Papers by Denis Constales
Archiv der Mathematik, 2006
This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dira... more This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. 1. Introduction. Eigensolutions to higher dimensional Dirac type systems play an important role in mathematics, physics and the applied sciences and are studied by numerous authors. One very elegant way to describe higher dimensional Dirac equations is the setting of Clifford algebras, which endow finite dimensional quadratic vector spaces with an additional multiplication operation. This in turn permits us to describe the Dirac operator as one compact entity. The use of Clifford algebras, in fact, permitted a very clear way of describing the structure of the solutions to the associated systems of PDE's. Its associated analysis, often called Clifford analysis, provides powerful methods to solve the related boundary value problems. Following e.g. [12], [11], [9] and others, any eigensolution of the Euclidean Dirac operator in R n to an arbitrary non-zero complex eigenvalue λ ∈ C\{0} can be described in the simple Mathematics Subject Classification (2000): 30G35, 33C10. 1) Financial support from the R&D unit Matemática a Aplicações (UIMA) of the University of Aveiro, sponsored through the Portuguese Foundation for Science and Technology (FCT) and co-financed by the European Community fund FEDER, gratefully acknowledged. 2) Financial support from BOF/GOA 01GA0405 of Ghent University gratefully acknowledged.
Mathematica Bohemica, 2001
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Journal of Mathematical Physics, 2020
In this paper we determine solutions for the Lévy-Leblond operator or a parabolic Dirac operator ... more In this paper we determine solutions for the Lévy-Leblond operator or a parabolic Dirac operator in terms of hypergeometric functions and spherical harmonics. We subsequently generalise our approach to a wider class of Dirac operators depending on 4 parameters.
Journal of Mathematical Analysis and Applications, 2018
In this paper, a new method is developed to obtain explicit and integral expressions for the kern... more In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the (κ, a)-generalized Fourier transform for κ = 0. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag-Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the (κ, a)-generalized Fourier transform are obtained as well.
Mathematical Methods in the Applied Sciences, 2016
In this paper, we introduce a new generalization of the Helgason-Fourier transform using the angu... more In this paper, we introduce a new generalization of the Helgason-Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role.
In this paper, we propose an efficient method for the identification of soil parameters in unsatu... more In this paper, we propose an efficient method for the identification of soil parameters in unsaturated porous media, using measurements from infiltration experiments. The infiltration is governed by Richard's nonlinear equation expressed in terms of effective saturation. The soil retention and hydraulic permeability functions are expressed using the Van Genuchten-Mualem ansatz in terms of the soil parameters. The mathematical algorithm is based on a transformation of Richard's equation to a system of ordinary differential equations completed by the governing equation for the movement of the wetness front. This system can be efficiently solved by specialized packages for the solution of stiff systems of ODE. The unknown parameters are determined using the optimization approach of minimizing a cost functional for the discrepancy between the model output and the measurements. The gradient and Hessian of the solution with respect to soil parameter vector are determined using automatic differentiation. Several numerical experiments are included.
Applied Mathematics and Computation, 2006
We present a new numerical scheme for the approximate solution of the Black-Scholes partial diffe... more We present a new numerical scheme for the approximate solution of the Black-Scholes partial differential equation describing the pricing of American call options. The corresponding mathematical problem is a free boundary problem for a convection-diffusion equation. The concept of our numerical solution is suitable for sensitivity analysis with respect to all parameters, for which we have used automatic differentiation implemented in the ODE solver LSODA-C.
AMERICAN MATHEMATICAL MONTHLY, 1998
SASVARI, Z, and Denis Constales. 1998. “The Remainder in the Logarithm Series.” American Mathemat... more SASVARI, Z, and Denis Constales. 1998. “The Remainder in the Logarithm Series.” American Mathematical Monthly 105 (1): 77–78. ... SASVARI, Z., & Constales, D. (1998). The remainder in the logarithm series. AMERICAN MATHEMATICAL MONTHLY, 105(1), 77–78. ... SASVARI Z, Constales D. The remainder in the logarithm series. AMERICAN MATHEMATICAL MONTHLY. 1998;105(1):77–8. ... SASVARI, Z, and Denis Constales. “The Remainder in the Logarithm Series.” AMERICAN MATHEMATICAL MONTHLY 105.1 (1998): 77–78. Print.
Ghent University Ghent University Academic Bibliography. ...
The American Mathematical Monthly, 2001
The American Mathematical Monthly, 1998
International Journal of Chemical Reactor Engineering, 2018
Homotopy techniques in nonlinear problems are getting increasingly popular in engineering practic... more Homotopy techniques in nonlinear problems are getting increasingly popular in engineering practice. The main reason is because the homotopy method deforms continuously a difficult problem under study into a simple problem, which then can be easy to solve. This study explores several homotopy approaches to obtain semi- or approximate analytical solutions for various cases involving mechanistic phenomena such as aggregation and breakage. The well-established approximate analytical methods namely, the Homotopy Perturbation Method (HPM), the Homotopy Analysis Method (HAM), and the more recent forms of homotopy approaches such as the Optimal Homotopy Asymptotic Method (OHAM) and the Homotopy Analysis Transform Method (HATM) have been used to solve using a general mathematical framework based on population balances. In this study, several test cases have been discussed such as conditions in which the aggregation kernel is not only constant, but also sum or product dependent. Furthermore c...
