Partial Fraction Expansion Research Articles

A q-theorem of Pólya using Hurwitz partial fraction method

AbstractWe establish a q-counterpart of the method of partial fraction developed by Hurwitz-Pólya to investigate the zeros of q-cosine and q-sine transforms, where $$q \in (0,1)$$ q ∈ ( 0 , 1 ) is a fixed number. We prove reality and simplicity of the zeros and give a precise description of their distribution. The conditions imposed on both q and the integrand are less restrictive than previously assumed in the literature. A direct infinite partial fraction expansion is obtained via q-sampling theory.

Various Methods of Realization for Fractional-Order Elements

Fractional-order circuits are finding applications in control systems, signal processing, and other allied fields. As a result, realizing fractional-order elements or fractance devices is an essential research topic. A unique fractance device is realized in this study using continued fraction expansion (CFE) and partial fraction expansion (PFE) formulas. Continued fraction expansion is used to calculate the rational approximation. For simulation, the fourth order rational approximation for fractional order, =-1/2, -1/3, -1/4 is used. All simulations are carried out with the help of the MATLAB and TINA software packages. The theoretical and simulation results are in agreement.

A Hybrid Approximation Method for Integer-Order Approximate Realization of Fractional-Order Derivative Operators

The use of fractional-order (FO) calculus for the solution of different problems in many fields has increased recently. However, the usage of FO system models in practice brings some difficulties. The FO operator, fractance device, is usually realized via several integer-order approximation methods, which have pros and cons in the aspect of operation frequency, time response and stability region. These methods may not meet all performance expectations. In this regard, author proposes an efficient hybrid integer-order approximation method for FO derivative operator without causing any additional difficulty in realization. The proposed method combines Matsuda and modified stability boundary locus (M-SBL) approximation methods. The advantage of each method is combined in a single hybrid function by considering root mean square error (RMSE) rates for step response. The performance of hybrid transfer function is analyzed in comparison with Matsuda, Oustaloup, continued fraction expansion (CFE) and M-SBL transfer functions for both frequency and time response. Analog realization of the proposed model is performed experimentally via partial fraction expansion method. Analog design is verified via both Multisim simulations and experimental results. The improvements due to the hybrid behavior and the consistency of experimental results with theoretical and simulation results demonstrate the practicality and usefulness of the hybrid model.

Special Case of Partial Fraction Expansion with Laplace Transform Application

Special Case of Partial Fraction Expansion with Laplace Transform Application

On the Design Flow of the Fractional-Order Analog Filters Between FPAA Implementation and Circuit Realization

This work explicitly states the design flows of the fractional-order analog filters that researchers throughout the literature have used. Two main flows are studied: the FPAA implementation and the circuit realization. Partial-fraction expansion representation is used to prepare the approximated fractional-order response for implementation on FPAA. The generalization of the second-order active RC analog filters based on opamp from the integer-order domain to the fractional-order domain is presented. The generalization is studied from both mathematical and circuit realization points of view. It is found that the great benefit of the fractional-order domain is that it adds more degrees of freedom to the filter design process. Simulation and experimental results match the expected theoretical analysis.

Implicit finite difference wave equation forward modeling based on partial fraction expansion

Implicit finite difference wave equation forward modeling based on partial fraction expansion

On a class of Hilbert-type inequalities in the whole plane involving some classical kernel functions

AbstractIn this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.

On the experimental vibroacoustic modal analysis of a plate-cavity system

This paper proposes a methodology to the experimental vibroacoustic modal analysis on a coupled plate-cavity system. The method is based on the nonlinear curve fitting to the Frequency Response Functions (FRFs) obtained through a roving hammer test. These FRFs are divided into two groups: one is for the acceleration of the plate; the other is for the sound pressure inside the cavity. The modal parameters are estimated independently in each group using both the single/multi-mode all-curve and the all-mode single-curve fitting techniques. Such a way overcomes the identification difficulties when the natural frequencies of the coupled system slightly change over the roving hammer process. Meanwhile, the mode shapes extraction is eased from the different scales between structural and acoustic responses. The two groups of FRFs are fitted by two different types of FRF models, respectively. The results are in agreement with respect to the natural frequencies and modal damping ratios. Particularly, the FRF models are in alternative forms instead of the conventional ones, and they fit the measured FRFs very well with the estimation outcomes applied. Furthermore, the identified modal parameters are verified by the counterpart estimation based on the conventional FRF models formulated by partial fraction expansions. It is shown that the proposed FRF models contain fewer unknown parameters but do not downgrade the estimation accuracy, making the curve fitting process more practical and efficient. Hence, the two-group strategy with the proposed FRF models provides a practical and efficient way for the experimental modal analysis of vibroacoustic systems.

