The generalization of the generalized Al-Oboudi differential operator

In this paper, firstly we define the generalization of the generalized Al-Oboudi differential operator. Then we also define new classes of analytic and p-valently starlike and convex functions with complex order by means of this new general differential operator. Our main purpose is to determine coefficient bounds for functions in certain subclasses of this classes, which are introduced here by means of a family of nonhomogeneous Cauchy-Euler differential equations. Relevant connections of some of the results obtained with those in earlier works are also provided.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

Polynomial approximations in the complex plane

Good polynomial approximations for analytic functions are potentially useful but are in short supply. A new approach introduced here involves the Lanczos τ-method, with perturbations proportional to Faber or Chebyshev polynomials for specific regions of the complex plane. The results show that suitable forms of the τ-method, which are easy to use, can produce near-minimax polynomial approximations for functions which satisfy linear differential equations with polynomial coefficients. In particular, some accurate approximations of low degree for Bessel functions are presented. An appendix describes a simple algorithm which generates polynomial approximations for the Bessel function J_ν(z) of any given order ν.     

Expansions in Series of Products of Eigenfunctions

A method is presented to establish expansions of analytic functions in series of m-fold products of special functions of Mathematical Physics. The idea is to “multiply” vector-valued solutions of first order differential systems in a suitable way and to construct the first order differential system which the “product” satisfies. Then an expansion theorem for the corresponding Floquet eigenvalue problem can be proved.     

Ordered differential fields

We introduce the abstract notion of an ordered differential field and show that some of the basic asymptotic growth properties of differentiable functions can be discussed within this setting. However, in order to ensure that log x → ∞ as x → ∞, we have to assume that the field of constants is archimedean.     

Fractal oscillations for a class of second order linear differential equations of Euler type

The s-dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied. The main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of Euler type. Next, it will be shown that the dimensional number s only depends on a positive real parameter α appearing in a singular term of the main equation. It continues some recent results on the rectifiable and unrectifiable oscillations given in Paši? [M. Paši?, Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl. 335 (2007) 724-738] and Wong [J.S.W. Wong, On rectifiable oscillation of Euler type second order linear differential equations, Electron. J. Qual. Theory Differ. Equ. 20 (2007) 1-12].     

Specialization of Appell's functions to univariate hypergeometric functions

Univariate specializations of Appell's hypergeometric functions F₁, F₂, F₃, F₄ satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2 and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.     

The generalization of Meyer-König and Zeller operators by generating functions

In this paper, theorems are proved concerned with some approximation properties of generating functions type Meyer-König and Zeller operators with the help of a Korovkin type theorem. Secondly, we compute the rates of convergence of these operators by means of the modulus of continuity, Peetre's K-functional and the elements of the modified Lipschitz class. Also we introduce the rth order generalization of these operators and we obtain approximation properties of them. In the last part, we give some applications to the differential equations.     

Differential analysis of matrix convex functions

We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18-42]. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of the following order n + 1.     

Coefficient bounds for a subclass of starlike functions of complex order

In the present work, we determine coefficient bounds for functions in certain subclass of starlike functions of complex order b, which are introduced here by means of a multiplier differential operator. Several corollaries and consequences of the main results are also considered.     

An incremental and parametrical algorithm for convex-concave fractional programming with a single constraint

A problem of the maximization of the ratio of a concave function to a convex function is considered, subject to an upper bound on a single convex constraint function; all these functions are assumed to be differentiable. An incremental algorithm is defined, which solves the problem parametrically for different values of the constraint function by the solution of a set of ordinary first order differential equations. If K is the number of variables in the problem and B(K) is an upper bound—dependent of K —of the time needed to evaluate any function value or any first or second order derivative, the complexity of the algorithm is of the order O[(B(K)K + K)a], where a is the number of integration steps applied in the solution of the differential equations. In particular, a cost-effectiveness resource allocation problem with separable functions is solved numerically in a time of the order O[Ka] if B(K) is independent of K; an example of such a problem is given with analytically solvable differential equations.     

Weighted pseudo almost periodic functions and applications

This Note introduces a new class of functions called weighted pseudo almost periodic functions, which generalize in a natural fashion the classical pseudo almost periodic functions due to C. Zhang. Properties of those weighted pseudo almost periodic functions are discussed including a composition result of weighted pseudo almost periodic functions, which plays a crucial role for the solvability of some weighted pseudo almost periodic semilinear differential and partial differential equations. To cite this article: T. Diagana, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in L _p .     

Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on by-passing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _α (h ^α D _x ^α )f(x).     

The generalized Bochner condition about classical orthogonal polynomials revisited

We bring a new proof for showing that an orthogonal polynomial sequence is classical if and only if any of its polynomial fulfils a certain differential equation of order 2k, for some k?1. So, we build those differential equations explicitly. If k=1, we get the Bochner's characterization of classical polynomials. With help of the formal computations made in Mathematica, we explicitly give those differential equations for k=1,2 and 3 for each family of the classical polynomials. Higher order differential equations can be obtained similarly.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.     

On the growth of meromorphic solutions of the Schwarzian differential equations

We study the properties of meromorphic solutions of the Schwarzian differential equations in the complex plane by using some techniques from the study of the class Wp. We find some upper bounds of the order of meromorphic solutions for some types of the Schwarzian differential equations. We also show that there are no wandering domains nor Baker domains for meromorphic solutions of certain Schwarzian differential equations.     

Some results in discontinuous fields and wave propogation

We present various applications of recently developed techniques which are an interplay between the differential geometry and the theory of distributions. We apply them to the microlocal theory, derive the general order transport equations for the strength of wave fronts, obtain the values of the jumps of the general Nth-order derivatives of the harmonic functions across the potential layers, and present the explicit formulas for an arbitrary order derivative of functions of the radial distance r.     

On quadratic integral inequalities of the second order

The aim of this paper is to generalize the uniform method of obtaining integral inequalities in order to derive inequalities involving a function h, its first and second derivatives with weights. Such inequalities have been considered before by others, but other methods were applied. Our method makes it possible to obtain, in a natural way, the equality conditions important in differential equations. Moreover it allows us to avoid some assumptions on weights that have to be given in other methods. Then the inequality will be examined in order to simplify the boundary conditions for h. These considerations will be followed by examples with Chebyshev weight functions and constant weights with the classical Hardy, Littlewood, Polya inequality as a special case.     

On an analog of the Riesz theorem and the basis property of the system of root functions of a differential operator in L p : II

We consider an ordinary differential operator of arbitrary order, obtain necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of the system of root functions of the given operator in L _p .