Special Issue "The Fractional View of Complexity"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (31 August 2019).

Special Issue Editors

Prof. Dr. J. A. Tenreiro Machado
Website
Guest Editor
Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, R. Dr. Roberto Frias, 4249-015 Porto, Portugal
Interests: nonlinear dynamics; complexity; fractional calculus; modeling; control; entropy; genomics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractal analysis and fractional differential equations have been proven as useful tools for describing the dynamics of complex phenomena characterized by long memory and spatial heterogeneity. Although there is a general agreement about the relation between both theories, the formal mathematical arguments supporting their relation are still being developed.

The fractional derivative of real order appears as the degree of structural heterogeneity between homogeneous and inhomogeneous domains. A purely real derivative order would imply a system with no characteristic scale, where a given property would hold regardless of the scale of the observations. However, in real-world systems, physical cut-offs may prevent the invariance spreading over all scales and, therefore, complex-order derivatives could yield more realistic models.

Information theory addresses the quantification and communication of information. Entropy and complexity are concepts that often emerge in relation to systems composed of many elements that interact with each other, which appear intrinsically difficult to model.

This Special Issue focuses on the synergies of fractals or fractional calculus and information theory tools, such as entropy, when modeling complex phenomena in engineering, physics, life, and social sciences. Submissions addressing novel issues as well as those on more specific topics, illustrating the broad impact of entropy-based techniques in fractality, fractionality, and complexity are welcome.

Papers should fit the scope of the journal Entropy and, therefore, must include some content on Information Theory and/or Entropy, both in the text and references.

The main topics of interest include (but are not limited to):

- Complex dynamics

- Fractional calculus and its applications

- Fractals and chaos

- Nonlinear dynamical systems

- Entropy and Information Theory

- Evolutionary computing

- Advanced control systems

- Finance and economy dynamics

- Biological systems and bioinformatics

- Nonlinear waves and acoustics

- Image and signal processing

- Transportation systems

- Geosciences

- Astronomy and cosmology

- Nuclear physics


Prof. Dr. J. A. Tenreiro Machado
Prof. Dr. António M. Lopes
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fractals
  • Fractional calculus
  • Complex systems
  • Dynamics
  • Entropy
  • Information theory

Published Papers (17 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Editorial

Jump to: Research

Editorial
The Fractional View of Complexity
Entropy 2019, 21(12), 1217; https://doi.org/10.3390/e21121217 - 13 Dec 2019
Viewed by 766
Abstract
Fractal analysis and fractional differential equations have been proven as useful tools for describing the dynamics of complex phenomena characterized by long memory and spatial heterogeneity [...] Full article
(This article belongs to the Special Issue The Fractional View of Complexity)

