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Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables)
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
Recall that, in the language of mathematics, an infinite series is a summation with infinitely many terms. For example, ∑k=1∞1k=1+12+13+14+15+⋯
Modelling and simulation of fluid flow through stenosis and aneurysm blood vessel: a computational hemodynamic analysis
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
J. V. Ramana Reddy, Hojin Ha, S. Sundar
In the HPM method, the solution will obtain as the summation of an infinite series, which converges in general. However, they converge will also be checked through a procedure that is reported in the present article. The method is introduced by He (2004). The accuracy of the solution is obtained by taking more terms in the infinite summation. The constructive approach has been used to get the solution of the modelled problem like the solution starts from the linear system solution, and step by step, the system’s complexity will be addressed. Elnaqeeb et al. (2016) studied the nanofluid model and implemented the homotopy perturbation method. In this article, it has been shown the HPM results of Elnaqeeb et al. (2016) studied problem well agrees with the analytical results obtained by Elnaqeeb et al. (2016).
Differential flatness based design of robust controllers using polynomial chaos for linear systems
Published in International Journal of Control, 2023
Oladapo Ogunbodede, Tarunraj Singh
Polynomial Chaos approximation is represented as an infinite series in the limit and has the form where are the polynomial basis functions and are the associated coefficients which can be time varying when developing surrogate models for dynamic systems. Depending on the level of accuracy required, the expansion is terminated with a finite number of terms such that To illustrate the steps needed to derive the surrogate model using polynomial chaos, a system with a scalar random variable and other deterministic system parameters and inputs denoted as is given as: where Y is the output variable of interest. The output is expanded as a polynomial in the random variable as: The choice of basis function for is dependent on the distribution of . For a system with more than one random variable of interest, the final set of basis functions are derived from a tensor product of the univariate polynomial basis.
Using modified nodal analysis in cavity-mode resonances printed circuit board power bus structures with the segmentation method
Published in Journal of the Chinese Institute of Engineers, 2019
Impedance matrices of a power bus can be measured using the modal expansion method (Lei, Techentin, and Gilbert 1999). In our expansion method, the bottom and top planes of the power bus structure are assumed as perfect electric conductors and perfect magnetic conductors (PMCs) around four sidewalls. Then, we presumed that the dielectric substrate was electrically thin. Finally, we considered the transverse magnetic modes in the PCBs. In our expansion method, the two-port impedance matrix could be computed through double-infinite-series summation. Figure 1 displays a rectangular power bus on a substrate. The side lengths of the power bus are Wx and Wy, and the thickness of the power bus is h. In the cavity model of the PCB, the side lengths Wx and Wy >> h, h ≪ λ (the wavelength), and the main field is composed of Ez, Hx, and Hy. Here, E denotes the electric field, H denotes the magnetic field, and the subscripts indicate the field directions. Furthermore, the four sidewalls consist of PMCs. The external port impedance matrix can be obtained using the method of (Liu, Oo, and Li 2006).