A new collection of real world applications of fractional calculus in science and engineering

Sun, Hongwei; Zhang, Yong; Băleanu, Dumitru; Chen, Wen; Chen, YangQuan

doi:10.1016/j.cnsns.2018.04.019

Cited by 1,139 publications

(479 citation statements)

References 112 publications

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“…Fractional-order calculus is a very useful tool with interdisciplinary applications in electrical engineering, biology and biomedicine, control systems, and signal processing. [1][2][3] As well, the development of fractional-order circuits which include filters, 4-8 oscillators, 9,10 biological tissue emulators, [11][12][13][14][15] and analog controllers 16,17 is of significant interest. The most appropriate and important building blocks for implementing these types of circuits are the fractional-order differentiators and integrators described in the frequency domain via the operator s ±r ; 0 < r < 1.…”

Section: Introductionmentioning

confidence: 99%

“…where the values of coefficients A i (i = 0, 1, ... n) and B i (i = 0, 1, ... n) in (1) are readily available in the literature. For the circuit implementation of the expression in (1), several solutions were presented in the literature, but all these solutions (see for example, the solution described in John et al, 8 which relies on using a multi-feedback topology) require connecting in cascade first-order or second-order transfer function sections. An alternative solution which avoids this cascadability was recently described in AbdelAty et al, 24 where a sum of first-order high-pass filter (HPF) transfer functions is used.…”

Section: Introductionmentioning

confidence: 99%

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Partial fraction expansion–based realizations of fractional‐order differentiators and integrators using active filters

Bertsias

Psychalinos

Maundy

et al. 2019

Circuit Theory & Apps

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Summary Approximations of the fractional‐order differentiator and integrator operators s±r are proposed in this work. These approximations target the realization of these operators using standard active filter transfer functions. Hence, circuit implementations in integrated circuit form or in discrete component form are significantly facilitated. Complementary metal‐oxide‐semiconductor (CMOS) realizations of the proposed approximations are given and validated via simulations using the AMS 0.35 μm CMOS technology, while experimental results using operational amplifier circuits are tested and confirm the proposed theory.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Partial fraction expansion–based realizations of fractional‐order differentiators and integrators using active filters

Bertsias

Psychalinos

Maundy

et al. 2019

Circuit Theory & Apps

View full text Add to dashboard Cite

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“…Due to that the single variable function is the theoretical foundation of multivariable functions, in this paper, we will study the q ‐rung orthopair single variable fuzzy functions ( q‐ ROSVFFs) to ensure the integrality of functions. Moreover, the calculus has been closely associated with practical applications, such as engineering, finance, economics, and so forth. Therefore, it is very necessary to study the calculus of continuous single variable membership and nonmembership functions under q ‐rung orthopair fuzzy continuous environment.…”

Section: Introductionmentioning

confidence: 99%

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Zhang

2019

Int J Intell Syst

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The q‐rung orthopair fuzzy set ( q‐ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q‐rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q‐rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q‐rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q‐rung orthopair single variable fuzzy function ( q‐ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q‐ROSVFFs, and offer the relationship between the limit of q‐ROSVFF and that of q‐ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q‐ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q‐ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q‐ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q‐ROSVFFs.

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“…Because of the frequent appearance of fractional calculus in physics, science, engineering, mathematics and other applications, fractional calculus has become a hot spot for many researches . At present, most researchers have studied the development of numerical methods for fractional differential and integral equations.…”

Section: Introductionmentioning

confidence: 99%

“…In Wang and Zhang [17], the existence of positive solutions for a class of high order nonlinear fractional differential equations with integral boundary conditions and parameters is certified. In this paper, the two-dimensional block-pulse functions (2D-BPFs) are proposed for solving the numerical solution of Equation (1). Section 2 introduces the mathematical foundation of fractional calculus.…”

mentioning

confidence: 99%

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Xie

Ren

et al. 2019

Numerical Methods Partial

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In the current study, an approximate scheme is established for solving the fractional partial differential equations (FPDEs) with Volterra integral terms via two‐dimensional block‐pulse functions (2D‐BPFs). According to the definitions and properties of 2D‐BPFs, the original problem is transformed into a system of linear algebra equations. By dispersing the unknown variables for these algebraic equations, the numerical solutions can be obtained. Besides, the proof of the convergence of this system is given. Finally, several numerical experiments are presented to test the feasibility and effectiveness of the proposed method.

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A new collection of real world applications of fractional calculus in science and engineering

Cited by 1,139 publications

References 112 publications

Partial fraction expansion–based realizations of fractional‐order differentiators and integrators using active filters

Partial fraction expansion–based realizations of fractional‐order differentiators and integrators using active filters

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

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