Research

Jump to: Editorial

Article
Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes
Entropy 2019, 21(10), 973; https://doi.org/10.3390/e21100973 - 05 Oct 2019
Cited by 3 | Viewed by 808
Abstract
Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order β , 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α , 0 < α 2 . [...] Read more.
Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order β , 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α , 0 < α 2 . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension n N if 0 < β 1 , 0 < α 2 and additionally for 1 < β α 2 in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders β and α of the time and spatial derivatives and on the space dimension n and is given by the expression β n α t , t being the time variable. Even if it is an increasing function in β , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition 0 < β 1 , i.e., only the slow and the conventional diffusion can be described by this equation. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Article
Parallel Implementation of Modeling of Fractional-Order State-Space Systems Using the Fixed-Step Euler Method
Entropy 2019, 21(10), 931; https://doi.org/10.3390/e21100931 - 24 Sep 2019
Cited by 1 | Viewed by 942
Abstract
This paper presents new results in implementation of parallel computing in modeling of fractional-order state-space systems. The methods considered in the paper are based on the Euler fixed-step discretization scheme and the Grünwald-Letnikov definition of the fractional-order derivative. Two different parallelization approaches for [...] Read more.
This paper presents new results in implementation of parallel computing in modeling of fractional-order state-space systems. The methods considered in the paper are based on the Euler fixed-step discretization scheme and the Grünwald-Letnikov definition of the fractional-order derivative. Two different parallelization approaches for modeling of fractional-order state-space systems are proposed, which are implemented both in Central Processing Unit (CPU)- and Graphical Processing Unit (GPU)-based hardware environments. Simulation examples show high efficiency of the introduced parallelization schemes. Execution times of the introduced methodology are significantly lower than for the classical, commonly used simulation environment. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Fractional Refined Composite Multiscale Fuzzy Entropy of International Stock Indices
Entropy 2019, 21(9), 914; https://doi.org/10.3390/e21090914 - 19 Sep 2019
Cited by 5 | Viewed by 1022
Abstract
Fractional refined composite multiscale fuzzy entropy (FRCMFE), which aims to relieve the large fluctuation of fuzzy entropy (FuzzyEn) measure and significantly discriminate different short-term financial time series with noise, is proposed to quantify the complexity dynamics of the international stock indices in the [...] Read more.
Fractional refined composite multiscale fuzzy entropy (FRCMFE), which aims to relieve the large fluctuation of fuzzy entropy (FuzzyEn) measure and significantly discriminate different short-term financial time series with noise, is proposed to quantify the complexity dynamics of the international stock indices in the paper. To comprehend the FRCMFE, the complexity analyses of Gaussian white noise with different signal lengths, the random logarithmic returns and volatility series of the international stock indices are comparatively performed with multiscale fuzzy entropy (MFE), composite multiscale fuzzy entropy (CMFE) and refined composite multiscale fuzzy entropy (RCMFE). The empirical results show that the FRCMFE measure outperforms the traditional methods to some extent. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Model Order Reduction: A Comparison between Integer and Non-Integer Order Systems Approaches
Entropy 2019, 21(9), 876; https://doi.org/10.3390/e21090876 - 09 Sep 2019
Cited by 8 | Viewed by 1015
Abstract
In this paper, classical and non-integer model order reduction methodologies are compared. Non integer order calculus has been used to generalize many classical control strategies. The property of compressing information in modelling systems, distributed in time and space, and the capability of describing [...] Read more.
In this paper, classical and non-integer model order reduction methodologies are compared. Non integer order calculus has been used to generalize many classical control strategies. The property of compressing information in modelling systems, distributed in time and space, and the capability of describing long-term memory effects in dynamical systems are two features suggesting also the application of fractional calculus in model order reduction. In the paper, an open loop balanced realization is compared with three approaches based on a non-integer representation of the reduced system. Several case studies are considered and compared. The results confirm the capability of fractional order systems to capture and compress the dynamics of high order systems. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Memories of the Future. Predictable and Unpredictable Information in Fractional Flipping a Biased Coin
Entropy 2019, 21(8), 807; https://doi.org/10.3390/e21080807 - 18 Aug 2019
Cited by 2 | Viewed by 1468
Abstract
Some uncertainty about flipping a biased coin can be resolved from the sequence of coin sides shown already. We report the exact amounts of predictable and unpredictable information in flipping a biased coin. Fractional coin flipping does not reflect any physical process, being [...] Read more.
Some uncertainty about flipping a biased coin can be resolved from the sequence of coin sides shown already. We report the exact amounts of predictable and unpredictable information in flipping a biased coin. Fractional coin flipping does not reflect any physical process, being defined as a binomial power series of the transition matrix for “integer” flipping. Due to strong coupling between the tossing outcomes at different times, the side repeating probabilities assumed to be independent for “integer” flipping get entangled with one another for fractional flipping. The predictable and unpredictable information components vary smoothly with the fractional order parameter. The destructive interference between two incompatible hypotheses about the flipping outcome culminates in a fair coin, which stays fair also for fractional flipping. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Graphical abstract

