution is sought. The solution should satisfy more conditions than the given conditions associated with the differential equations, which makes the problem over-determined. To solve the over-determined system, the least squares method is adopted. For the spectral method, the weighted norm method with different scaling factors and the mixed formulation are used. For the radial basis function method, the weighted shape parameter method is presented. Numerical results show that the least squares collocation method provides an accurate solution with high efficacy and that better accuracy is obtained with the spectral method.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

Elements of harmonic analysis related to the third basic zero order Bessel function

This paper is devoted to the study of some q-harmonic analysis related to the third q-Bessel function of order zero. We establish a product formula leading to a q-translation with some positive kernel. As an application, we provide a q-analogue of the continuous wavelet transform related to this harmonic analysis.     

On equilibrium problem of abstract economy, generalized quasi-variational inequality, and an optimization problem in locally H-convex spaces

In this paper, the equilibrium existence problem for abstract economies, the existence problem for solution of generalized quasi-variational inequality, and an optimization problem in locally H-convex spaces are researched.     

Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko-Dubrovsky equations

In this paper, new exact solutions with two arbitrary functions of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by means of the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. It is shown that the Exp-function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.     

Hardy-Littlewood maximal function of τ-measurable operators

We define the Hardy-Littlewood maximal function of τ-measurable operators and obtain weak (1,1)-type and (p,p)-type inequalities for the Hardy-Littlewood maximal function.     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.     

q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.     

Certain subclasses of p-valent functions involving the Dziok-Srivastava operator

Using the methods of differential subordination, we investigate some inclusion relationships of certain subclasses of analytic and p-valent functions which are defined by means of Dziok-Srivastava operator.     

Subordination properties of multivalent functions involving an extended fractional differintegral operator

The object of this article is to investigate inclusion, radius, and other various properties of subclasses of multivalent analytic functions, which are defined by using an extended version of the Owa-Srivastava fractional differintegral operator Ω(λ, p).     

Approximate solutions of nonlinear fractional equations

Fractional exponential that are invariant under fractional derivatives, elementary and special fractional functions are introduced. Approximate solutions to fractional Burgers equation, by using the homotopy perturbation method, are obtained. Furthermore, real integral representations for some H-functions are found that may be very helpful in numerical computations.     

On p-adic interpolating function for q-Euler numbers and its derivatives

In this paper we study a two-variable p-adic q-l-function lp,q(s,t|χ) for Dirchlet's character χ, with the property that

     

Remark on: “Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation” [Appl. Math. Comput. (2009) doi:10.1016/j.amc.2009.07.009]

By means of the Exp-function method, Inan and Ugurlu [Appl. Math. Comput. (2009) doi:10.1016/j.amc.2009.07.009] reported eight expressions for being solutions to the two equations studied. In fact, all of them can be easily simplified to constants.     

Asymptotic behaviours and generalized Toeplitz operators

We study some generalized Toeplitz operators associated to operators T on a Hilbert space H, for which there exists the limit of {‖Tnh‖} for every h∈H. We refer to the asymptotic limit ST of such a T, in the sense of [L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997) 571-605; L. Kerchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged) 68 (2002) 373-400; G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems, J. Operator Theory 53 (1) (2005) 101-140; C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997], and we give some conditions of ergodicity for T. Also, certain results of Douglas [R.G. Douglas, On the operator equation S^∗XT=X and related topics, Acta Sci. Math. (Szeged) 30 (1969) 19-32] involving generalized Toeplitz operators are extended in our more general setting, and we apply these results to ρ-contractions.     

On multivalued complementarity problems in Banach spaces

In this paper, we prove a new existence result for multivalued complementarity problems in Banach spaces. Our result represents a refinement and improvement of the previously known results.