Heaviside Operational Calculus Research Articles

Applying unrigorous mathematics: Heaviside's operational calculus

Applications of unrigorous mathematics are relatively common in the history and current practice of physics but underexplored in existing philosophical work on applications of mathematics. I argue that perspicuously representing some of the most philosophically interesting aspects of these cases requires us to go beyond the most prominent accounts of the role of mathematics in scientific representations, namely versions of the mapping account. I defend an alternative, the robustly inferential conception (RIC) of mathematical scientific representations, which allows us to represent the relevant practices more naturally. I illustrate the advantages of RIC by considering one such case, Heaviside's use of his unrigorous operational calculus to produce and apply an early generalization of Ohm's law in terms of “resistance operators.”

Self-similar network model for fractional-order neuronal spiking: implications of dendritic spine functions

Fractional-order derivatives have been widely used to describe the spiking patterns of neurons, without considering their self-similar dendritic structures. In this study, a self-similar resistor–capacitor network is proposed to relate the spiny dendritic structure with fractional spiking properties. In order to achieve this goal, two types of networks comprising recursively staggered resistors and capacitors were developed to model the functional properties of smooth and spiny dendrites, respectively. Their overall electrotonic properties can be described by fractional order temporal operators derived by Heaviside operational calculus. According to this operator method, spiking patterns of spiny dendrites were controlled by the standard 0.5-order derivative, whereas an exponential modulation term was added in the governing fractional operator of the smooth dendrites. The application of these fractional operators in a leaky integrate-and-fire model reveals that the dendritic spine plays an important role in alternations of the spiking properties, including first-spike latency, firing rate adaptation, and afterhyperpolarization conductance. Further, the multilevel assembly of this network indicates that the fractional spiking behaviors of spiny neurons might originate from their hierarchical substructures, thereby highlighting possible functional consequences of alterations to dendritic self-similarity.

Abstraction and operator characterization of fractal ladder viscoelastic hyper-cell for ligaments and tendons

This study investigates the viscoelastic behaviour of soft bio-fibres in association with their fractal structures. A spring-dashpot fractal network with the self-similar topology, named the Π-type fractal ladder hyper-cell (FLHC), is abstracted from the micro/nano-structure of ligaments and tendons (LTs). Its constitutive operator is derived by the Heaviside operational calculus, which is of intrinsic fractional order. In terms of this operator, the long-term viscoelastic relaxation of bio-fibres arising from the fractal ladder topology is expounded. In addition, the fractional-order viscoelastic constitutive equation is obtained based on the FLHC of LTs, and its results are consistent with those of available human knee and spinal LT relaxation experiments. Results on the constitutive equation of FLHCs are formulated into two propositions. The multidisciplinary invariance and implications from the fractal ladder pattern of bio-fibres are also discussed.

An Easy Approach to Distributions and Operational Calculus

The Mikusinski space is introduced as the smallest extension of the space of classical continuous functions in which the integration operator is bijective. (This can be viewed as a very small part of the field of Mikusinski operators.) It is shown that the space of Schwartz distributions (of finite order) can be defined as a quotient of the Mikusinski space. A remarkable property of Mikusinski functions is that they admit multiplication by all rational functions. It is demonstrated that this multiplication provides a natural simple basis for Heaviside's operational calculus.

Determination of high power synchronous generator subtransient reactances based on the waveforms for a steady state two-phase short-circuit

In the paper, there is presented an analysis of the waveforms at the stator winding terminals of a high power synchronous generator for a steady state two-phase short-circuit. The analysis was performed by the finite element method based on a 2D generator model taking into account higher spatial and time harmonics of the magnetic field as well as the impact of eddy currents in the rotor damping circuits. The used FE model was experimentally verified. For the steady state two-phase short-circuit, the waveforms determined by the FE method were compared with the approximated waveforms calculated using analytical formulas. The analytical formulas were derived from the machine equations written in the stator fixed coordinate system (α, β, 0), using normalized transformation of coordinates and the Heaviside operational calculus. The higher time harmonics obtained based on the FFT analysis of the waveforms of the short-circuit current and open phase voltage were used for determining subtransient reactances in the d and q axes of a 200MW generator. The thus determined generator reactances were compared with the values determined by other methods.