Article
On the Complexity Analysis and Visualization of Musical Information
Entropy 2019, 21(7), 669; https://doi.org/10.3390/e21070669 - 09 Jul 2019
Cited by 3 | Viewed by 1493
Abstract
This paper considers several distinct mathematical and computational tools, namely complexity, dimensionality-reduction, clustering, and visualization techniques, for characterizing music. Digital representations of musical works of four artists are analyzed by means of distinct indices and visualized using the multidimensional scaling technique. The results [...] Read more.
This paper considers several distinct mathematical and computational tools, namely complexity, dimensionality-reduction, clustering, and visualization techniques, for characterizing music. Digital representations of musical works of four artists are analyzed by means of distinct indices and visualized using the multidimensional scaling technique. The results are then correlated with the artists’ musical production. The patterns found in the data demonstrate the effectiveness of the approach for assessing the complexity of musical information. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
An Entropy Formulation Based on the Generalized Liouville Fractional Derivative
Entropy 2019, 21(7), 638; https://doi.org/10.3390/e21070638 - 28 Jun 2019
Cited by 9 | Viewed by 1111
Abstract
This paper presents a new formula for the entropy of a distribution, that is conceived having in mind the Liouville fractional derivative. For illustrating the new concept, the proposed definition is applied to the Dow Jones Industrial Average. Moreover, the Jensen-Shannon divergence is [...] Read more.
This paper presents a new formula for the entropy of a distribution, that is conceived having in mind the Liouville fractional derivative. For illustrating the new concept, the proposed definition is applied to the Dow Jones Industrial Average. Moreover, the Jensen-Shannon divergence is also generalized and its variation with the fractional order is tested for the time series. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method
Entropy 2019, 21(6), 557; https://doi.org/10.3390/e21060557 - 03 Jun 2019
Cited by 19 | Viewed by 1819
Abstract
In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition [...] Read more.
In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Complexity Analysis of Escher’s Art
Entropy 2019, 21(6), 553; https://doi.org/10.3390/e21060553 - 31 May 2019
Cited by 2 | Viewed by 2176
Abstract
Art is the output of a complex system based on the human spirit and driven by several inputs that embed social, cultural, economic and technological aspects of a given epoch. A solid quantitative analysis of art poses considerable difficulties and reaching assertive conclusions [...] Read more.
Art is the output of a complex system based on the human spirit and driven by several inputs that embed social, cultural, economic and technological aspects of a given epoch. A solid quantitative analysis of art poses considerable difficulties and reaching assertive conclusions is a formidable challenge. In this paper, we adopt complexity indices, dimensionality-reduction and visualization techniques for studying the evolution of Escher’s art. Grayscale versions of 457 artworks are analyzed by means of complexity indices and represented using the multidimensional scaling technique. The results are correlated with the distinct periods of Escher’s artistic production. The time evolution of the complexity and the emergent patterns demonstrate the effectiveness of the approach for a quantitative characterization of art. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Adaptive Synchronization Strategy between Two Autonomous Dissipative Chaotic Systems Using Fractional-Order Mittag–Leffler Stability
Entropy 2019, 21(4), 383; https://doi.org/10.3390/e21040383 - 10 Apr 2019
Cited by 8 | Viewed by 1286
Abstract
Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two [...] Read more.
Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
New Texture Descriptor Based on Modified Fractional Entropy for Digital Image Splicing Forgery Detection
Entropy 2019, 21(4), 371; https://doi.org/10.3390/e21040371 - 05 Apr 2019
Cited by 13 | Viewed by 1904
Abstract
Forgery in digital images is immensely affected by the improvement of image manipulation tools. Image forgery can be classified as image splicing or copy-move on the basis of the image manipulation type. Image splicing involves creating a new tampered image by merging the [...] Read more.
Forgery in digital images is immensely affected by the improvement of image manipulation tools. Image forgery can be classified as image splicing or copy-move on the basis of the image manipulation type. Image splicing involves creating a new tampered image by merging the components of one or more images. Moreover, image splicing disrupts the content and causes abnormality in the features of a tampered image. Most of the proposed algorithms are incapable of accurately classifying high-dimension feature vectors. Thus, the current study focuses on improving the accuracy of image splicing detection with low-dimension feature vectors. This study also proposes an approximated Machado fractional entropy (AMFE) of the discrete wavelet transform (DWT) to effectively capture splicing artifacts inside an image. AMFE is used as a new fractional texture descriptor, while DWT is applied to decompose the input image into a number of sub-images with different frequency bands. The standard image dataset CASIA v2 was used to evaluate the proposed approach. Superior detection accuracy and positive and false positive rates were achieved compared with other state-of-the-art approaches with a low-dimension of feature vectors. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations
Entropy 2019, 21(4), 335; https://doi.org/10.3390/e21040335 - 28 Mar 2019
Cited by 25 | Viewed by 2664
Abstract
In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are [...] Read more.
In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
A Fractional-Order Partially Non-Linear Model of a Laboratory Prototype of Hydraulic Canal System
Entropy 2019, 21(3), 309; https://doi.org/10.3390/e21030309 - 21 Mar 2019
Cited by 4 | Viewed by 1354
Abstract
This article addresses the identification of the nonlinear dynamics of the main pool of a laboratory hydraulic canal installed in the University of Castilla La Mancha. A new dynamic model has been developed by taking into account the measurement errors caused by the [...] Read more.
