Search

This paper is devoted to introduce a numerical simulation with a theoretical study for flow of a Newtonian fluid over an impermeable stretching sheet which embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by a non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing PDEs are transformed into a system of coupled non-linear ODEs which are using appropriate boundary conditions for various physical parameters. The proposed method is based on replacement of the unknown function by truncated series of well known shifted Legendre expansion of functions. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients. Thus, by solving this system of equations, the shifted Legendre coefficients are obtained. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, thermal conductivity parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. Comparison of obtained numerical results is made with previously published results in some special cases, and excellent agreement is noted. The results attained in this paper confirm the idea that proposed method is powerful mathematical tool and it can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.

In this paper, we propose two classes of derivative-free Newton-like methods for solving system of nonlinear equations based on double Newton step. We also give the local convergence analysis of the iterative methods. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.

In this paper, we investigate boundedness, polar decomposition and spectral decomposition of weighted conditional expectation type operators on \(L^2(\Sigma )\). Some examples are provided to illustrate concrete application of the main results of the paper.

In this paper, we introduce a new class of fractal approximants as a fixed points of the Read–Bajraktarevic operator defined on a suitable function space. In the development of our fractal approximants, we used the suitable bounded linear operators defined on the space $${\mathcal {C}}(I)$$ of continuous functions and $$\alpha $$ -fractal functions. The convergence of the proposed fractal approximants towards the continuous function f does not need any condition on the scaling vector. Owing to this reason, the proposed fractal approximants approximate the function f without losing their fractality. We establish constrained approximation by a new class of fractal polynomials. In particular, our constrained fractal polynomials preserve positivity and fractality of the original function simultaneously whenever the original function is positive and irregular. Calculus of the proposed fractal approximants is studied using suitable bounded linear operators defined on the space $${\mathcal {C}}^r(I)$$ of all real-valued functions on the compact interval I that are r-times differentiable with continuous r-th derivative. We identify the IFS parameters so that our $$\alpha $$ -fractal functions preserve fundamental shape properties such as monotonicity and convexity in addition to the smoothness of f in the given compact interval.

In this paper, we consider the nonsymmetric algebraic Riccati equation whose four coefficient matrices form an M-matrix K. When K is a regular M-matrix, the Riccati equation is known to have a minimal nonnegative solution. We present two new numerical methods which can be applied directly in the case where K is a regular M-matrix. Furthermore, we find that the alternately linearized implicit iteration method is also feasible. In addition, we study the monotone convergence property of the proposed methods. Numerical experiments show that the above three numerical iteration methods are feasible and effective for solving the nonsymmetric algebraic Riccati equation.

In this paper, we study the limit of the iterates of a large class of linear bounded operators preserving constants. We obtain in addition the limit of the iterates of algebraic version of the trigonometric Jackson integrals. The proofs are based on spectral theory of linear operators.

Analytical solution of the two-dimensional integral equations are usually difficult. In many cases, approximate solutions are required. In this paper, we study the approximate solution for two-dimensional Fredholm integral equations of the first kind by two-dimensional wavelet. First, definition and the properties of one-dimensional Franklin wavelet must be presented. Next, integral equations converted via regularization method into the second kind, then, using the idea of wavelet Galerkin method, we will find an approximate solution. Finally, the convergence and efficiency of this method will be discussed with some examples which indicate the ability and accuracy of the method.

In this paper, we demonstrate under some general assumptions, that a generator of any uniformly bounded composition operator, mapping spaces of bounded variation (Waterman) functions into other spaces of this type, must be an affine function in the functional variable.

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology., The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of $${\mathbb{R}}$$ interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in $${\mathcal{L}^{2}(-1,1)}$$ with respect to the weight function $${\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}$$ , $${r > -1}$$ and $${s > -1}$$ . In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of $${\alpha}$$ -fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions $${\mathcal{L}_{\rho}^{2}(-1,1)}$$ is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.

This paper deals with general \({n\times n}\) upper-triangular operator matrices with given diagonal entries. The characterizations of perturbations of their point spectra, residual spectra and continuous spectra are given, based on the space decomposition method.