Control-Oriented Modeling of All-Solid-State Batteries Using Physics-Based Equivalent Circuits

Considered as one of the ultimate energy storage technologies for electrified transportation, the emerging all-solid-state batteries (ASSBs) have attracted immense attention due to their superior thermal stability, increased power and energy densities, and prolonged cycle life. To achieve the expected high performance, practical applications of ASSBs require accurate and computationally efficient models for the design and implementation of many onboard management algorithms, so that the ASSB safety, health, and cycling performance can be optimized under a wide range of operating conditions. A control-oriented modeling framework is thus established in this work by systematically simplifying a rigorous partial differential equation (PDE) based model of the ASSBs developed from underlying electrochemical principles. Specifically, partial fraction expansion and moment matching are used to obtain ordinary differential equation based reduced-order models (ROMs). By expressing the models in a canonical circuit form, excellent properties for control design such as structural simplicity and full observability are revealed. Compared to the original PDE model, the developed ROMs have demonstrated high fidelity at significantly improved computational efficiency. Extensive comparisons have also been conducted to verify its superiority to the prevailing models due to the consideration of concentration-dependent diffusion and migration. Such ROMs can thus be used for advanced control design in future intelligent management systems of ASSBs.

Partial Fraction Expansion of Meromorphic Maps

Partial fraction expansion of a meromorphic function or map is a way of expressing it as an infinite series of rational functions and polynomials. This representation is particularly useful in the summation of series in mathematical analysis and has at its heart the Mittag-Leffler’s expansion theorem which holds for arbitrary domains. In this article, the Mittag-Leffler’s expansion theorem was used to obtain partial fraction expansions of the functions tan⁡z and coth⁡z which are meromorphic in the entire complex plane. The expansion of tan⁡z was then used to obtain the Laurent series for it where except for the odd coefficients, all the even coefficients in the series were shown to vanish identically. By comparing these coefficients with those of the well-known Maclaurin series for tan⁡z the exact sum of the odd terms in the important p-series ∑_(n=1)^∞▒1/n^p (p=2,4,6) were obtained in closed form and consequently, those of the series ∑_(n=1)^∞▒1/n^p (p=2,4,6) for all values of n. The partial fraction expansion of coth⁡z was also gainfully employed in finding the exact sum (in closed form) of the p-series ∑_(n=1)^∞▒1/n^2 and ∑_(n=-∞)^∞▒〖1/(n^2+z^2 ) (z≠ni,n=0,1,2,…)〗 to also show its usefulness in application. Finally, since closed form expressions were obtained for those series found from the partial fraction expansions of the functions considered in this article, we suggest that further research in this area should focus on the application of the Mittag-Leffler’s theorem to other meromorphic functions such as sec⁡z, cot⁡z etc to find similar results on the summation of series as done here.

Proof of the functional equation for the Riemann zeta-function

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.

Frequency Domain Convolution of Rational Transfer Functions

The convolution of two rational transfer functions is also rational, but a formula for the convolution has never been derived. This paper introduces a formula for the convolution of two rational functions in the frequency domain by two new methods. The first method involves a partial fraction expansion of the rational transfer functions where the problem gets reduced to the sum of the convolution of the partial fractions of the two functions, each of which can be solved by a new formula. Since the calculation of the roots of a high-order polynomial can be very time-consuming, we also demonstrate new methods for performing the convolution without calculating these roots or undergoing partial fraction expansion. The convolution of two rational Laplace transform denominators can be calculated using their resultant, while that of the two rational Z-transform transfer functions can be found using Newton’s identities. The numerator can be easily found by multiplying the numerator with the initial values of the power series of the result.

An experimental analog circuit realization of Matsuda’s approximate fractional-order integral operators for industrial electronics

Analog circuit realization of fractional order (FO) elements is a significant step for the industrialization of FO control systems because of enabling a low-cost, electric circuit realization by means of standard industrial electronics components. This study demonstrates an effective operational amplifier-based analog circuit realization of approximate FO integral elements for industrial electronics. To this end, approximate transfer function models of FO integral elements, which are calculated by using Matsuda’s approximation method, are decomposed into the sum of low-pass filter forms according to the partial fraction expansion. Each partial fraction term is implemented by using low-pass filters and amplifier circuits, and these circuits are combined with a summing amplifier to compose the approximate FO integral circuits. Widely used low-cost industrial electronics components, which are LF347N opamps, resistor and capacitor components, are used to achieve a discrete, easy-to-build analog realization of the approximate FO integral elements. The performance of designed circuit is compared with performance of Krishna’s FO circuit design and performance improvements are shown. The study presents design, performance validation and experimental verification of this straightforward approximate FO integral realization method.

