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Content available remote On Two Families of Implicit Interval Methods of Adams-Moulton Type
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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.
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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.
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Content available remote On nonlinear differential equations in generalized Musielak-Orlicz spaces
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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.
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