Start-up flow of a viscoelastic fluid in a pipe with a fractional Maxwell’s model

Unidirectional start-up flow of a viscoelastic fluid in a pipe with a fractional Maxwell’s model is studied. The flow starting from rest is driven by a constant pressure gradient in an infinite long straight pipe. By employing the method of variable separations and Heaviside operational calculus, we obtain the exact solution, from which the flow characteristics are investigated. It is found that the start-up motion of a fractional Maxwell’s fluid with parameters α and β, tends to be at rest as time goes to infinity, except the case of β=1. This observation, which also can be predicted from the mechanics analogue of fractional Maxwell’s model, agrees with the classical work of Friedrich and it indicates that a fractional Maxwell’s fluid presents solid-like behavior if β≠1 and fluid-like behavior if β=1. For an arbitrary viscoelastic model, a conjecture is proposed to give an intuitive way judging whether it presents fluid-like or solid-like behavior. Also oscillations may occur before the fluid tends to the asymptotic behavior stated above, which is a common phenomenon for viscoelastic fluids.

Mechanical Analogies of Fractional Elements

A Ffactional element model describes a special kind of viscoelastic material. Its stress is proportional to the fractional-order derivative of strain. Physically the mechanical analogies of fractional elements can be represented by spring-dashpot fractal networks. We introduce a constitutive operator in the constitutive equations of viscoelastic materials. To derive constitutive operators for spring-dashpot fractal networks, we use Heaviside operational calculus, which provides explicit answers not otherwise obtainable simply. Then the series-parallel formulas for the constitutive operator are derived. Using these formulas, a constitutive equation of fractional element with 1/2-order derivative is obtained. Finally we find the way to derive the constitutive equations with other fractional-order derivatives and their mechanical analogies.

Inverse laplace transformation of rational functions. 1

The Laplace transform is one of an array of integral transforms gathered under the topic of operational mathematics. Laplace transform and the Fourier transform are the two integral transforms most utilized by the typical physicist or engineer. Rediscovered more than two centuries ago by P.S. de Laplace - Euler discovered it first - the Laplace transform has enjoyed a prominent place in scientists' and engineers' toolboxes for only the past 60-or-so years. It is the modern form of Heaviside's operational calculus, placing his calculus on firm mathematical ground and extending its rules and methods. The case for the inverse transformation when H(s) is rational is discussed

Class E power amplifier steady-state solution as series in 1∕Q

The Heaviside operational calculus is used to obtain general expressions for the charge and current in the output circuit of a class E power amplifier. The class E conditions of zero charge and current at the switching point are applied, together with the steady-state requirement that the solution should be periodic. For the case of 50% duty cycle, circuit parameters can be obtained in the form of power series in Q -1 . Series are given for component values, power, efficiency and total harmonic power, in some cases as a function of normalised frequency offset.

Heaviside Operational Rules Applicable to Electromagnetic Problems - Abstract

The Heaviside operational calculus is known to be an effective means for obtaining first-hand solutions to various electromagnetic problems. Basic problems can often be reduced to the compact form of a pseudo-differential operator operating on the delta function. The problem is then shifted to the interpretation of the result in terms of calculable functions. The present article reviews methods helping in the interpretation process and a table of useful operational rules is given as an appendix. Operator formulation of two-dimensional multipole fields and sources is discussed with examples on the synthesis of radiation patterns by multipole sources.

Heaviside Operational Method for Delta Function Sequences

Delta function sequences are regular functions approaching the singular delta function. They represent more realistic physical models for concentrated sources than the idealized delta function. It is shown that delta function sequences can be formed in a straightforward manner through the Heaviside operational calculus, by defining quasi-differential broadening operators and operating on the singular delta function. Different operators and corresponding delta sequences are studied in one and three dimensions. Operator approach for removing the singularity of the Green function is considered as an application.

Image theory for reflected TE/TM wave in waveguide

The image principle is extended to the time-harmonic problem of TE/TM wave propagation and reflection in a waveguide. The fictitious image generating the reflected field is derived with the aid of Heaviside operational calculus and a transmission-line model of the waveguide. The operational calculus reveals that the image of a point-like source in front of the waveguide discontinuity is another point-like source in the mirror-image position and a line source extending from the mirror-image position to infinity. The image derived with operational calculus turns out to be independent of the waveguide's transverse geometry.

Electrostatic image method for the anisotropic half space

Electrostatic image theory, well known for a point charge in front of an isotropic dielectric half space, is generalised to the case where the half space may be anisotropic, characterised by a symmetric and positive-definite permittivity dyadic. The theory is derived through Heaviside operational calculus involving manipulations with pseudodifferential operators. As a result, the image of a point charge is seen to consist of a point charge at the mirror image point plus a continuous distribution of surface charge filling an angular sector the apex of which lies at the image point. The surface charge is shown to obey a simple analytic law. The angle of the image sector depends on the amount of transverse anisotropy of the half space. For decreasing anisotropy, the angle becomes small and the angular image can be approximated by a quadrifilar line charge coinciding with the result recently obtained by these authors for the case of small anisotropy. Numerical tests confirm the validity of the image expression. The result can be readily applied to a wide variety of electrostatic problems involving sources and obstacles in front of an anisotropic half space, making a numerical development of a Green function for the anisotropic half space obsolete.

