Klaus Gürlebeck

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• Prof. Dr.
• Professor Emeritus at Bauhaus-Universität Weimar

One of the most important renewable energy technologies used nowadays are wind power turbines. In this paper, we are interested in identifying the operating status of wind turbines, especially rotor blades, by means of multiphysical models. It is a state-of-the-art technology to test mechanical structures with ultrasonic-based methods. However, due...

The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling...

This paper presents numerical analysis of the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice. Additionally, to provide estimates in interior and exterior domains, two different regularisations of the discrete fundamental solution are considered. Estimates for the absolute difference and lp‐estimates are cons...

A growing complexity of modern engineering problems emphasises the need to methods working with discrete mathe- matical structures, such as methods of discrete potential theory and discrete function theory, are gaining more and more popularity nowadays. However, solution procedure of boundary value problems by these methods requires at first a cons...

Discrete double-layer potentials are powerful tools to describe the solution of discrete Dirichlet problems for the Laplace equation. In discrete potential theory there are already theorems known, regarding the existence and uniqueness of the solution of Dirichlet problems. The results on the solution of these finite difference Dirichlet problems a...

Rapid advancements of modern technologies put high demands on mathematical modelling of engineering systems. Typically, systems are no longer "simple" objects, but rather coupled systems involving multiphysics phenomena, the modelling of which involves coupling of models that describe different phenomena. After constructing a mathematical model, it...

Micropolar elasticity is a refined version of the classical elasticity. Equations of micropolar elasticity are not given only by a single differential equation w.r.t. a vector field of displacement, but by a coupled system of differential equations connecting fields of displacements and rotations. However, construction of solution methods for bound...

In this paper, we consider homogeneous polynomial solutions for the classical Maxwell equations in a time-space domain with time variable \(t<t_0\), where \(t_0\) is an arbitrary constant. The technique is motivated by the study of the so-called generalized Maxwell operators, which are constructed as conformally invariant differential operators in...

Induction heating is a process of heat generation which uses metal conductors and the Joule effect. The induction heating process has many applications in industry, such as metal melting, preheating for forging operations, hardening, and welding. A model of induction heating is given by a coupled system of partial differential equations relating te...

Holomorphic functions are the key tool to construct representation formulae for the solutions for a manifold of plane problems, especially for the flow of a viscous fluid modelled by the Stokes system. Three-dimensional representation formulae can be constructed by tools of hypercomplex analysis, i.e. by working with monogenic functions playing the...

The p-Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p-Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p-Laplace equation for...

Reliable modelling in structural engineering is crucial for the serviceability and safety of structures. A huge variety of aerodynamic models for aeroelastic analyses of bridges poses natural questions on their complexity and thus, quality. Moreover, a direct comparison of aerodynamic models is typically either not possible or senseless, as the mod...

In a recent paper of ours, we proved that a special “harmonic conjugation” operator is not bounded in weighted Bergman spaces of quaternion‐valued functions in the 3D ball. In the present paper, we prove that, in contrast to the Bergman spaces case, the same operator is bounded in weighted Dirichlet spaces of quaternion‐valued functions in the 3D b...

Holomorphic functions are the key tool to construct representation formulae for the solutions for a manifold of plane problems, especially for the flow of a viscous fluid modelled by the Stokes system. Three-dimensional representation formulae can be constructed by tools of hypercomplex analysis, i.e. by working with monogenic functions playing the...

Micropolar elasticity is a refined version of the classical elasticity. Equations of micropolar elasticity are not given only by a single differential equation w.r.t. a vector field of displacement, but by a coupled system of differential equations connecting fields of displacements and rotations. However, construction of solution methods for bound...

We present in this paper a boundary value problem of induction heating. The idea is to solve this problem numerically by means of finite difference potentials. Due to the geometric restrictions we have to consider a rectangular lattice, and therefore, an extension of existing results in discrete potential theory is required.

The p-Laplace equation is a nonlinear generalization of the well-known Laplace equation. It appears in many problems; for instance in the theory of non-Newtonian fluids and fluid dynamics or in rock fill dam problems, as well as in special problems of image restoration and image processing. The idea in this paper is to apply a hypercomplex integral...

The fundamental problem of a holomorphic extension of functions of one complex variable has a great importance in applications. This problem is well studied in the classical complex analysis, particularly, a lot of results related to the conditions for a holomorphic extension of a function into a domain from its boundary values are available. By th...

