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In this paper, energy slope averaging in the one-dimensional steady gradually varied flow model is considered. For this purpose, different methods of averaging the energy slope between cross-sections are used. The most popular are arithmetic, geometric, harmonic and hydraulic means. However, from the formal viewpoint, the application of different averaging formulas results in different numerical integration formulas. This study examines the basic properties of numerical methods resulting from different types of averaging.
Content available remote On nonlinear differential equations in generalized Musielak-Orlicz spaces
We consider ordinary differential equations u′(t)+(I−T)u(t)=0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.
The evolution of three-dimensional disturbances in a magnetohydrodynamic Couette flow is investigated using the initial-value problem approach. The general solution to the linearized equations governing three-dimensional disturbances is obtained by using two-dimensional Fourier transformation and other transformations rather than the traditional normal mode approach. The governing stability equation is solved using both the Fourier method and perturbation method. In the Fourier approach, the stability equation is reduced to Mathieu's equation and a periodic solution is obtained. Perturbation solution is obtained for small values of Alfvén velocity. Here Green's function method is employed to obtain the time evolution of linearized disturbances. A measure of disturbance energy is obtained in the case of square wave pulse for velocity and the magnetic field. The time evolution of the three-dimensional disturbances is obtained in terms of the two Green's function representations, one in the form of a Fourier sine series and the other in the form of sine hyperbolic functions representing the energy of a single component and the total energy of a single component. It is shown graphically that the total energy and the sum of first five components of energy are similar but are of different magnitudes.
Content available remote On integrated Volterra integrodifferential equation of higher order
In this paper we study the existence and other properties of solutions of a certain iterated Volterra integrodifferential equation of higher order. The tools employed in the analysis are based on application of the Leray-Schauder alternative and a certain integral inequality which provides explicit bound on the unknown function.
Content available remote Approximate controllability for systems described by right invertible operators
In this paper, we deal with the approximate controllability for linear systems described by right invertible operators in an infinite dimensional Banach space.
The paper presents some improvements of the Lohner's algorithm for the rigorous enclosure of trajectories of an ODE.
The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment t = 0 the value of the desired function x(t) is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree N is introduced into the investigation. Then the term "polynomial quasisolution" (PQ-solution) is understood in the sense of appearance of the residual Δ(t) = O(tN), when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis.
This paper is to deal with the controllability of the linear system described by right invertible operators with constrained controls in Banach space.
Content available remote On Two Families of Implicit Interval Methods of Adams-Moulton Type
In our previous paper [1] we have presented implicit interval methods of Adams-Moulton type. It appears that two families of these types of methods exist. We compare both families of methods and present a numerical example.
Content available remote Ordinary differential equations in Banach lattices
In the present paper initial value problems x+f(t,x), t EI=[0,1], x(0)=x_0 in Banach lattices will be investigated with respect to order and topological properties of their solution sets S(f)
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