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The purpose of this study is to originally present noise analysis of electrical circuits defined on fractal set.
Abstract
Purpose
The purpose of this study is to originally present noise analysis of electrical circuits defined on fractal set.
Design/methodology/approach
The fractal integrodifferential equations of resistor-inductor, resistor-capacitor, inductor-capacitor and resistor-inductor-capacitor circuits subjected to zero mean additive white Gaussian noise defined on fractal set have been formulated. The fractal time component has also been considered. The closed form expressions for crucial stochastic parameters of circuit responses have been derived from these equations. Numerical simulations of power spectral densities based on the derived autocorrelation functions have been performed. A comparison with those without fractal time component has been made.
Findings
We have found that the Hausdorff dimension of the middle b Cantor set strongly affects the power spectral densities; thus, the average powers of noise induced circuit responses and the inclusion of fractal time component causes significantly different analysis results besides the physical measurability of electrical quantities.
Originality/value
For the first time, the noise analysis of electrical circuit on fractal set has been performed. This is also the very first time that the fractal time component has been included in the fractal calculus-based circuit analysis.
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The purpose of this paper is to propose a novel nonlocal fractal calculus scheme dedicated to the analysis of fractal electrical circuit, namely, the generalized nonlocal fractal…
Abstract
Purpose
The purpose of this paper is to propose a novel nonlocal fractal calculus scheme dedicated to the analysis of fractal electrical circuit, namely, the generalized nonlocal fractal calculus.
Design/methodology/approach
For being generalized, an arbitrary kernel function has been adopted. The condition on order has been derived so that it is not related to the γ-dimension of the fractal set. The fractal Laplace transforms of our operators have been derived.
Findings
Unlike the traditional power law kernel-based nonlocal fractal calculus operators, ours are generalized, consistent with the local fractal derivative and use higher degree of freedom. As intended, the proposed nonlocal fractal calculus is applicable to any kind of fractal electrical circuit. Thus, it has been found to be a more efficient tool for the fractal electrical circuit analysis than any previous fractal set dedicated calculus scheme.
Originality/value
A fractal calculus scheme that is more efficient for the fractal electrical circuit analysis than any previous ones has been proposed in this work.
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The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor.
Abstract
Purpose
The purpose of this paper is to test the capability to properly analyze the electrical circuits of a novel constitutive relation of capacitor.
Design/methodology/approach
For ceteris paribus, the constitutive relations of the resistor and inductor have been reformulated by following the novel constitutive relation of capacitor. The responses of RL, RC, LC and RLC circuits defined on the fractal set described by these definitions have been derived by means of the fractal calculus and fractal Laplace transformation. A comparative Hamiltonian formalism-based analysis has been performed where the circuits described by the conventional and the formerly proposed revisited constitutive relations have also been considered.
Findings
This study has found that the novel constitutive relations give unreasonable results unlike the conventional ones. Like such previous revisited constitutive relations, an odd Hamiltonian has been obtained. On the other hand, the conventional constitutive relations give a reasonable Hamiltonian.
Originality/value
To the best of the author’s knowledge, for the first time, the analysis of fractal set defined electrical circuits by means of unconventional constitutive relations has been performed where the deficiency of the tested capacitive constitutive relation has been pointed out.
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The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to…
Abstract
Purpose
The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus.
Design/methodology/approach
The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited.
Findings
The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead.
Originality/value
For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.
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The purpose of this paper is to derive a new fractal active low-pass filter (LPF) within the local fractional derivative (LFD) calculus on the Cantor set (CS).
Abstract
Purpose
The purpose of this paper is to derive a new fractal active low-pass filter (LPF) within the local fractional derivative (LFD) calculus on the Cantor set (CS).
Design/methodology/approach
To the best of the author’s knowledge, a new fractal active LPF within the LFD on the CS is proposed for the first time in this work. By defining the nondifferentiable (ND) lumped elements on the fractal set, the author successfully extracted its ND transfer function by applying the local fractional Laplace transform. The properties of the ND transfer function on the CS are elaborated in detail.
