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A FRACTAL MODIFICATION OF THE PSEUDO-PARABOLIC EQUATION AND ITS GENERALIZED FRACTAL VARIATIONAL PRINCIPLE 伪抛物方程的分形修正及其广义分形变分原理
Fractals Pub Date : 2024-02-14 DOI: 10.1142/s0218348x24500373
Kang-Jia Wang, Shuai Li, Peng Xu, Feng Shi
{"title":"A FRACTAL MODIFICATION OF THE PSEUDO-PARABOLIC EQUATION AND ITS GENERALIZED FRACTAL VARIATIONAL PRINCIPLE","authors":"Kang-Jia Wang, Shuai Li, Peng Xu, Feng Shi","doi":"10.1142/s0218348x24500373","DOIUrl":"https://doi.org/10.1142/s0218348x24500373","url":null,"abstract":"In this work, a new fractal pseudo-parabolic equation is derived by means of He’s fractal derivative. The semi-inverse method (SIM) is employed to develop the generalized fractal variational principle (GFVP), which can reveal the energy conservation law in the fractal space and provide some new insights on the study of the variational method.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"54 19","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139778868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A FRACTAL MODIFICATION OF THE PSEUDO-PARABOLIC EQUATION AND ITS GENERALIZED FRACTAL VARIATIONAL PRINCIPLE 伪抛物方程的分形修正及其广义分形变分原理
Fractals Pub Date : 2024-02-14 DOI: 10.1142/s0218348x24500373
Kang-Jia Wang, Shuai Li, Peng Xu, Feng Shi
{"title":"A FRACTAL MODIFICATION OF THE PSEUDO-PARABOLIC EQUATION AND ITS GENERALIZED FRACTAL VARIATIONAL PRINCIPLE","authors":"Kang-Jia Wang, Shuai Li, Peng Xu, Feng Shi","doi":"10.1142/s0218348x24500373","DOIUrl":"https://doi.org/10.1142/s0218348x24500373","url":null,"abstract":"In this work, a new fractal pseudo-parabolic equation is derived by means of He’s fractal derivative. The semi-inverse method (SIM) is employed to develop the generalized fractal variational principle (GFVP), which can reveal the energy conservation law in the fractal space and provide some new insights on the study of the variational method.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"150 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139838702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
PHYSICS-INFORMED DEEP AI SIMULATION FOR FRACTAL INTEGRO-DIFFERENTIAL EQUATION 分形积分微分方程的物理信息深度 AI 仿真
Fractals Pub Date : 2024-01-31 DOI: 10.1142/s0218348x24500221
XUEJUAN LI, RUI ZHAO
{"title":"PHYSICS-INFORMED DEEP AI SIMULATION FOR FRACTAL INTEGRO-DIFFERENTIAL EQUATION","authors":"XUEJUAN LI, RUI ZHAO","doi":"10.1142/s0218348x24500221","DOIUrl":"https://doi.org/10.1142/s0218348x24500221","url":null,"abstract":"<p>Fractal integro-differential equations (IDEs) can describe the effect of local microstructure on a complex physical problem, however, the traditional numerical methods are not suitable for solving the new-born models with the fractal integral and fractal derivative. Here we show that deep learning can be used to solve the bottleneck. By the two-scale transformation, the fractal IDE is first approximately converted to its traditional integro-differential partner, which is further converted to a differential equation system by introducing an auxiliary variable to remove the integral operation. Moreover, a flexible adaptive technology is adopted to deal with the loss weights of a deep learning neural network. A fractal Volterra IDE is used to show the effectiveness and simplicity of this new physics-informed deep AI simulation model. All results indicate the AI simulation model has good robustness and convergence, and the fractal Volterra IDE might explore the different properties of viscoelasticity for a porous medium.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
EXPLORING INSECTS FREE FLIGHT: ENHANCING THE DIPTERAN FLIGHT MODEL TO INCLUDE FRACTAL EFFECTS 探索昆虫的自由飞行：增强双翅目昆虫的飞行模型，使其包含分形效应
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500154
ALEX ELÍAS-ZÚÑIGA, OSCAR MARTÍNEZ-ROMERO, DANIEL OLVERA-TREJO, IMPERIO ANEL PERALES-MARTÍNEZ, LUIS MANUEL PALACIOS-PINEDA
{"title":"EXPLORING INSECTS FREE FLIGHT: ENHANCING THE DIPTERAN FLIGHT MODEL TO INCLUDE FRACTAL EFFECTS","authors":"ALEX ELÍAS-ZÚÑIGA, OSCAR MARTÍNEZ-ROMERO, DANIEL OLVERA-TREJO, IMPERIO ANEL PERALES-MARTÍNEZ, LUIS MANUEL PALACIOS-PINEDA","doi":"10.1142/s0218348x24500154","DOIUrl":"https://doi.org/10.1142/s0218348x24500154","url":null,"abstract":"<p>This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect’s anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
MULTIPARENT FRACTAL IMAGE CODING-BASED METHODS FOR SALT-AND-PEPPER NOISE REMOVAL 基于多子分形图像编码的椒盐噪声消除方法
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500129
