Martin Hermann - Academia.edu (original) (raw)

Papers by Martin Hermann

In this chapter we consider nonlinear monotone ODEs with Dirichlet boundary conditions. Using a n... more In this chapter we consider nonlinear monotone ODEs with Dirichlet boundary conditions. Using a non-equidistant grid we construct an EDS on a three-point stencil and prove the existence and uniqueness of its solution. Moreover, on the basis of the EDS we develop an algorithm for the construction of a three-point TDS of rank \(\bar{m}\,=\,2[(m\,+\,1)/2],{\rm{where}\,{m}}\,\varepsilon\,\mathbb{N}\) is a given natural number and [·] denotes the entire part of the argument in brackets. We prove the existence and uniqueness of the solution of the TDS and determine the order of accuracy. Numerical examples are given which confirm the theoretical results.

International series of numerical mathematics, 2011

This chapter deals with a generalization of the results from the previous chapter to the case of ... more This chapter deals with a generalization of the results from the previous chapter to the case of systems of second-order ODEs with a monotone operator.

International journal of applied nonlinear science, 2018

In the applications, the homotopy analysis method (HAM) is an often used method to determine an a... more In the applications, the homotopy analysis method (HAM) is an often used method to determine an analytical approximate solution of lower-dimensional nonlinear ordinary differential equations. This approximation consists of an infinite series which depends on an auxiliary real parameter h. This parameter must be adjusted such that the series converges towards the exact solution of the given problem. In this paper we propose a computational approach, which is based on the residual of the truncated series, to determine an optimal value hopt or an optimal region for h. Using the numerical computing environment MATLAB, we describe several possibilities how this approach can be realised. Finally, by means of three examples (an IVP, a two-point BVP, as well as a BVP on an infinite interval) we show how this mathematically sophisticated strategy can be applied and we present the optimal parameter hopt for each example.

In this last chapter we present a variety of mathematical exercises and the corresponding sample ... more In this last chapter we present a variety of mathematical exercises and the corresponding sample solutions by which the reader can test and deepen the knowledge acquired in the previous chapters of this book.

Physica D: Nonlinear Phenomena, Jul 1, 1999

ABSTRACT We study the solution field M of a parameter dependent nonlinear two-point boundary valu... more ABSTRACT We study the solution field M of a parameter dependent nonlinear two-point boundary value problem presented by Troger and Steindl [H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991]. This problem models the buckling of a thin-walled spherical shell under a uniform external static pressure. The boundary value problem is formulated as an abstract operator equation T(x,λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T, we obtain detailed informations about the structure of M. These theoretical results are used to compute efficiently interesting parts of M with numerical standard techniques. Bifurcation diagrams, a stability diagram and pictures of deformed shells are presented.

International Journal of Bifurcation and Chaos, Oct 1, 2015

The spring-mass model is a frequently used gait template to describe human and animal locomotion.... more The spring-mass model is a frequently used gait template to describe human and animal locomotion. In this study, we transform the spring-mass model for running into a boundary value problem and use it for the computation of bifurcation points. We show that the analysis of the region of stable solutions can be reduced to the calculation of its boundaries. Using the new bifurcation approach, we investigate the influence of asymmetric leg parameters on the stability of running. Like previously found in walking, leg asymmetry does not necessarily restrict the range of stable running and may even provide benefits for system dynamics.

Journal of Applied Mathematics and Mechanics, 1995

Computational Methods in Applied Mathematics, 2008

In this paper, we study nonlinear two-point boundary value problems (BVPs) which depend on an ext... more In this paper, we study nonlinear two-point boundary value problems (BVPs) which depend on an external control parameter. In order to determine numeri-cally the singular points (turning or bifurcation points) of such a problem with so-called extended systems and to realize branch switching, some information on the type of the singularity is required. In this paper, we propose a strategy to gain numerically this information. It is based on strongly equivalent approximations of the corresponding Liapunov — Schmidt reduced function which are generated by a simplified Newton method. The graph of the reduced function makes it possible to determine the type of singularity. The efficiency of our numerical-graphical technique is demonstrated for two BVPs.

International series of numerical mathematics, 2011

Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. B... more Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. Both techniques are based on the successive solution of IVPs on small subintervals and are theoretically supported by a posteriori error estimates. However, the advantage of our difference methods is that a unified theory of a priori estimates can be established.

