Ordinary Differential Equation Research Papers (original) (raw)

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Recent papers in Ordinary Differential Equation

We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point... more

We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.

An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of... more

An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of Runge-Kutta-Fehlberg type. Criteria are proposed as well as a method for the realization of the choice of an 'optimum' scheme. The effectiveness of the presented approach to problems in the field of satellite dynamics is illustrated by results from a numerical experiment. These results refer to a case when a satisfactory global stability of the solution for all treated cases is available. The effectiveness has been evaluated as good, especially when solving multi-variable problems in the sphere of simulation modelling.

In epidemiological models of infectious diseases the basic reproduction number mathcalR_0{\mathcal{R}_0}mathcalR_0 is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian... more

In epidemiological models of infectious diseases the basic reproduction number mathcalR_0{\mathcal{R}_0}mathcalR0 is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for mathcalR0{\mathcal{R}_0}mathcalR_0 can be obtained from the digraphs in the reduction process.

It is known from Lie's works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of... more

It is known from Lie's works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of variables, the Riccati equation. For planar complex polynomial differential systems, the classical Darboux integrability theory exists based on the fact that a sufficient number of invariant algebraic curves permits the construction of a first integral or an inverse integrating factor. In this paper, we present a generalization of the Darboux integrability theory based on the definition of generalized cofactors.

Many problems in Biology and Engineering are governed by anisotropic reaction–diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately... more

Many problems in Biology and Engineering are governed by anisotropic reaction–diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction–diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of... more

The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential ...

In this article we use Adomian decomposition method, which is a well-known method for solving functional equations now-a-days, to solve systems of differential equations of the first order and an ordinary differential equation of any... more

In this article we use Adomian decomposition method, which is a well-known method for solving functional equations now-a-days, to solve systems of differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one. Theoretical considerations are being discussed, and convergence of the method for theses systems is addressed. Some examples are presented to show the ability of the method for linear and non-linear systems of differential equations.

In this paper we describe the development of parallel software for the numerical solution of boundary value ordinary differential equations (BVODEs). The software, implemented on two shared memory, parallel architectures, is based on a... more

In this paper we describe the development of parallel software for the numerical solution of boundary value ordinary differential equations (BVODEs). The software, implemented on two shared memory, parallel architectures, is based on a modification of the MIRKDC package, which employs discrete and continuous mono-implicit Runge-Kutta schemes within a defect control algorithm. The primary computational costs are associated with the

This paper is concerned with the development of a control system for tracking of a desired temperature-time profile for accelerated cooling processes in hot rolling of steel. Control of temperature during cooling is essential for... more

This paper is concerned with the development of a control system for tracking of a desired temperature-time profile for accelerated cooling processes in hot rolling of steel. Control of temperature during cooling is essential for achieving desired mechanical and ...

Nonlinear ordinary differential equations with superposition formulas corresponding to the exceptional Lie group G 2ℂ and its two maximal (complex) parabolic subgroups are determined. The G 2-invariance of a third-order skewsymmetric... more

Nonlinear ordinary differential equations with superposition formulas corresponding to the exceptional Lie group G 2ℂ and its two maximal (complex) parabolic subgroups are determined. The G 2-invariance of a third-order skewsymmetric tensor is exploited. The obtained ODEs have polynomial nonlinearities of order 2 in one case and of order 4 in the other.

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