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11 – 20 of 313Jerzy Bochnia and Slawomir Blasiak
The purpose of this paper was to verify the possibility of applying differential calculus of incomplete order to describe relaxation of the material obtained using selective laser…
Abstract
Purpose
The purpose of this paper was to verify the possibility of applying differential calculus of incomplete order to describe relaxation of the material obtained using selective laser sintering (SLS) technology.
Design/methodology/approach
The samples were made using the incremental technology for three print directions. Relaxation tests were conducted. The theoretical curves, which are the solution of the equation describing the five-parameter Maxwell-Wiechert model for derivatives in relation to the total time of complete order and fractional order, were adjusted to the obtained experimental curves.
Findings
The SLS technology creates new possibilities regarding modelling polymeric elements which might be applied as functional models (products). Therefore, it is necessary to conduct an in-depth study of their properties, including relaxation properties, which is associated with the necessity to use proper mathematical tools to describe those properties. The differential calculus of incomplete order was applied herein to describe the anisotropy of relaxation properties because of the print direction in relation to the relaxation curves adjusted with the five-parameter Maxwell-Wiechert model.
Research limitations/implications
As a result of the conducted considerations, the authors obtained the dependencies describing the anisotropy of relaxation properties with the use of coefficients alpha and beta, which stand for the derivative order of the differential equation, whereas coefficient kappa stands for the translation coefficient which is an innovative application of this type of mathematical apparatus.
Practical implications
The developed method might be applied to describe the anisotropy of a broader group of materials manufactured with the use of incremental technologies.
Originality/value
The application of the differential calculus of incomplete order to describe the anisotropy of the materials manufactured from polyamide powder using the SLS technology is a distinctive feature of this paper. A crucial cognitive element of the conducted research is the fact which confirms that the dynamic viscosity coefficients have the greatest impact on the anisotropy of material properties depending on the print directions.
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This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics…
Abstract
Purpose
This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.
Design/methodology/approach
Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.
Findings
HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.
Originality/value
The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.
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Syed Tauseef Mohyud‐Din, Ahmet Yıldırım and Eda Yülüklü
The purpose of this paper is to present numerical solutions for the space‐ and time‐fractional Korteweg‐de Vries (KdV) equation using homotopy analysis method (HAM). The space and…
Abstract
Purpose
The purpose of this paper is to present numerical solutions for the space‐ and time‐fractional Korteweg‐de Vries (KdV) equation using homotopy analysis method (HAM). The space and time‐fractional derivatives are described in the Caputo sense. The paper witnesses the extension of HAM for fractional KdV equations.
Design/methodology/approach
This paper presents numerical solutions for the space‐ and time‐fractional KdV equation using HAM. The space and time‐fractional derivatives are described in the Caputo sense.
Findings
In this paper, the application of homotopy analysis method was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The homotopy analysis method was clearly a very efficient and powerful technique in finding the solutions of the proposed equations.
Originality/value
In this paper, the application of HAM was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The HAM was clearly very efficient and powerful technique in finding the solutions of the proposed equations. The obtained results demonstrate the reliability of the algorithm and its wider applicability to fractional nonlinear evolution equations. Finally, the recent appearance of nonlinear fractional differential equations as models in some fields such as the thermal diffusion in fractal media makes it necessary to investigate the method of solutions for such equations and the authors hope that this paper is a step in this direction.
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Ahmet Yıldırım and Alev Kelleci
This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.
Abstract
Purpose
This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.
Design/methodology/approach
The realistic numerical solutions were obtained in a form of rapidly convergent series with easily computable components.
Findings
The figures show the effectiveness and good accuracy of the proposed method.
Originality/value
The paper obtains realistic numerical solutions in a form of rapidly convergent series with easily computable components. It shows the effectiveness and good accuracy of the proposed method.
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Muhammad Nadeem and Shao-Wen Yao
This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that…
Abstract
Purpose
This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that the results are very close to the exact solution.
Design/methodology/approach
First with the help of some basic properties of fractional derivatives, a scheme that has the capability to solve fractional partial differential equations is constructed. Then, TFH-L and W-L equations with variable coefficients are solved by this scheme, which yields results very close to the exact solution. The derived results demonstrate that this scheme is very effective. Finally, the convergence of this method is discussed.
Findings
A traditional method is combined with the Laplace transform to construct this scheme. To decompose the nonlinear terms, this paper introduces the homotopy perturbation method with He’s polynomials and thus the solution is provided in the form of a series that converges to the exact solution very quickly.
Originality/value
The proposed approach is original and very effective because this approach is, to the authors’ knowledge, used for the first time very successfully to tackle the fractional partial differential equations, which are of great interest.
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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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Hamed Tirandaz and Ali Karami-Mollaee
The purpose of this paper is to propose a novel and secure image transmission based on the unpredictable behavior of the chaotic systems.
Abstract
Purpose
The purpose of this paper is to propose a novel and secure image transmission based on the unpredictable behavior of the chaotic systems.
Design/methodology/approach
The proposed approach includes two main contributions: synchronization scheme and transmission scheme. The synchronization scheme benefits the advantage of the fractional-order active synchronization method. A new control law is derived to asymptotically synchronize the underlined fractional-order Bloch chaotic system. The validity of the proposed synchronization scheme is proved by the Lyapunov stability theorem. Then, a novel image transmission scheme is designed to transfer image data via chaotic signals, which modulates the encrypted data in the sender signals and demodulates it at the receiver side.
Findings
Numerical simulations are provided to show the validity and effectiveness of the proposed image transmission system. Furthermore, the performance of the image transmission system is evaluated using some illustrative examples and their corresponding statistical tests. The results demonstrate the effectiveness of the proposed method in comparison with other proposed methods in this subject.
Originality/value
A new chaos-based image transmission system is developed based on the synchronization of Bloch chaotic system. The introduced transmission system is interesting and could be applicable to any kind of secure image/video transmission.
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Mostafa Abbaszadeh, AliReza Bagheri Salec and Shurooq Kamel Abd Al-Khafaji
The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is…
Abstract
Purpose
The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.
Design/methodology/approach
The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.
Findings
A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.
Originality/value
The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.
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Bo Tang, Xuemin Wang, Leilei Wei and Xindong Zhang
This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and…
Abstract
Purpose
This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions.
Design/methodology/approach
The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method.
Findings
The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations.
Originality/value
The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.
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P. Veeresha, D.G. Prakasha and Jagdev Singh
The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM).
Abstract
Purpose
The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM).
Design/methodology/approach
The proposed technique (q-HATM) is the graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana-Baleanu (AB) operator.
Findings
The fixed point hypothesis considered to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional-order model. To illustrate and validate the efficiency of the future technique, the authors analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order.
Originality/value
To illustrate and validate the efficiency of the future technique, we analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. The obtained results elucidate that, the proposed algorithm is easy to implement, highly methodical, as well as accurate and very effective to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.
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