Findings
The comparative results between the fractal active LPF (for γ = ln2/ln3) and the classic one (for γ = 1) on the amplitude–frequency and phase–frequency characteristics show that the proposed method is correct and effective, and is expected to shed light on the theory study of the fractal electrical systems.
Originality/value
To the best of the author’s knowledge, the fractal active LPF within the LFD calculus on the CS is proposed for the first time in this study. The proposed method can be used to study the other problems in the fractal electrical systems, and is expected to shed a light on the theory study of the fractal electrical systems.
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Hong-Yan Liu, Ji-Huan He and Zheng-Biao Li
Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable…
Abstract
Purpose
Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus.
Design/methodology/approach
This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given.
Findings
Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations.
Originality/value
Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.
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Keywords
Kang-Jia Wang and Jing-Hua Liu
As a powerful mathematical analysis tool, the local fractional calculus has attracted wide attention in the field of fractal circuits. The purpose of this paper is to derive a new
Abstract
Purpose
As a powerful mathematical analysis tool, the local fractional calculus has attracted wide attention in the field of fractal circuits. The purpose of this paper is to derive a new
Design/methodology/approach
A new
Findings
The characteristics of the
Originality/value
To the best of the authors’ knowledge, this paper, for the first time ever, proposes the
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Keywords
The purpose of this paper is the coupled nonlinear fractal Schrödinger system is defined by using fractal derivative, and its variational principle is constructed by the fractal…
Abstract
Purpose
The purpose of this paper is the coupled nonlinear fractal Schrödinger system is defined by using fractal derivative, and its variational principle is constructed by the fractal semi-inverse method. The approximate analytical solution of the coupled nonlinear fractal Schrödinger system is obtained by the fractal variational iteration transform method based on the proposed variational theory and fractal two-scales transform method. Finally, an example illustrates the proposed method is efficient to deal with complex nonlinear fractal systems.
Design/methodology/approach
The coupled nonlinear fractal Schrödinger system is described by using the fractal derivative, and its fractal variational principle is obtained by the fractal semi-inverse method. A novel approach is proposed to solve the fractal model based on the variational theory.
Findings
The fractal variational iteration transform method is an excellent method to solve the fractal differential equation system.
Originality/value
The author first presents the fractal variational iteration transform method to find the approximate analytical solution for fractal differential equation system. The example illustrates the accuracy and efficiency of the proposed approach.
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Xiankang Luo and Muhammad Nadeem
This study aims to investigate the approximate solution of the time fractional time-fractional Newell–Whitehead–Segel (TFNWS) model that reflects the appearance of the stripe…
Abstract
Purpose
This study aims to investigate the approximate solution of the time fractional time-fractional Newell–Whitehead–Segel (TFNWS) model that reflects the appearance of the stripe patterns in two-dimensional systems. The significant results of plot distribution show that the proposed approach is highly authentic and reliable for the fractional-order models.
Design/methodology/approach
The Laplace transform residual power series method (ℒT-RPSM) is the combination of Laplace transform (ℒT) and residual power series method (RPSM). The ℒT is examined to minimize the order of fractional order, whereas the RPSM handles the series solution in the form of convergence. The graphical results of the fractional models are represented through the fractional order α.
Findings
The derived results are obtained in a successive series and yield the results toward the exact solution. These successive series confirm the consistency and accuracy of ℒT-RPSM. This study also compares the exact solutions with the graphical solutions to show the performance and authenticity of the visual solutions. The proposed scheme does not require the restriction of variables and produces the numerical results in terms of a series. This strategy is capable to handle the nonlinear terms very easily for the TFNWS model.
Originality/value
This paper presents the original work. This study reveals that ℒT can perform the solution of fractional-order models without any restriction of variables.
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Kaiyu He, Muhammad Nadeem, Siddra Habib, H.M. Sedighi and Duohui Huang
The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system.
Abstract
Purpose
The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system.
Design/methodology/approach
First, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution.
Findings
An example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system.
Originality/value
The work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work.
Details