Natural convective heat **transfer** in porous media has received considerable attention during the past few decades. This interest can be attributed to its wide range of applications in ceramic processing, nuclear reactor cooling **system**, crude oil drilling, chemical reactor design, ground water pollution and ﬁltration processes. External natural convection in a porous medium adjacent to heated bodies was analyzed by Nield and Bejan [39], Merkin [33, 34], Minkowycz and Cheng [35, 36, 37], Pop and Cheng [12, 47], Ingham and Pop [29]. All through these studies, it is assumed that the boundary layer approximations are applicable and the coupled set of governing equations are solved by numerical methods. Also, [1, 51] worked out this problem. Parand [40] Compared two common collocation approaches based on radial basis **functions** for the case of heat **transfer** equations arising in porous medium.

Show more
16 Read more

The purpose of the paper though is not to discuss quadrature in particular. It is just an example application that does not require much extra introduction of new terminology and notation. The main purpose however is to give a general framework on which to build for the many applications of ORFs. Just like orthogonal polynomials are used in about every branch of mathematics, ORFs can be used with the extra freedom to exploit the location of the poles. For example, it can be shown that the ORFs can be used to solve multipoint moment problems as well as more general **rational** interpolation problems where locations of the poles inside and outside the circle are important for the engineering applications like **system** identification, model reduction, filtering, etc. When modelling the **transfer** function of a linear **system**, poles should be chosen inside as well as outside the disk to guarantee that the transient as well as the steady state of the **system** is well modelled. It would lead us too far to also include the interpolation properties of multipoint Pad´ e approximation and the related applications in several branches of engineering. We only provide the basics in this paper so that it can be used in the context of more applied papers.

Show more
49 Read more

A **transfer** course designed for students preparing for trigonometry, statistics, or calculus. The focus is on the analysis of piecewise, polynomial, **rational**, exponential, logarithmic, power **functions** and their properties. These **functions** will be explored symbolically, numerically and graphically in real life applications and mathematical results will be analyzed and interpreted in the given context. The course will also include transformations, symmetry, composition, inverse **functions**, regression, the binomial theorem and an introduction to sequences and series.

Show more
second kind. Next, under certain conditions on the poles in A, we prove that the ϕ (1) j\1 form an orthonormal **system** of ratio- nal **functions** with respect to a Hermitian positive-definite inner product. Finally, we give a relation between associated **rational** **functions** of different order, independent of whether they form an orthonormal **system**.

Remark 3.1. It is worth to mention here that, the **system** (3.10) can be solved by applying an iterative method, like the Newton’s method. Having solved the nonlinear **system** (3.10) analytically for very small N , say N = 1, we see that some components of solution of this **system** are close to zero. Therefore, for larger values of N the initial guesses for a i ’s and b i ’s of the Newton’s method can be estimated by values

11 Read more

In an earlier paper [3] the authors gave another construction of the general **rational** Γ-inner function h = (s, p) of degree n, starting from diﬀerent data, to wit, the royal nodes of h and the zeros of s. One step in the construction in [3] is to perform a Fejér–Riesz factorization of a non-negative trigonometric polynomial, whereas, in contrast, the construction in this paper can be carried out entirely in **rational** arithmetic.

34 Read more

3.3 Similarity of the search coils, between the three components and between the four spacecraft As the aim of the Cluster mission is to perform three- dimensional measurements, this implies the ability to com- bine the data of the different spacecraft either to derive quan- tities as a curl to get e.g. small-scale currents or to apply the so-called K-filtering method (Pinçon and Lefeuvre, 1991) to disentangle possible different waves modes in turbulent spec- tra, it was a requirement to produce four experiments as sim- ilarly as possible (see e.g. Fig. 4 in Cornilleau-Wehrlin et al., 1997). An example is given in Fig. 6 below, where it can be seen that the relative difference in response in amplitude of the **transfer** function in the frequency range 0.1–180 Hz is less than 2 %. The normalised differences Bx–By, Bx–Bz and By–Bz are overplotted in red, green and blue respec- tively. For other spacecraft and for 10 Hz filter output, the normalised differences have the same order of magnitude. Note that the differences start to increase around 180 Hz, where the low-pass filters start to be efficient.