American Journal of Computational and Applied Mathematics, 2013

The Mathematical Gazette, 1988

Computational Methods in Applied Mathematics, 2007

The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the... more The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

The eigenfleld of an in∞ated/de∞ated stretched circular membrane, which is clamped to a circular ... more The eigenfleld of an in∞ated/de∞ated stretched circular membrane, which is clamped to a circular cylindrical cavity fllled by a liquid, is examined. The preprint presents basic mathematical formulations, preliminary mathematical results and proposes a numerical approach.

Exact and Truncated Difference Schemes for Boundary Value ODEs, 2011

One of the important fields of application for modern computers is the numerical solution of dive... more One of the important fields of application for modern computers is the numerical solution of diverse problems arising in science, engineering, industry, etc. Here, mathematical models have to be solved which describe e.g. natural phenomena, industrial processes, nonlinear vibrations, nonlinear mechanical structures or phenomena in hydrodynamics and biophysics. A lot of such mathematical models can be formulated as initial value problems (IVPs) or boundary value problems (BVPs) for systems of nonlinear ordinary differential equations (ODEs). However, it is not possible in general to determine the solution of nonlinear problems in a closed form. Therefore the exact solution must be approximated by numerical techniques.

Exact and Truncated Difference Schemes for Boundary Value ODEs, 2011

Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. B... more Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. Both techniques are based on the successive solution of IVPs on small subintervals and are theoretically supported by a posteriori error estimates. However, the advantage of our difference methods is that a unified theory of a priori estimates can be established.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2011

Computational Methods in Applied Mathematics, 2008

International Journal of Bifurcation and Chaos, 2015

The spring-mass model is a frequently used gait template to describe human and animal locomotion.... more The spring-mass model is a frequently used gait template to describe human and animal locomotion. In this study, we transform the spring-mass model for running into a boundary value problem and use it for the computation of bifurcation points. We show that the analysis of the region of stable solutions can be reduced to the calculation of its boundaries. Using the new bifurcation approach, we investigate the influence of asymmetric leg parameters on the stability of running. Like previously found in walking, leg asymmetry does not necessarily restrict the range of stable running and may even provide benefits for system dynamics.

Nonlinear Ordinary Differential Equations, 2016

Nonlinear ordinary differential equations (ODEs) are encountered in various fields of mathematics... more Nonlinear ordinary differential equations (ODEs) are encountered in various fields of mathematics, physics, mechanics, chemistry, biology, economics, and numerous applications.

In this chapter we consider nonlinear monotone ODEs with Dirichlet boundary conditions. Using a n... more In this chapter we consider nonlinear monotone ODEs with Dirichlet boundary conditions. Using a non-equidistant grid we construct an EDS on a three-point stencil and prove the existence and uniqueness of its solution. Moreover, on the basis of the EDS we develop an algorithm for the construction of a three-point TDS of rank \(\bar{m}\,=\,2[(m\,+\,1)/2],{\rm{where}\,{m}}\,\varepsilon\,\mathbb{N}\) is a given natural number and [·] denotes the entire part of the argument in brackets. We prove the existence and uniqueness of the solution of the TDS and determine the order of accuracy. Numerical examples are given which confirm the theoretical results.

International series of numerical mathematics, 2011

This chapter deals with a generalization of the results from the previous chapter to the case of ... more This chapter deals with a generalization of the results from the previous chapter to the case of systems of second-order ODEs with a monotone operator.

International journal of applied nonlinear science, 2018

In the applications, the homotopy analysis method (HAM) is an often used method to determine an a... more In the applications, the homotopy analysis method (HAM) is an often used method to determine an analytical approximate solution of lower-dimensional nonlinear ordinary differential equations. This approximation consists of an infinite series which depends on an auxiliary real parameter h. This parameter must be adjusted such that the series converges towards the exact solution of the given problem. In this paper we propose a computational approach, which is based on the residual of the truncated series, to determine an optimal value hopt or an optimal region for h. Using the numerical computing environment MATLAB, we describe several possibilities how this approach can be realised. Finally, by means of three examples (an IVP, a two-point BVP, as well as a BVP on an infinite interval) we show how this mathematically sophisticated strategy can be applied and we present the optimal parameter hopt for each example.

In this last chapter we present a variety of mathematical exercises and the corresponding sample ... more In this last chapter we present a variety of mathematical exercises and the corresponding sample solutions by which the reader can test and deepen the knowledge acquired in the previous chapters of this book.