Looking back [Ernst Berg ideas on Heavyside's calculus

Quite recently, in a major American journal on electrical engineering education, a rather forceful suggestion was made for (re)integrating Heaviside's operational calculus in the training of electrical engineers. The author stressed the better physical motivation of this method, its conceptual parallelism to the well-known complex number method in AC circuit analysis, less mathematical intricacy and the particular appropriateness to initial value problems. In the light of this proposal, it might merit some interest to look at the first wave of the revival of Heaviside's calculus some 70 years ago. >

A unified theory of lumped circuits and differential systems based on Heaviside operators and causality

The paper demonstrates that a coherent and logical foundation for circuits and systems can, in an entirely elementary manner, be based upon the Heaviside operational calculus and causality. It is commonly believed that the operational calculus is both difficult and nonrigorous; in fact, it is neither. Further, it is more general than the Laplace transform in that it involves integration over only a finite interval and thus introduces no convergence questions as does the latter. Since it analyzes circuits and systems directly in the time domain, the Heaviside method is more intuitive and direct to apply. Furthermore, it provides a theme, a motif, linking all of the major concepts of circuits and systems. It is argued here that circuit analysis and system theory are currently taught as disparate disciplines, both being presented as a collection of isolated topics. The Heaviside theory, on the other hand, permits the two to be taught in an integrated fashion-with circuits providing concrete examples and system theory the abstract and general mathematical methodology. >

Heaviside's operational calculus applied to electrical circuit problems

Heaviside's operational calculus approach is compared with the one-sided Laplace transform method augmented with distribution theory analysis as applied to electrical circuit problems. It is seen that there is a high degree of commonality to these methods, but also several significant problem-solving and pedagogical advantages to Heaviside's method. In particular, the Heaviside operational calculus is better physically motivated, conceptually parallel to the well-established complex sinusoidal circuit analysis (the jw method), and less mathematically intricate, and may with ease also handle initial value problems. Moreover, the mathematical theory of the impulse or delta (t) function, defined as the convolution identity and developed using Heaviside's operational approach, compares very favorably with the quite extensive mathematical machinery of the theory of distributions. >

The ascendancy of the Laplace transform and how it came about

The modern Laplace transform is relatively recent. It was first used by Bateman in 1910, explored and codified by Doetsch in the 1920s and was first the subject of a textbook as late as 1937. In the 1920s and 1930s it was seen as a topic of front-line research; the applications that call upon it today were then treated by an older technique — the Heaviside operational calculus. This, however, was rapidly displaced by the Laplace transform and by 1950 the exchange was virtually complete. No other recent development in mathematics has achieved such ready popularisation and acceptance among the users of mathematics and the designers of undergraduate curricula.

Heaviside's operational calculus: Oliver's revenge

The Heaviside operational calculus is rigorously developed from general linear system considerations. This approach is abstract, yet quite simple and allows the most general results to be reached. Both stable and nonstable linear systems are considered but the stable single input-output systems are emphasized, as they are of primary interest. Extension to multivariate systems is immediate.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of high-voltage waveforms in 3-phase 12-pulse type x-ray units

Simple equivalent circuits of an X-ray unit are first given using linear lumped electric circuit elements, then modified Heaviside operational calculus is applied. The case of small tube currents, such as those used in fluoroscopy, are treated, at which abnormal voltages can occur at closing times. Theoretical waveforms of anode and cathode voltages and those of currents at mA-meter positions (which are rectified currents in secondary windings in a high-voltage transformer) coincide with experimental results in the following typical two cases: (a) abnormal voltages appearing at simultaneous 3-phase closing and (b) abnormal voltages being suppressed at 2-step (from single-phase to 3-phase) closing. Effects of the two types of closing are discussed.

A new symbolic calculus for the response of nonlinear systems

We present a new operational calculus for computing the response of nonlinear systems to various deterministic excitations. The use of a new tool: noncommutative generating power series, allows us to derive, by simple algebraic manipulations, the Volterra functional series of the solution of a large class of nonlinear forced differential equations. The symbolic calculus introduced appears as a natural generalization to the nonlinear domain, of the well known Heaviside operational calculus. Moreover, this method has the advantage of allowing the use of a computer.