Show more
25 Read more

There is a similar connection between orthogonal **rational** matrix-valued **functions** and solving certain interpolation problems of Nevanlinna-Pick type for matrix-valued Carath´eodory **functions** (i.e., matrix interpolation problems which are studied with other methods, e.g., in [2, 8, 15]). But it takes more technical eﬀort to verify such a connection in that case. The main task of this paper is to go some steps towards generalizing the re- sults presented in [33] to the matrix case. In fact, we provide particular formulae starting from the recurrence relations for orthogonal **rational** matrix-valued **functions** stated in [26]. In a forthcoming work, these formulae will finally play a key role by solving interpo- lation problems of Nevanlinna-Pick type for matrix-valued Carath´eodory **functions** in D via orthogonal **rational** matrix-valued **functions** including an interrelation between the parameters which appear in the recurrence relations studied in the present paper and the parameters which appear in the algorithm discussed in [24, Section 5].

Show more
37 Read more

The time response represents how the state of a dynamic **system** changes in time when subjected to a particular input since the models has been derived of differential equation ,some integration must be performed in order to determine the time response of the **system** . fortunately ,MATLAB provides many useful resources for calculating time responses for many types of inputs. MATLAB provides tools for automatically choosing optimal PID gains[3].

Since we will be dealing with many **functions** it is convenient to name vari- ous **functions** (usually with letters f, g, h, etc). Often we will implicitly assume that a domain and codomain are given without specifying these explicitly. If the range can be determined and the codomain is not given explicitly, then we take the codomain to be the range. If the range can not easily be determined and the codomain is not explicitly given, then the codomain should be taken to be a ‘simple’ set which clearly contains the range.

426 Read more

Write the original **rational** function as a sum of fractions. Each of the terms in the sum gets one of the original factors as its denominator. The numerator of each new fraction is an arbitrary polynomial of degree one less than the degree of the denominator.

Trigonometric Substitution and Integration of Rational Functions by Partial Fractions.. Mathematics 54–Elementary Analysis 2.[r]

28 Read more

The problems considered in this thesis are: The problem of realisation of nonproper transfer functions in singular system form, and the closely related problem of canonical forms for sin[r]

266 Read more

There are various window **functions** which are used for FIR filter design. As we know Finite-duration impulse response filters (FIR) is based on frequency response specifications and the filter implementation is done by taking Fourier transform of the desired frequency response specifications. Basically following are the types of window **functions** which are used in FIR filter designing.

In this paper a closed shape matrix **rational** model for the calculation of step and limited incline reactions of Resistance Inductance Capacitance (RLC) interconnects in VLSI circuits is displayed. This model permits the numerical estimation of deferral and overshoot in loss VLSI interconnects. The proposed technique depends on the U-change, which gives **rational** capacity approximation to getting inactive interconnect model. With the decreased request loss interconnect exchange capacity, step and limited slope reactions are acquired and line deferral and flag overshoot are assessed. The assessed deferral and overshoot qualities are contrasted and the Euder strategy, Pade technique and HSPICE W-component model. The half postpone results are in good agreement with those of HSPICE inside 0.5% mistake while the overshoot blunder is inside 1% for a 2 mm long interconnect. For global lines of length more than 5 mm in SOC (framework on chip) applications, the proposed technique is observed to be almost four times more exact than existing methods.

Show more
The generalized order ρα, β, f of the rate of growth of entire **functions** f was introduced by ˇSeremeta 8, who obtained a characterization of ρα, β, f in terms of the coeﬃcients of the power series of f. In 8, the relationship between the generalized order of entire **functions** f and the degree of polynomial approximation of f was studied. The coeﬃcient characterization of a generalized order of the rate of growth of **functions** analytic in a disk has been discussed in several papers 9–12. The degree of **rational** approximation of entire **functions** of a finite generalized order is investigated in 6.

Show more
11 Read more

terval 0 ≤ x, t ≤ a < ∞ . This variant or improvement for the method gave us a faster and more accurate method compared to the other methods. In addition, an interesting feature of this method is to ﬁnd the analytical solutions if the equation has an exact solution that is a **rational** **functions**. Illustrative examples are used to demonstrate the applicability and the eﬀectiveness of the proposed technique.

14 Read more

In this thesis, we shall add to the list of these known functions, some special class of continuous functions whose best -uniform rational approximations and the quantity En,mf can be gi[r]

164 Read more

2 Rational functions are continuous on their respective domains.. 2 A piecewise-defined function may be discontinuous at endpoints of intervals..[r]

49 Read more

A homoclinic solution to the Higg **system** (1) was presented in [8] by assistance of Hirota’s bilinear technique. (G’/G)-expansion and He’s semi inverse methods were also used to set trigonometric and hyperbolic function solutions covering the complex ones [9]. Solutions determined by exp (−𝜑(𝜀))-expansion method in forms of **rational**, trigonometric and hyperbolic **functions** were summarized in [10].

15 Read more