Physica D: Nonlinear Phenomena, Jul 1, 1999

ABSTRACT We study the solution field M of a parameter dependent nonlinear two-point boundary valu... more ABSTRACT We study the solution field M of a parameter dependent nonlinear two-point boundary value problem presented by Troger and Steindl [H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991]. This problem models the buckling of a thin-walled spherical shell under a uniform external static pressure. The boundary value problem is formulated as an abstract operator equation T(x,λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T, we obtain detailed informations about the structure of M. These theoretical results are used to compute efficiently interesting parts of M with numerical standard techniques. Bifurcation diagrams, a stability diagram and pictures of deformed shells are presented.

International Journal of Bifurcation and Chaos, Oct 1, 2015

The spring-mass model is a frequently used gait template to describe human and animal locomotion.... more The spring-mass model is a frequently used gait template to describe human and animal locomotion. In this study, we transform the spring-mass model for running into a boundary value problem and use it for the computation of bifurcation points. We show that the analysis of the region of stable solutions can be reduced to the calculation of its boundaries. Using the new bifurcation approach, we investigate the influence of asymmetric leg parameters on the stability of running. Like previously found in walking, leg asymmetry does not necessarily restrict the range of stable running and may even provide benefits for system dynamics.

Journal of Applied Mathematics and Mechanics, 1995

Computational Methods in Applied Mathematics, 2008

In this paper, we study nonlinear two-point boundary value problems (BVPs) which depend on an ext... more In this paper, we study nonlinear two-point boundary value problems (BVPs) which depend on an external control parameter. In order to determine numeri-cally the singular points (turning or bifurcation points) of such a problem with so-called extended systems and to realize branch switching, some information on the type of the singularity is required. In this paper, we propose a strategy to gain numerically this information. It is based on strongly equivalent approximations of the corresponding Liapunov — Schmidt reduced function which are generated by a simplified Newton method. The graph of the reduced function makes it possible to determine the type of singularity. The efficiency of our numerical-graphical technique is demonstrated for two BVPs.

International series of numerical mathematics, 2011

Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. B... more Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. Both techniques are based on the successive solution of IVPs on small subintervals and are theoretically supported by a posteriori error estimates. However, the advantage of our difference methods is that a unified theory of a priori estimates can be established.

American Journal of Computational and Applied Mathematics, 2013

The Mathematical Gazette, 1988

Computational Methods in Applied Mathematics, 2007

The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the... more The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

The eigenfleld of an in∞ated/de∞ated stretched circular membrane, which is clamped to a circular ... more The eigenfleld of an in∞ated/de∞ated stretched circular membrane, which is clamped to a circular cylindrical cavity fllled by a liquid, is examined. The preprint presents basic mathematical formulations, preliminary mathematical results and proposes a numerical approach.

Exact and Truncated Difference Schemes for Boundary Value ODEs, 2011

One of the important fields of application for modern computers is the numerical solution of dive... more One of the important fields of application for modern computers is the numerical solution of diverse problems arising in science, engineering, industry, etc. Here, mathematical models have to be solved which describe e.g. natural phenomena, industrial processes, nonlinear vibrations, nonlinear mechanical structures or phenomena in hydrodynamics and biophysics. A lot of such mathematical models can be formulated as initial value problems (IVPs) or boundary value problems (BVPs) for systems of nonlinear ordinary differential equations (ODEs). However, it is not possible in general to determine the solution of nonlinear problems in a closed form. Therefore the exact solution must be approximated by numerical techniques.

Exact and Truncated Difference Schemes for Boundary Value ODEs, 2011

Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. B... more Note that the EDS and TDS are very similar to the multiple shooting method [2, 35, 39, 40, 79]. Both techniques are based on the successive solution of IVPs on small subintervals and are theoretically supported by a posteriori error estimates. However, the advantage of our difference methods is that a unified theory of a priori estimates can be established.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2011

Computational Methods in Applied Mathematics, 2008

International Journal of Bifurcation and Chaos, 2015

The spring-mass model is a frequently used gait template to describe human and animal locomotion.... more The spring-mass model is a frequently used gait template to describe human and animal locomotion. In this study, we transform the spring-mass model for running into a boundary value problem and use it for the computation of bifurcation points. We show that the analysis of the region of stable solutions can be reduced to the calculation of its boundaries. Using the new bifurcation approach, we investigate the influence of asymmetric leg parameters on the stability of running. Like previously found in walking, leg asymmetry does not necessarily restrict the range of stable running and may even provide benefits for system dynamics.

Nonlinear Ordinary Differential Equations, 2016

Nonlinear ordinary differential equations (ODEs) are encountered in various fields of mathematics... more Nonlinear ordinary differential equations (ODEs) are encountered in various fields of mathematics, physics, mechanics, chemistry, biology, economics, and numerous applications.