This article addresses the identification of the nonlinear dynamics of the main pool of a laboratory hydraulic canal installed in the University of Castilla La Mancha. A new dynamic model has been developed by taking into account the measurement errors caused by the different parts of our experimental setup: (a) the nonlinearity associated to the input signal, which is caused by the movements of the upstream gate, is avoided by using a nonlinear equivalent upstream gate model, (b) the nonlinearity associated to the output signal, caused by the sensor’s resolution, is avoided by using a quantization model in the identification process, and (c) the nonlinear behaviour of the canal, which is related to the working flow regime, is taken into account considering two completely different models in function of the operating regime: the free and the submerged flows. The proposed technique of identification is based on the time-domain data. An input pseudo-random binary signal (PRBS) is designed depending on the parameters of an initially estimated linear model that was obtained by using a fundamental technique of identification. Fractional and integer order plus time delay models are used to approximate the responses of the main pool of the canal in its different flow regimes. An accurate model has been obtained, which is composed of two submodels: a first order plus time delay submodel that accurately describes the dynamics of the free flow and a fractional-order plus time delay submodel that properly describes the dynamics of the submerged flow. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Descriptions of Entropy with Fractal Dynamics and Their Applications to the Flow Pressure of Centrifugal Compressor
Entropy 2019, 21(3), 266; https://doi.org/10.3390/e21030266 - 08 Mar 2019
Cited by 6 | Viewed by 1439
Abstract
In this study, some important intrinsic dynamics have been captured after analyzing the relationships between the dynamic pressure at an outlet of centrifugal compressor and fractal characteristics, which is one of powerful descriptions in entropy to measure the disorder or complexity in the [...] Read more.
In this study, some important intrinsic dynamics have been captured after analyzing the relationships between the dynamic pressure at an outlet of centrifugal compressor and fractal characteristics, which is one of powerful descriptions in entropy to measure the disorder or complexity in the nonlinear dynamic system. In particular, the fractal dynamics of dynamic pressure of the flow is studied, as the centrifugal compressor is in surge state, resulting in the dynamic pressure of flow and becoming a serious disorder and complex. First, the dynamic pressure at outlet of a centrifugal compressor with 800 kW is tested and then obtained by controlling the opening of the anti-surge valve at the outlet, and both the stable state and surge are initially tested and analyzed. Subsequently, the fractal dynamics is introduced to study the intrinsic dynamics of dynamic pressure under various working conditions, in order to identify surge, which is one typical flow instability in centrifugal compressor. Following fractal dynamics, the Hurst exponent, autocorrelation functions, and variance in measure theories of entropy are studied to obtain the mono-fractal characteristics of the centrifugal compressor. Further, the multi-fractal spectrums are investigated in some detail, and their physical meanings are consequently explained. At last, the statistical reliability of multi-fractal spectrum by modifying the original data has been studied. The results show that a distinct relationship between the dynamic pressure and fractal characteristics exists, including mono-fractal and multi-fractal, and such fractal dynamics are intrinsic. As the centrifugal compressor is working under normal condition, its autocorrelation function curve demonstrates apparent stochastic characteristics, and its Hurst exponent and variance are lower. However, its autocorrelation function curve demonstrates an apparent heavy tail distribution, and its Hurst exponent and variance are higher, as it is working in an unstable condition, namely, surge. In addition, the results show that the multi-fractal spectrum parameters are closely related to the dynamic pressure. With the state of centrifugal compressor being changed from stable to unstable states, some multi-fractal spectrum parameters Δα, Δf(α), αmax, and f(αmin) become larger, but αmin in the multi-fractal spectrum show the opposite trend, and consistent properties are graphically shown for the randomly shuffled data. As a conclusion, the proposed method, as one measure method for entropy, can be used to feasibly identify the incipient surge of a centrifugal compressor and design its surge controller. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Entropy Analysis of Soccer Dynamics
Entropy 2019, 21(2), 187; https://doi.org/10.3390/e21020187 - 16 Feb 2019
Cited by 16 | Viewed by 1490
Abstract
This paper adopts the information and fractional calculus tools for studying the dynamics of a national soccer league. A soccer league season is treated as a complex system (CS) with a state observable at discrete time instants, that is, at the time of [...] Read more.
This paper adopts the information and fractional calculus tools for studying the dynamics of a national soccer league. A soccer league season is treated as a complex system (CS) with a state observable at discrete time instants, that is, at the time of rounds. The CS state, consisting of the goals scored by the teams, is processed by means of different tools, namely entropy, mutual information and Jensen–Shannon divergence. The CS behavior is visualized in 3-D maps generated by multidimensional scaling. The points on the maps represent rounds and their relative positioning allows for a direct interpretation of the results. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Article
Approximation to Hadamard Derivative via the Finite Part Integral
Entropy 2018, 20(12), 983; https://doi.org/10.3390/e20120983 - 18 Dec 2018
Cited by 3 | Viewed by 1054
Abstract
In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful [...] Read more.
In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique. Full article
(This article belongs to the Special Issue The Fractional View of Complexity)
Show Figures

Figure 1

Back to TopTop