The inversion problem for a given polynomial set, \({\{P_n\}_{n\geq0}}\), expanded in a basis \({\{\mathfrak{B}_{n}\}}\) is the problem of finding the coefficients \({I_{k}(n)}\) in the expansion $${\mathfrak{B}_n(x) } = \sum_{k=0}^{n}I_k(n) P_{k}(x).$$ In this paper, we state a theorem solving this problem for each family of polynomials defined only by its explicit representation. The obtained coefficients are expressed by means of a simple recurrence relation with two indexes. We illustrate the method by producing explicit formulas for some known polynomial sets.

In this paper, we use the techniques of measure of weak noncompactness and Henstock–Kurzweil–Pettis integrals to discuss the existence theorem of weak solutions for a class of the nonlinear integral equations and obtain a new result, which improves and extends some relevant results for differential and integral equations in Banach spaces.

In this paper, an efficient wavelet Galerkin method based on the stochastic operational matrix of second kind Chebyshev wavelet is proposed for solving stochastic Ito-Volterra integral equations. Convergence and error analysis of the presented wavelets method are investigated. The numerical results are compared with exact solution and those of other existing methods.

In this paper, we investigate the stability of regular, semi-Fredholm, finite essential ascent and finite essential descent linear relations under small and commuting finite rank perturbations as well as the behavior of the nullity, the defect and the index under such perturbations.

In this paper, we shall discuss the family of biharmonic mappings for which maximum principle holds. As a consequence of our study, we present Schwarz Lemma for the family of biharmonic mappings. Also we discuss the univalency of certain class of biharmonic mappings., Comment: 16 pages; 10 figures; The article is with a journal

The purpose of the present paper is to investigate some subordination, other properties and inclusion relations for functions in certain subclasses of multivalent functions which are defined by the linear operator containing the generalized Mittag–Leffler function.

In this paper, a high-order B-spline collocation method on a uniform mesh is presented for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions: $$\begin{aligned} (p(x)y')'= & {} p(x)f(x,y), \quad 0

In this paper, by composite previous-current-step idea, we propose two numerical schemes for solving the Ito stochastic differential systems. Our approaches, which are based on the Euler–Maruyama method, solve stochastic differential systems with strong sense. The mean-square convergence theory of these methods are analyzed under the Lipschitz and linear growth conditions. The accuracy and efficiency of the proposed numerical methods are examined by linear and nonlinear stochastic differential equations.

In this paper, a new approximation method for fractional differential equations based on Mittag-Leffler function is developed. Finite Mittag-Leffler function and its fractional-order derivatives are investigated. An efficient technique for solving linear and nonlinear fractional order differential equations is developed. The proposed method combines Mittag-Leffler collocation method and optimization technique. Error estimation of the approximation is stated and proved. We present numerical results and comparisons of previous treatments to demonstrate the efficiency and applicability of the proposed method. Making use of small number of unknowns, the resulting solution converges to the exact one in the linear case and it has a very small error in the nonlinear case.

We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschitz functions enjoying the mentioned property almost everywhere on the boundary are also exhibited. It is also investigated the algebraic size of the family of functions in the disc algebra that either do not preserve Borel sets on the unit circle or possess the Cantor boundary behavior on the disc.

In this paper, we establish explicit upper and lower bounds for the ratio of the arithmetic and geometric means of the first n prime numbers, which improve the current best estimates. Furthermore, we prove several conjectures related to this ratio stated by Hassani. To do this, we use explicit estimates for the prime counting function, Chebyshev’s $$\vartheta $$ -function, and the sum of the first n primes.

Fractal interpolation has become popular in 2D and 3D data visualization to model a wide variety of physical phenomena. The purpose of this paper is to present a promising new class of positivity preserving $$\mathcal {C}^1$$ -rational cubic spline fractal interpolation surfaces (FISs) to stitch surface data arranged over a rectangular grid. We develop the rational cubic FIS as a combination of blending functions and the rational fractal boundary curves that are constructed along the grid lines of the interpolation domain. We investigate the stability results of the rational cubic FIS with respect to its independent and dependent variables, and the corresponding first order partial derivatives at knots. We identify the scaling factors and shape parameters so that the positivity feature of given data along the grid lines is translated to the corresponding rational fractal boundary curves. In particular, our rational cubic FIS is positive whenever the corresponding fractal boundary curves are positive. Numerical examples are provided to illustrate: (i) the construction of positive rational cubic FISs for a given positive surface data, (ii) the effects on the shape of the positive rational cubic FIS due to the changes in the scaling factors or/and shape parameters, and (iii) the stability results of the proposed rational cubic FIS.