Discrete function theory is a natural extension of the continuous theory to functions defined on lattices. The idea of the discrete function theory is to work directly with discretised domains (lattices) and to transfer all important properties from the continuous case to the discrete level. In the field of boundary value problems it is more benefi...

In this article we study a Dirichlet problem for a hypercomplex Beltrami equation. We prove the existence of a unique solution of the problem and give a representation formula for the solution.

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first...

Interpolation is an important tool for many practical applications, and very often it is beneficial to interpolate not only with a simple basis system, but rather with solutions of a certain differential equation, e.g. elasticity equation. A typical example for such type of interpolation are collocation methods widely used in practice. It is known,...

Interpolation is an important tool for many practical applications, and very often it is beneficial to interpolate not only with a simple basis system, but rather with solutions of a certain differential equation, e.g. elasticity equation. A typical example for such type of interpolation are collocation methods widely used in practice. It is known,...

The fundamental problem of a holomorphic extension of functions of one complex variable has a great importance in applications. This problem is well studied in the classical complex analysis, particularly, a lot of results related to the conditions for a holomorphic extension of a function into a domain from its boundary values are available. By th...

In this paper we present error estimates for a continuous coupling of an analytical and a numerical solution for a boundary value problem with a singularity. A solution of the Lame-Navier equation with a singularity caused by a crack is considered as an example. The analytical solution near a singularity is constructed by using complex function the...

We discuss computational aspects of the inverse and ill-posed problem of identifying residual stresses in steel structures under thermal loading. This corresponds to an inverse source problem in linear thermo-elasticity. The studies aim in investigating whether thermal loadings for the excitation of structures are sufficient in order to detect reli...

One of the open problems in hypercomplex analysis is the interpolation of monogenic functions by monogenic polynomials. We consider the case of monogenic functions defined in a domain of with values in the algebra of quaternions. The idea is to interpolate these functions by a special system of monogenic polynomials, the so-called pseudo complex p...

Antiplane stress state of a piecewise-homogeneous elastic body with a semi-infinite crack along the interface is considered. The longitudinal displacements along one of the crack edges on a finite interval, adjacent to the crack tip, are known. Shear stresses are applied to the body along the crack edges and at infinity. The problem reduces to a Ri...

In this short section we shall introduce a class of mappings in \(\mathbb{C} \;\mathrm{and}\; \mathbb{B}\) named after the German mathematician AUGUST FERDINAND MÖBIUS (1790–1868). In \(\it C l(n)\) this is also possible, but it is a bit more difficult, the reader is referred to our book [118].

In the classical two-dimensional Vekua theory the so-called T-operator plays an essential role. This operator is nothing else than a two-dimensional weakly singular integral operator over a domain in the complex plane, which is a right inverse to the Cauchy–Riemann operator.

The classical Maxwell equations were discovered in the second half of the 19th century as a result of the stormy development of research in electricity and magnetism. The study of these questions has attracted generations of physicists and mathematicians, yet some of their secrets are still hidden. Adopting a historical point of view, we will use s...

In his famous dissertation from 1851, Bernhard Riemann (1826–1866) formulated the following problem: In a given bounded domain of the complex plane, determine a holomorphic function, if a relation is prescribed between the boundary values of its real part and its imaginary part. In the case of a linear relation this problem was first considered in...

Within this book we shall use the well-known complex numbers in the plane, the quaternions in three and four dimensions, and Clifford numbers in higher dimensions. The definition for real Clifford numbers can be seen as a basis for quaternions and complex numbers.

In this chapter, we consider classes of fluid flow problems on the sphere and in ball shells with given initial and boundary value conditions. We focus our attention on the corresponding Navier-Stokes equations and their linearizations – the socalled forecasting equations. Shallow water equations are rather similar to this set of equations, and we...

Hypercomplex Fourier transforms, i.e., quaternion, Clifford, and geometric algebra Fourier transforms (QFT, CFT, GAFT) [50,145,147,154] have proven very useful tools for applications in fields like non-marginal color image processing, image diffusion, electromagnetism, multi-channel processing, vector field processing, shape representation, linear...

In this chapter we consider scales of spaces of holomorphic quaternion-valued functions which are generalizations of complex versions of the Bloch space and Dirichlet space. Firstly, we introduce these spaces over the unit disc in the complex plane. After that we formulate possible generalizations for quaternion-valued holomorphic functions in the...

