We present numerical solutions of the semi-empirical model of self-propagating fluid pulses (auto-pulses) through the channel simulating an artificial artery. The key mechanism behind the model is the active motion of the walls in line with the earlier model of Roberts. Our model is autonomous, nonlinear and is based on the partial differential equation describing the displacement of the wall in time and along the channel. A theoretical plane configuration is adopted for the walls at rest. For solving the equation we used the One-dimensional Integrated Radial Basis Function Network (1D-IRBFN) method. We demonstrated that different initial conditions always lead to the settling of pulse trains where an individual pulse has certain speed and amplitude controlled by the governing equation. A variety of pulse solutions is obtained using homogeneous and periodic boundary conditions. The dynamics of one, two, and three pulses per period are explored. The fluid mass flux due to the pulses is calculated.
We analyse a model of the fluid flow between elastic walls simulating arteries actively interacting with the blood. Lubrication theory for the flow is coupled with the pressure and shear stress from the walls. The resulting nonlinear partial differential equation describes the displacement of the walls as a function of the distance along the flow and time. The equation is solved numerically using the one-dimensional integrated radial basis function network method. A solution in the form of a self-sustained train of pulses is obtained. Numerical experiments demonstrate the process of formation of the train from randomly chosen initial conditions. Dependence of the pulses on the boundary conditions is explored.
In this article, an efficient analytical technique, called Sumudu variational iteration method (SVIM), is used to obtain the solution of fractional partial differential equations arising in mathematical physics. The fractional derivatives are described in terms of Caputo sense. This method is the combination of the Sumudu transform (ST) and variational iteration method (VIM). The solution of the suggested technique is represented in a series form, which is convergent to the exact solution of the given problems. Furthermore, the results of the present method have shown close relations with the exact approaches of the investigated problems. Illustrative examples are discussed, showing the validity of the current method. The attractive and straightforward procedure of the present method suggests that this method can easily be extended for the solutions of other nonlinear fractional-order partial differential equations.
Slow variations in the phase of oscillators coupled by diffusion is generally described by a partial differential equation involving infinitely many terms. We consider the case of nonlocal coupling and numerically evaluate the ranges of parameters leading to different forms of a finite truncation of the equation, namely a form based on nonlinear excitation and a form based on linear excitation-the Kuramoto-Sivashinsky equation.
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