AIP Conference Proceedings, 2012
The present paper is a continuation of our paper [4]. The problem to be studied is the non-impact... more The present paper is a continuation of our paper [4]. The problem to be studied is the non-impact of algebraic property on the convergence properties of the μ-th root base of special monogenic polynomials. These convergence properties are the investigation of the relation between the effectiveness in closed balls of a given base and that of its μ-th root base, and also the relation between their rate of increase. Our result which is concerned with effectiveness in closed balls is quite different from its corresponding one for the square ...
In this paper we study the structure of the solutions to higher dimensional Dirac type equations ... more In this paper we study the structure of the solutions to higher dimensional Dirac type equations generalizing the known λ-hyperholomorphic functions, where λ is a complex parameter. The structure of the solutions to the system of partial differential equations (D- λ) f=0 show a close connection with Bessel functions of first kind with complex argument. The more general system of partial differential equations that is considered in this paper combines Dirac and Euler operators and emphasizes the role of the Bessel functions. However, contrary to the simplest case, one gets now Bessel functions of any arbitrary complex order.
The quaternionic operator calculus can be applied very elegantly to solve many important boundary... more The quaternionic operator calculus can be applied very elegantly to solve many important boundary value problems arising in fluid dynamics and electrodynamics in an analytic way. In order to set up fully explicit solutions. In order to apply the quaternionic operator calculus to solve these types of boundary value problems fully explicitly, one has to evaluate two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. Recently the theory of quaternionic holomorphic multiperiodic functions and automorphic forms provided new impulses to set up explicit representation formulas for large classes of hyperbolic polyhedron type domains. These include block shaped domains, wedge shaped domains (with or without additional rectangular restrictions) and circular s...
In this paper we consider the time independent Klein-Gordon equation on some conformally flat 3-t... more In this paper we consider the time independent Klein-Gordon equation on some conformally flat 3-tori with given boundary data. We set up an explicit formula for the fundamental solution. We show that we can represent any solution to the homogeneous Klein-Gordon equation on the torus as finite sum over generalized 3-fold periodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore we prove Cauchy and Green type integral formulas and set up a Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator. These in turn are used to set up explicit formulas for the solution to the inhomogeneous version of the Klein-Gordon equation on the 3-torus.
The Journal of Geometric Analysis, 2021
The monogenic Hua-Radon transform is defined as an orthogonal projection on holomorphic functions... more The monogenic Hua-Radon transform is defined as an orthogonal projection on holomorphic functions in the Lie sphere. Extending the work of Sabadini and Sommen, J. Geom. Anal., 29 (2019), 2709-2737, we determine its reproducing kernel. Integrating this kernel over the Stiefel manifold yields a linear combination of the zonal spherical monogenics. Using the reproducing properties of those monogenics we obtain an inversion for the monogenic Hua-Radon transform. Contents 1. Introduction 1 2. Preliminaries 2 2.1. Clifford algebras 2 2.2. Clifford analysis 3 2.3. The Lie ball 4 3. The monogenic Hua-Radon transform 5 4. Technical lemmas 9 5. The kernel of the monogenic Hua-Radon transform 12 6. Inversion of the monogenic Hua-Radon transform 16 7. Conclusions 22 Appendix A. Computation of λ α k from Theorem 5.2 23 Appendix B. Computation of the constants γ α,k from Proposition 6.4 26 Acknowledgements 29 References 29
Complex Variables and Elliptic Equations, 2016
In this paper, we study the generalized Clifford-Fourier transform introduced in [6] using the La... more In this paper, we study the generalized Clifford-Fourier transform introduced in [6] using the Laplace transform technique. We give explicit expressions in the even dimensional case, we obtain polynomial bounds for the kernel functions and establish a generating function.
Journal of Fourier Analysis and Applications, 2016
In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fouri... more In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.
Computational Technologies for Fluid/Thermal/Structural/Chemical Systems With Industrial Applications, Volume 2, 2004
A pointwise simultaneous solution algorithm based on dual time stepping was developed by De Wilde... more A pointwise simultaneous solution algorithm based on dual time stepping was developed by De Wilde et al. (2002). With increasing grid aspect ratios, the efficiency of the point method quickly drops. Most realistic flow cases, however, require high grid aspect ratio grids, with the highest grid spacing in the streamwise direction. In this direction, the stiffness is efficiently removed by applying preconditioning (Weiss and Smith, 1995). In the direction perpendicular to the stream wise direction, stiffness remains because of the viscous and the acoustic terms. To resolve this problem, a line method is presented. All nodes in a plane perpendicular to the stream wise direction, a so-called line, are solved simultaneously. This allows a fully implicit treatment of the fluxes in the line, removing the stiffness in the line wise directions. Calculations with different grid aspect ratios are presented to investigate the convergence behavior of the line method. The line method presented is...
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Papers by Denis Constales