WEIJIE LIANG, XIAOYI LI, ZHIHUI TU, JIAN LU
{"title":"MULTIPARENT FRACTAL IMAGE CODING-BASED METHODS FOR SALT-AND-PEPPER NOISE REMOVAL","authors":"WEIJIE LIANG, XIAOYI LI, ZHIHUI TU, JIAN LU","doi":"10.1142/s0218348x24500129","DOIUrl":"https://doi.org/10.1142/s0218348x24500129","url":null,"abstract":"<p>Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
MEASURING STRUCTURAL CHANGES OF RECURRENCE PATTERNS IN MULTIFRACTAL AND MULTISCALE ASPECTS BY GENERALIZED RECURRENCE LACUNARITY 通过广义递归缺陷测量多分形和多尺度递归模式的结构变化
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500178
XUEGENG MAO, ZEZHOU LIU, JINZHAO LIU, WANRU XIE, PENGJIAN SHANG, ZHIWEI SHAO
{"title":"MEASURING STRUCTURAL CHANGES OF RECURRENCE PATTERNS IN MULTIFRACTAL AND MULTISCALE ASPECTS BY GENERALIZED RECURRENCE LACUNARITY","authors":"XUEGENG MAO, ZEZHOU LIU, JINZHAO LIU, WANRU XIE, PENGJIAN SHANG, ZHIWEI SHAO","doi":"10.1142/s0218348x24500178","DOIUrl":"https://doi.org/10.1142/s0218348x24500178","url":null,"abstract":"<p>Recurrence lacunarity has been recently proposed to detect dynamical state transitions over various temporal scales. In this paper, we combine suggested distribution moments and introduce multifractal recurrence lacunarity to unearth rich information of trajectories in phase space. By considering generalized moments, it provides an enhanced measurement to account for differences of black pixels in the recurrence plot at various scales. Numerical simulations have proved that the proposed method is able to differentiate varying types of time series and provide further insights of inherent features including stochastic series, chaotic maps and series contaminated interference components. In real-world applications, it performs well on quantifying the subtle structural changes of financial time series. In addition, it is intriguing to confirm that corrugation signals possess much more vivid information of heterogeneity in terms of recurrence plots than normal ones.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"144 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
RESEARCH ON THE K-DIMENSION OF THE SUM OF TWO CONTINUOUS FUNCTIONS AND ITS APPLICATION 关于两个连续函数之和的 k 维数及其应用的研究
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500300
Y. X. CAO, N. LIU, Y. S. LIANG
{"title":"RESEARCH ON THE K-DIMENSION OF THE SUM OF TWO CONTINUOUS FUNCTIONS AND ITS APPLICATION","authors":"Y. X. CAO, N. LIU, Y. S. LIANG","doi":"10.1142/s0218348x24500300","DOIUrl":"https://doi.org/10.1142/s0218348x24500300","url":null,"abstract":"<p>In this paper, we have done some research studies on the fractal dimension of the sum of two continuous functions with different <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span>-dimensions and approximation of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi></math></span><span></span>-dimensional fractal functions. We first investigate the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span>-dimension of the linear combination of fractal function whose <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>K</mi></math></span><span></span>-dimension is <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi></math></span><span></span> and the function satisfying Lipschitz condition is still <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi></math></span><span></span>-dimensional. Then, based on the research of fractal term and the Weierstrass approximation theorem, an approximation of the <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>s</mi></math></span><span></span>-dimensional continuous function is given, which is composed of finite triangular series and partial Weierstrass function. In addition, some preliminary results on the approximation of one-dimensional and two-dimensional fractal continuous functions have been given.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"149 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
THE MULTI-PARAMETER FRACTAL–FRACTIONAL INEQUALITIES FOR FRACTAL (P,m)-CONVEX FUNCTIONS 反褶（P，m）-反褶函数的多参数反褶-反褶不等式
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500257
XIAOMAN YUAN, HÜSEYIN BUDAK, TINGSONG DU