In this paper, we will study the dependence of eigen-pairs $$(\lambda _k(\rho ), \varphi _k(x,\rho ))$$ of weighted Dirichlet eigenvalue problem on weights $$\rho $$ . It will be shown that $$\lambda _k(\rho )$$ and $$\varphi _k(x,\rho )$$ are completely continuous (CC) in $$\rho $$ . Precisely, when $$\rho _n$$ is weakly convergent to $$\rho $$ in some Lebesgue space, $$\lambda _k(\rho _n)$$ is convergent to $$\lambda _k(\rho )$$ . As for the convergence of eigenfunctions, since eigenvalues may have multiple eigenfunctions, it will be shown that the distance from $$\varphi _k(x,\rho _n)$$ to the eigen space $$V_k(\rho )$$ of $$\lambda _k(\rho )$$ is tending to zero. As applications, the CC dependence of solutions of linear inhomogeneous equations and the CC dependence of the heat kernels on coefficients will be given.

[EN] The directional k-step Newton methods (k a positive integer) is developed for solving a single nonlinear equation in n variables. Its semilocal convergence analysis is established by using two different approaches (recurrent relations and recurrent functions) under the assumption that the first derivative satisfies a combination of the Lipschitz and the center-Lipschitz continuity conditions instead of only Lipschitz condition. The convergence theorems for the existence and uniqueness of the solution for each of them are established. Numerical examples including nonlinear Hammerstein-type integral equations are worked out and significantly improved results are obtained. It is shown that the second approach based on recurrent functions solves problems failed to be solved by first one using recurrent relations. This demonstrates the efficacy and applicability of these approaches. This work extends the directional one and two-step Newton methods for solving a single nonlinear equation in n variables. Their semilocal convergence analysis using majorizing sequences are studied in Levin (Math Comput 71(237): 251-262, 2002) and Ioannis (Num Algorithms 55(4): 503-528, 2010) under the assumption that the first derivative satisfies the Lipschitz and the combination of the Lipschitz and the center-Lipschitz continuity conditions, respectively. Finally, the computational order of convergence and the computational efficiency of developed method are studied., The authors thank the referees for their fruitful suggestions which have uncovered several weaknesses leading to the improvement in the paper. A. Kumar wishes to thank UGC-CSIR(Grant no. 2061441001), New Delhi and IIT Kharagpur, India, for their financial assistance during this work.

In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann–Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.

This paper deals with a special Hermite–Fejer interpolation process based at the zeros of generalized Freud polynomials which are orthogonal with respect to the weight $$w(x)=|x|^\alpha e^{-|x|^\beta }$$ , $$x\ne 0, \ \alpha>-1, \ \beta >1$$ . We prove its uniform convergence for functions belonging to a suitable space of functions equipped with a weighted uniform norm.

This paper is aimed to extend the scheme of self scaling, appropriate for the quasi-Newton methods, to the two-step quasi-Newton methods. The scaling scheme has been performed during the main approach of updating the current Hessian approximation and prior to the computation of the next quasi-Newton direction whenever necessary. Global convergence property of the new method is explored on uniformly convex functions with the standard Wolfe line search. Preliminary numerical testing has been performed showing that this technique improves the performance of the two-step method substantially.

The aim of this paper is to explore the stability of several Weyl type theorems (including Weyl’s theorem, a-Weyl’s theorem, Browder’s theorem and a-Browder’s theorem) under compact perturbations in the setting of Hilbert space. It is completely determined when these spectral properties are invariant under compact perturbations. As an application, Weyl type theorems for Toeplitz operators are discussed.

Let $$n\in \mathbb {N}$$ , A and B be Banach algebras and let B be a right A-module. We say that a linear mapping $$\varphi :A\longrightarrow B$$ is a pseudo n-Jordan homomorphism if there exists an element $$w\in A$$ such that $$\varphi (a^nw)=\varphi (a)^n\cdot w$$ , for every $$a\in A$$ and $$n\ge $$ 2. In this paper, among other things, we show that under some conditions if a linear mapping $$\varphi $$ is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) $$(n + 1)$$ -Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.