A vector field is a notion in multidimensional analysis. Specifically, to point x in a domain \(G\;\subset\;\mathbb{R}^n\) one assigns a vector \({\bf{ u}}(x)\;=\;(u_1,\ldots,u_n)^T\). Vector fields play an important role in the description of physical relevant equations in the plane and space. In particular, vector fields describe the intensity an...

In 1812 D. Poisson discovered that for many applied problems the Laplace equation is only valid outside the relevant domain G. In the journal Bulletin de la Societé Philosphique he published one year later the first paper on an equation of the type \(-\Delta u\;=\;f \quad \mathrm{in} \;G\;\subset\;I\!\!R^n ,\) which now bears his name.

In this chapter we will study initial-boundary value problems and their treatment by methods of quaternionic analysis in combination with classical analytic numerical techniques. We start with a brief discussion of strategies for the treatment of time-dependent parabolic problems. Three methods should be considered here: the horizontal method of li...

Solution of any engineering problem starts with a modelling process, which typically involves a choice among different kinds of models. To create a realistic model, one has to think carefully about the modelling process. Particularly in the case of coupled problems when several models are coupled together to represent a given physical phenomenon. T...

Monogenic functions are typically approximated by help of monogenic polynomials. Different systems of monogenic polynomials have been developed by several authors in the last years. One of available constructions are the so called system of Pseudo-Complex Polynomials (PCP). PCP are 3D monogenic polynomials which have a structure similar to integer...

Solution of any engineering problem starts with a modelling process which typically involves a choice among different kinds of models. To create a realistic model one has to think carefully about the modelling process. Particularly in the case of coupled problems when several models are coupled together to represent a given physical phenomenon. Thi...

The p-Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p-Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p-Laplace equation for...

This book presents applications of hypercomplex analysis to boundary value and initial-boundary value problems from various areas of mathematical physics. Given that quaternion and Clifford analysis offer natural and intelligent ways to enter into higher dimensions, it starts with quaternion and Clifford versions of complex function theory includin...

A short overview of results of the thesis ”Evaluation of the coupling between an analytical and a numerical solution for boundary value problems with singularities”.

Generalizing the complex one-dimensional function theory the class of quaternion-valued functions, defined in domains of \(\mathbb{R}^{4}\), will be considered. The null solutions of a generalized Cauchy–Riemann operator are defined as the \(\mathbb{H}\)-holomorphic functions. They show a lot of analogies to the properties of classical holomorphic...

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In R3, similar formulae were already developed in recent papers, using quaternionic monogenic functions as a...

In complex analysis, every harmonic function admits a decomposition as a sum of a holomorphic and an anti-holomorphic function. However, this fact does not hold for paravector-valued harmonic functions, or so-called 𝒜-valued harmonic functions, in quaternion function theory. In previous articles, the authors proved that by taking into account ψ-hy...

This overview gives an insight in the new field of hypercomplex analysis in relation to harmonic analysis. The algebra of complex numbers is replaced by the non-commutative algebra of real quaternions or by Clifford algebras. This contribution is focused on the presentation of an operator calculus on the sphere as well as the discussion of monogeni...

We discuss the inverse and ill-posed problem of identifying residual stresses in steel structures. Therefore, steel specimen are subjected to thermal loadings. Due to these loadings deformations occure which are recorded by high resolution laser scanners. The latter serve as input for the inverse problem, where residual stresses are to be recovered...

The main goal of this paper is to improve the theoretical basis of coupling of an analytical and a finite element solution to the Lame-Navier equations in case of singularities caused by a crack. The main interest is to construct a continuous coupling between two solutions through the whole interaction interface. To realize this continuous coupling...

The purpose of this paper is to prove an interpolation theorem which arises in a method of coupling of a finite element and an analytical solution for boundary value problems with singularities.

Additive decompositions of harmonic functions play an important role in function theory and for the solution of partial differential equations. One of the best known results is the decomposition of harmonic functions as a sum of a holomorphic and an anti-holomorphic function. This decomposition can be generalized also to the analysis of quaternion-...

Additive decompositions of harmonic functions play an important role in function theory, and in its application to the solution of partial differential equations. One of the most fundamental results is the decomposition of a harmonic function into a sum of a holomorphic and an anti-holomorphic function. This decomposition can be generalized to the...