{"title":"THE MULTI-PARAMETER FRACTAL–FRACTIONAL INEQUALITIES FOR FRACTAL (P,m)-CONVEX FUNCTIONS","authors":"XIAOMAN YUAN, HÜSEYIN BUDAK, TINGSONG DU","doi":"10.1142/s0218348x24500257","DOIUrl":"https://doi.org/10.1142/s0218348x24500257","url":null,"abstract":"<p>Local fractional calculus theory and parameterized method have greatly assisted in the advancement of the field of inequalities. To continue its enrichment, this study investigates the multi-parameter fractal–fractional integral inequalities containing the fractal <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>P</mi><mo>,</mo><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-convex functions. Initially, we formulate the new conception of the fractal <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>P</mi><mo>,</mo><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-convex functions and work on a variety of properties. Through the assistance of the fractal–fractional integrals, the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>ℓ</mi></math></span><span></span>-fractal identity with multiple parameters is established, and from that, integral inequalities are inferred regarding twice fractal differentiable functions which are fractal <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>P</mi><mo>,</mo><mi>m</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-convex. Furthermore, a few typical and novel outcomes are discussed and visualized for specific parameter values, separately. It concludes with some applications in respect of the special means, the quadrature formulas and random variable moments, respectively.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
FRACTAL STUDY ON THE PERMEABILITY IN CHARGED MICRO-FRACTURED POROUS MEDIA 带电微裂隙多孔介质渗透性的分形研究
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500208
WENYAN LIU, YUZENG DUAN, BOQI XIAO, LIANG LUO, MINGQING ZOU, MINGCHAO LIANG
{"title":"FRACTAL STUDY ON THE PERMEABILITY IN CHARGED MICRO-FRACTURED POROUS MEDIA","authors":"WENYAN LIU, YUZENG DUAN, BOQI XIAO, LIANG LUO, MINGQING ZOU, MINGCHAO LIANG","doi":"10.1142/s0218348x24500208","DOIUrl":"https://doi.org/10.1142/s0218348x24500208","url":null,"abstract":"<p>Fractured porous media is of great significance to the exploration and development of unconventional reservoirs. In this paper, a fractal model for permeability through micro-fractured porous media with consideration of the electric double layer (EDL) effect is proposed based on the fractal theory. The present model indicates that the permeability is a function of the electrokinetic parameters and micro-structural parameters of fractured porous media, and each parameter has a clear physical meaning, the results from the proposed model are found to be in good agreement with experimental data. Moreover, factors influencing the permeability are also analyzed in detail. The results indicate that the more obvious EDL effects will lead to permeability becoming lower. The present fractal model for the permeability with the EDL effect can provide guidance for tight reservoir development or other micro-porous media transportation.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
ON THE ASYMPTOTIC STABILITY OF A NEW FRACTIONAL-ORDER SLIDING MODE CONTROL WITH APPLICATION TO ROBOTIC SYSTEMS 新分数阶滑动模态控制的渐近稳定性及其在机器人系统中的应用
Fractals Pub Date : 2024-01-27 DOI: 10.1142/s0218348x24500312
FATMA ABDELHEDI, RIM JALLOULI KHLIF, AHMED SAID NOURI, NABIL DERBEL
{"title":"ON THE ASYMPTOTIC STABILITY OF A NEW FRACTIONAL-ORDER SLIDING MODE CONTROL WITH APPLICATION TO ROBOTIC SYSTEMS","authors":"FATMA ABDELHEDI, RIM JALLOULI KHLIF, AHMED SAID NOURI, NABIL DERBEL","doi":"10.1142/s0218348x24500312","DOIUrl":"https://doi.org/10.1142/s0218348x24500312","url":null,"abstract":"<p>This paper presents an advanced control strategy based on Fractional-Order Sliding Mode Control (FO-SMC), which introduces a robust solution to significantly improve the reliability of robotic manipulator systems and increase its control performance. The proposed FO-SMC strategy includes a two-key term-based Fractional Sliding Function (FSF) that presents the main contribution of this work. Additionally, a fractional-order-based Lyapunov stability analysis is developed for a class of nonlinear systems to guarantee the asymptotic stability of the closed loop system. Four FSF-based versions of the designed FO-SMC are studied and discussed. Various scenarios of the proposed control strategy are tested on a 3-degree-of-freedom SCARA robotic arm and compared to recent FO-SMC works, demonstrating the effectiveness of the new proposed control strategy to (i) ensure the asymptotic stability, (ii) achieve a smooth start-up, (iii) cancel the static error, giving a good tracking trajectory, and (iv) reduce the control torques, yielding a consumed energy minimization.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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