The aim of this paper is to develop a family of iterative methods with accelerated convergence for solving restricted linear system of equations $$Ax = b$$ , $$x\in T$$ , where $$A \in \mathbb {C}^{m\times n}$$ , $$b \in \mathbb {C}^{m}$$ and T is a subspace of $$\mathbb {C}^{n}$$ . Necessary and sufficient convergence conditions along with the estimation of error bounds of derived approximations are established. The proposed family of iterative methods is applied to solve restricted linear system of equations for different subspaces T. In particular, least-squares solution of an overdetermined linear system and Krylov solution of a square singular linear system of equations are inspected. In addition, the iterative family is tested on a number of numerical examples, including randomly generated dense matrices, sparse singular matrices, and standard test matrices from the Matrix Computational Toolbox. The obtained results are compared with the existing iterative methods and with direct methods, such as the singular value decomposition technique and the QR factorization method.

This paper gives a characterization of the asymptotic limit AT associated to a contraction T that is similar to a normal operator (Theorem 2). Extensions from contractions to power bounded operators intertwined to a contraction with a \({\mathcal{C}_{0}}\). completely nonunitary part (not necessarily a normaloid contraction) are considered as well (Theorem 1). It is also given a characterization of the asymptotic limit AT for a hyponormal contraction T, and it is shown that if a hyponormal contraction has no nontrivial invariant subspace, then one of the defect operators is not finite-rank (Corollary 1).

Recently Salkuyeh et al. (Int J Comput Math 92:802–815, 2015) studied the generalized SOR (GSOR) iterative method for a class of complex symmetric linear system of equations. In this paper, we present an inexact variant of the GSOR method in which the conjugate gradient and the preconditioned conjugate gradient methods are regarded as its inner iteration processes at each step of the GSOR outer iteration. Moreover, we construct a new method called shifted GSOR iteration method which is obtained from combination of a shift-splitting iteration scheme and the GSOR iteration method. The convergence analysis of the proposed methods are presented. Some numerical experiments are given to show the performance of the methods and are compared with those of the inexact MHSS method.

Let X and Y be Banach spaces. When A and B are linear relations in X and Y, respectively, we denote by M C the linear relation in X × Y of the form $$ M_c = \Big(\begin{array}{ll} A & C \\ 0 & B \end{array}\Big)$$ , where 0 is the zero operator from X to Y and C is a bounded operator from Y to X. The goal of this paper is to present some necessary and sufficient conditions on A and B such that there exists a bounded operator C from Y to X for which M C is a Browder linear relation in X × Y.

This paper gives new contributions to atomic systems theory in Hilbert spaces. More precisely, we introduce and study the concept of continuous atomic systems for subspaces, and we give some examples to show differences between this and the discrete version. Moreover, we give a new characterization of continuous (resp. discrete) atomic systems. Finally, among other things, we discuss the relationship between continuous (resp. discrete) frame of subspaces (fusion frames) and continuous (resp. discrete) atomic systems. In particular, we show that the perturbation of a Parseval continuous (resp. discrete) frame of subspaces or identity operator, under certain conditions, gives us a continuous (resp. discrete) atomic system.

Consider the Banach space m of real bounded sequences, x, with $${\Vert x\Vert =\sup_{k}|x_{k}|}$$ . A positive linear functional L on m is called an S-limit if $${L(\chi _{K})=0}$$ for every characteristic sequence $${\chi _{K} }$$ of sets, K, of natural density zero. We provide regular sublinear functionals that both generate as well as dominate S-limits. The paper also shows that the set of S-limits and the collection of Banach limits are distinct but their intersection is not empty. Furthermore, we show that the generalized limits generated by translative regular methods is equal to the set of Banach limits. Some applications are also provided.

In this paper, we obtain L1-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In particular, these new formulae are useful to calculate directly some positive defined integrals as several examples show.

Let \({\mathfrak{M}}\) be a Hilbert C*-module on a C*-algebra \({\mathfrak{A}}\) and let \({End_\mathfrak{A}(\mathfrak{M})}\) be the algebra of all operators on \({\mathfrak{M}}\). In this paper, first the continuity of \({\mathfrak{A}}\)-module homomorphism derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) is investigated. We give some sufficient conditions on which every derivation on \({End_\mathfrak{A}(\mathfrak{M})}\) is inner. Next, we study approximately innerness of derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) for a σ-unital C*-algebra \({\mathfrak{A}}\) and full Hilbert \({\mathfrak{A}}\)-module \({\mathfrak{M}}\). Finally, we show that every bounded linear mapping on \({End_\mathfrak{A}(\mathfrak{M})}\) which behave like a derivation when acting on pairs of elements with unit product, is a Jordan derivation.

