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A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation

J. Zhu, Y.-T. Zhang, S. A. Newman, M. S. Alber (2009)

Mathematical Modelling of Natural Phenomena

Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently [Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation reaction-diffusion system was developed to describe the interaction...

A fixed point method to compute solvents of matrix polynomials

Fernando Marcos, Edgar Pereira (2010)

Mathematica Bohemica

Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.

A method for determining constants in the linear combination of exponentials

Jiří Cerha (1996)

Mathematica Bohemica

Shifting a numerically given function b 1 exp a 1 t + + b n exp a n t we obtain a fundamental matrix of the linear differential system y ˙ = A y with a constant matrix A . Using the fundamental matrix we calculate A , calculating the eigenvalues of A we obtain a 1 , , a n and using the least square method we determine b 1 , , b n .

A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems

Josef Dalík (1991)

Applications of Mathematics

A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation - ϵ u n + p u ' + q u = f are presented and analyzed theoretically. The positive number ϵ is supposed to be much less than the discretization step and the values of p , q . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

An algebraic theory of order

Philippe Chartier, Ander Murua (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified)...

Computing the differential of an unfolded contact diffeomorphism

Klaus Böhmer, Drahoslava Janovská, Vladimír Janovský (2003)

Applications of Mathematics

Consider a bifurcation problem, namely, its bifurcation equation. There is a diffeomorphism Φ linking the actual solution set with an unfolded normal form of the bifurcation equation. The differential D Φ ( 0 ) of this diffeomorphism is a valuable information for a numerical analysis of the imperfect bifurcation. The aim of this paper is to construct algorithms for a computation of D Φ ( 0 ) . Singularity classes containing bifurcation points with c o d i m 3 , c o r a n k = 1 are considered.

Decentralized control and synchronization of time-varying complex dynamical network

Wei-Song Zhong, Jovan D. Stefanovski, Georgi M. Dimirovski, Jun Zhao (2009)

Kybernetika

A new class of controlled time-varying complex dynamical networks with similarity is investigated and a decentralized holographic-structure controller is designed to stabilize the network asymptotically at its equilibrium states. The control design is based on the similarity assumption for isolated node dynamics and the topological structure of the overall network. Network synchronization problems, both locally and globally, are considered on the ground of decentralized control approach. Each sub-controller...

Discrete evolutions: Convergence and applications

Erich Bohl, Johannes Schropp (1993)

Applications of Mathematics

We prove a convergence result for a time discrete process of the form x ( t + h ) - x ( t ) = h V ( h , x ( t + α 1 ( t ) h ) , . . . , x ( t + α L ( t ) h ) ) t = T + j h , j = 0 , . . . , σ ( h ) - 1 under weak conditions on the function V . This result is a slight generalization of the convergence result given in [5].Furthermore, we discuss applications to minimizing problems, boundary value problems and systems of nonlinear equations.

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