An effective analog circuit design of approximate fractional-order derivative models of M-SBL fitting method

There is a growing interest in fractional calculus and Fractional Order (FO) system modeling in many fields of science and engineering. Utilization of FO models in real-world applications requires practical realization of FO elements. This study performs an analog circuit realization of approximate FO derivative models based on Modified Stability Boundary Locus (M-SBL) fitting method. This study demonstrates a low-cost and accurate analog circuit implementation of M-SBL fitting based approximate model of FO derivative elements for industrial electronics. For this purpose, a 4th order approximate derivative transfer function model of the M-SBL method is decomposed into the sum of first order low-pass filters form by using Partial Fraction Expansion (PFE) method, and the analog circuit design of the approximate FO derivative model is performed. Firstly, by using the final value theorem, authors theoretically show that the time response of the sum of first order low-pass filter form can converge to the time response of fractional order derivative operators. Then, the approximation performance of proposed FO derivative circuit design is validated for various input waveforms such as sinusoidal, square and sawtooth waveforms via Multisim simulations. Results indicate an accurate realization of the FO derivative in time response (an RMSE of 0.0241). The derivative circuit realization of the M-SBL fitting model in the form of the sum of first order low pass filters can yield a better time response approximation performance compared to the Continued Fraction Expansion (CFE) based ladder network realization of the approximate derivative circuit.

On a new generalization of some Hilbert-type inequalities

Abstract In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

Time-domain simulations of sound propagation in a flow duct with extended-reacting liners

A general methodology to simulate acoustic propagation in ducts with extended-reacting liners in the time domain is presented, including a generic perforated sheet on the air-material interface. The Linearized Euler Equations (LEE) with a mean flow profile are solved in the duct and the linearized equations on an equivalent fluid are solved in the liner material. The auxiliary differential equation method (ADE) is used to prevent the computation of convolution integrals, and leads to a formulation compatible with high-order numerical schemes. The methodology is illustrated for the case of liners consisting of rigid-frame porous materials and a prototype of locally-resonant acoustic metamaterials. A one-dimensional (1D) test case is first used to validate the algorithm and assess the numerical error. The numerical order of the algorithm is the expected one, independently of the interface, as long as the number of poles retained in the partial fraction expansions involved in the formulation is high enough. The algorithm is then applied to a realistic two-dimensional (2D) configuration in a duct with flow, and is used to illustrate the restrained validity of the locally-reacting approximation. Finally, the impact of the flow Mach number on the acoustic performance of porous and metamaterial extended-reacting liners is briefly assessed.

On a class of Hilbert-type inequalities in the whole plane related to exponent function

By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.

On a Hilbert-type inequality with homogeneous kernel involving hyperbolic functions

We first establish a Hilbert-type inequality and its equivalent Hardy form with best possible constant factors by constructing a homogeneous kernel function involving hyperbolic functions. Furthermore, we introduce Bernoulli numbers and the partial fraction expansions of trigonometric functions, and then we present several special and interesting Hilbert-type inequalities, in which the constant factors are represented by Bernoulli numbers and by some higher derivatives of trigonometric functions.

A method for inverting the Laplace transforms of two classes of rational transfer functions in control engineering

A dependable approach for inverting two classes of rational Laplace transforms, involving regular polynomials and partial sums with non-integer exponents is developed. Such types of Laplace transforms frequently emerge as the system transfer functions in the analysis of feedback control loops or process dynamics. The proposed method systematically translates the Laplace inversion problem into an integer or fractional order differential equation and yields the analytical inverse Laplace transform function utilizing the Adomian decomposition method. The method is especially useful in dealing with high-order rational transfer functions; where approaches based on the partial fraction expansion and the residue inversion theorem lose their practicality. The ready-to-use inversion formulas are presented and their usability and reliability is demonstrated through a number of case studies.

Implementation of a Fractional-Order Electronically Reconfigurable Lung Impedance Emulator of the Human Respiratory Tree

The fractional-order lung impedance model of the human respiratory tree is implemented in this paper, using Operational Transconductance Amplifiers. The employment of such active element offers electronic adjustment of the impedance characteristics in terms of both elements values and orders. As the MOS transistors in OTAs are biased in the weak inversion region, the power dissipation and the dc bias voltage of operation are also minimized. In addition, the partial fraction expansion tool has been utilized, in order to achieve reduction of the spread of the required time-constants and scaling factors. The performance of the proposed scheme has been evaluated, at post-layout level, using MOS transistors models provided by the 0.35 μ m Austria Mikro Systeme technology CMOS process, and the Cadence IC design suite.