A class of trigonometric polynomial basis functions over triangular domain with three shape parameters is constructed in this paper. Based on these new basis functions, a kind of trigonometric polynomial patch over triangular domain, which can be used to construct some surfaces whose boundaries are arcs of ellipse or parabola, is proposed. Without changing the control points, the shape of the trigonometric polynomial patch can be adjusted flexibly in a foreseeable way using the shape parameters. For computing the proposed trigonometric polynomial patch stably and efficiently, a practical de Casteljau-type algorithm is developed. Moveover, the conditions for G1 continuous smooth joining two trigonometric polynomial patches are deduced.

This paper presents functional identities fulfilled by the forms of the dual sequence of polynomial eigenfunctions of certain differential operators, belonging to the class of the two-orthogonal polynomial sequences. For a specific third-order lowering operator, the correspondent matrix differential identity is deduced, proving that the resultant polynomial sequence is a classical polynomial sequence in the Hahn’s sense. As an example, the vectorial relation fulfilled by the tuple of functionals (u 0, u 1) of a two-orthogonal polynomial sequences analogous to the classical Laguerre polynomials is given, treated in a work of Ben Cheikh and Douak.

The aim of this paper was to establish a new transcendence criterion for a class of quasi-periodic continued fractions of formal power series over a finite field depending only on the length of specific blocks appearing in the sequence of partial quotients.

In this paper, we study the uniform Holder continuity of the generalized Riemann function $${R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}$$ ) defined by $$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$ using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of $${R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}$$ tends to infinity.

This paper contributes a novel framework to obtain the numerical solution of second-order linear hyperbolic telegraph equations. Using special properties of the Euler polynomials for evaluating integral and derivatives, we convert the main problems into their associated Volterra integro-differential equations. The error bound of approximation is also given. Some numerical examples are given to demonstrate the performance of the method.

In this paper we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of the considered problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenelements of A are complete in H.

Let π be a set of primes. A subgroup H of a finite group G is said to be π-S-permutable in G if H permutes with every Sylow q-subgroup of G for all primes q ∈ π. The main aim of this paper is to establish structural results about the normal closure of π-S-permutable subgroups and p-subgroups permuting with all p′-subgroups for a single prime p. Our results stem from a recent article by Isaacs [5] and subsequent discussions with the authors about it.

In this paper, at first we investigate the existence and uniqueness of solutions for a nonlinear fractional integro-differential equation of pantograph type. Then using an algorithm based on Sinc basis functions, we approximate the solution numerically. The single and double exponential transformations are used, respectively. Some test problems have been solved to demonstrate the applicability and efficiency of the proposed methods.

In this paper, we consider a class of singularly perturbed elliptical problems with homogeneous boundary conditions. We consider a regularized iterative method for solving such problems. Convergence analysis and error estimate are derived. The regularization parameter is chosen according to an a priori strategy. We give numerical results to illustrate that the method is implementable compared with numerical methods such as Shishkin and finite element schemes. The study demonstrates that the iterated regularized scheme can be considered as an alternate method for solving singularly perturbed elliptical problems.

This paper is dedicated to presenting and analyzing a numerical algorithm for the solution of even-order boundary value problems. The proposed solutions are spectral and they depend on introducing a new matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the introduced matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm.

In this paper, we extend the iterative expression for the generalized spherical functions associated with the root systems of type A previously obtained (Sawyer in Trans Am Math Soc 349(9):3569–3584, 1997; Sawyer in Q J Math Oxf Ser (2) 50(197):71–86, 1999) beyond regular elements. We also provide a similar expression in the corresponding flat case. From there, we derive a Laplace-type representation for the generalized spherical functions associated with the root systems of type A in the Dunkl setting as well as in the trigonometric Dunkl setting. This representation leads us to describe precisely the support of the generalized Abel transform. Thanks to a recent result of Gallardo and Rejeb (Support properties of the intertwining and the mean value operators in Dunkls analysis. Preprint [hal01331693], pp 1–10, 2016) and Rejeb (Harmonic and subharmonic functions associated with root systems. Mathematics, Universite Francois-Rabelais de Tours, Universite de Tunis El Manar, 2015), which allows us to give the support for the Dunkl intertwining operator.