Abstract One of the most fruitful and elegant approach (known as Kolosov-Muskhelishvili formulas) for plane isotropic elastic problems is to use two complex-valued holomorphic potentials. In this paper, the algebra of real quaternions is used in order to propose in three dimensions, an extension of the classical Muskhelishvili formulas. The starti...

In this paper, the thermo-poroelasticity theory is used to investigate the quasi-static response of temperatures, pore pressure, stress, displacement, and fluid flux around a cylindrical borehole subjected to impact thermal and mechanical loadings in an infinite saturated poroelastic medium. It has been reported in literatures that coupled flow kno...

A complete orthogonal system for the monogenic $L_2$-space consisting of solid oblate spheroidal monogenics $\{\S^m_n\}$ in $\R^3$ is constructed by means of harmonic functions. The lack of symmetry leads to some differences compared with the orthogonal polynomial Appell system $\{A^l_k\}$ for spherical domains. Explicit representation formulae of...

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In this paper, we study ψ-hyperholomorphic functions in R3 where ψ ⊂ A is a structural set. A geometric mapping property which characterizes ψ-hyperholomorphic mappings is discussed. With this, the composition of monogenic functions and Möbius transformations are revisited. Besides, we find a new decomposition of A-valued harmonic functions by mean...

The main goal of this paper is to discuss the convergence of a mixed method for coupling analytical and finite element solutions for a boundary value problem with a singularity. As an example, we consider a crack-tip problem from the linear elastic fracture mechanics in a plane. The convergence proof is done by adaptation of the classical theory...

In continuation of recent studies, we discuss two constructive approaches for the generation of harmonic conjugates to find null solutions to the Riesz system in . This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. Our first algorithm for harmonic conjugates is based on special systems...

In this article, we first give the Plemelj formula for functions with parameter by following the classical method. Then by using the higher order Cauchy integral representation formulas, some properties for harmonic functions and bi-harmonic functions are presented, for example, the mean value theorem, the Painlevé theorem, etc. Finally, we conside...

This paper investigates the physical state of a concrete hinge in the case of an existing crack by combining analytical and finite element solutions. In a small region around the crack-tip in the hinge throat the analytical solution is constructed to obtain the more accurate description of displacement and stress fields. For the remaining parts of...

The classical theorem of Bloch (1924) asserts that if $f$ is a holomorphic function on a region that contains the closed unit disk $|z|\leq 1$ such that $f(0) = 0$ and $|f'(0)| = 1$, then the image domain contains discs of radius $\frac{3}{2}-\sqrt{2} > \frac{1}{12}$. The optimal value is known as Bloch's constant and $\frac{1}{12}$ is not the best...

This overview gives an insight in the new field of hypercomplex analysis in relation to harmonic analysis. The algebra of complex numbers is replaced by the non-commutative algebra of real quaternions or by Clifford algebras. This contribution is focused on the presentation of an operator calculus on the sphere as well as the discussion of monogeni...

The main goal of this paper is to solve a problem of coupling between analytical and finite element solutions for a boundary value problem with a singularity. We present the improvement of results from previous research by introducing the new interpolation function, which allows us to overcome the problem of discontinuity on the boundary between tw...

This note announces some results that will be presented in the forthcoming paper [10]. In continuation to these studies we discuss a constructive approach for the generation of harmonic conjugates to find nullsolutions to the Riesz system in R3. This class of solutions coincides with the subclass of monogenic functions with values in the reduced qu...

M-conformal mappings are given by a subclass of monogenic quasiconformal mappings. We compare some equivalent definitions and study then how an M-conformal mapping does change the solid angle in the special case of mappings from R 3 to R 3.

The main aim of this paper is to recall the notion of Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac and the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT constru...

The idea of this paper is to discuss some problems in coupling of analytical solutions and the finite element method for boundary value problems with a singularity. As an example we consider a problem from fracture mechanics in the plane.

This paper is focused on the first numerical tests for coupling between analytical solution and finite element method on the example of one problem of fracture mechanics. The calculations were done according to ideas proposed in [1]. The analytical solutions are constructed by using an orthogonal basis of holomorphic and anti-holomorphic functions....

The classical theorem of Bloch (1924) asserts that if $f$ is a holomorphic function on a region that contains the closed unit disk $|z|\leq 1$ such that $f(0) = 0$ and $|f'(0)| = 1$, then the image domain contains discs of radius $3/2-\sqrt{2} > 1/12$. The optimal value is known as Bloch's constant and 1/12 is not the best possible. In this paper w...

Based on the theory of porous medium, the general analytical solutions of quasi-static response fulfilling thermal non-equilibrium condition were obtained in Laplace transform space for one dimensional double-layered half space. The numerical approximations for the solutions were also compared. Through the analysis and calculation, the following co...

By using the solution to the Helmholtz equation Δu − λu = 0 (λ ≥ 0), the explicit forms of the so-called kernel functions and the higher order kernel functions are given. Then by the generalized Stokes formula, the integral representation formulas related with the Helmholtz operator for functions with values in C(V 3,3) are obtained. As application...

Abstract unavailable.

In this paper we study a system of biharmonic equations coupled by the boundary conditions. These boundary conditions contain some combinations of the values, div, curl, and grad of the solution. It is the aim of the paper also to demonstrate the application of Clifford analytic methods developed for second order elliptic problems to the solution o...

With the aid of Volterra multiplier, we study ecological equations for both tree system and cycle system. We obtain a set of sufficient conditions for the ultimate boundedness to nonautonomous n-dimensional Lotka-Volterra tree systems with continuous time delay. The criteria are applicable to cooperative model, competition model, and predator-prey...

The main goal of this article is to generalize Hadamard's real part theorem and invariant forms of Borel–Carathéodory's theorem from complex analysis to solutions of the Riesz system in the three-dimensional Euclidean space in the framework of quaternionic analysis.

We construct a stratified L-fuzzy topology from a given L-fuzzy topology, where L is a complete residuated lattice. We study the relationships between the L-fuzzy topology and the stratification of it.

The main goal of the paper is to deal with a special orthogonal system of polynomial solutions of the Riesz system in ℝ3. The restriction to the sphere of this system is analogous to the complex case of the Fourier exponential functions {e in θ } n≥0 on the unit circle and has the additional property that also the scalar parts of the polynomials f...

The main aim of this paper is to recall the notion of the Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac or the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT cons...

We propose a constructive method for the generation of harmonic conjugates to find solutions to the Riesz system from a harmonic square integrable function. This method does not need the solution of a Poisson equation nor any integration.

This article deals with some classes of fluid flow problems under given initial-value and boundary-value conditions. Using a quaternionic operator calculus, representations of solutions are constructed. For the case of a bounded velocity, a numerically stable semi-discretization procedure for the solution of the problem is presented.

In this paper we study the description of monogenic functions in R3 by their geometric mapping properties. The monogenic functions are considered at first as general quasi-conformal mappings. We consider the local mapping properties of a monogenic function and show that this class of functions can be defined as a special subclass of quasiconformal...

The Bohr theorem states that any function $f(z) = \sum_{n=0}^{\infty} a_{n} z^{n}$, analytic and bounded in the open unit disk, obeys the inequality $\sum_{n=0}^{\infty} |a_{n}| |z|^{n} < 1$ in the open disk of radius 1/3, the so-called Bohr radius. Moreover, the value 1/$ cannot be improved. In this paper we review some results related to this the...

Main goal of this paper is to generalize Hadamard's real part theorem and invariant forms of Borel-Carath\'eodory's theorem from complex analysis to solutions of the Riesz system in the three-dimensional Euclidean space in the framework of quaternionic analysis.

In this paper we study the harmonic conjugation problem in weighted Bergman spaces of quaternion-valued functions on the unit ball. For a scalar-valued harmonic function belonging to a Bergman space, harmonic conjugates in the same Bergman space are found.

The main goal of this paper is to generalize Bohr’s phenomenon from complex one-dimensional analysis to the three-dimensional Euclidean space in the framework of quaternionic analysis.

The great importance of conformal mapping methods in complex analysis suggests to look for a suitable higher dimensional analogue in the context of quaternionic analysis. As in the complex case, a serious study of the geometric properties of generalized holomorphic functions (or monogenic functions) requires to consider at first the local behavior...

Nonlinear analyses are characterised by approximations of the fundamental equa-tions in different quality. On a simple truss structure the influence of different formulations concerning the nonlinear kinematic equation is investigated using global deformation and en-ergy formulations as an indicator for the approximation quality.

In this paper, we present some new recurrence formulae to generate orthonormal systems of inner and outer solid spherical monogenics by means of a monogenic subset which is isomorphic to the anti-holomorphic z-powers of the complex one-dimensional case. The resulting systems are complete in the space of square integrable H-valued functions and have...