Test Functions Research Articles

An enhanced dung beetle optimizer with multiple strategies for robot path planning

In order to make up for the shortcomings of the original dung beetle optimization algorithm, such as low population diversity, insufficient of global exploration ability, being easy to fall into local optimization and unsatisfactory convergence accuracy, etc. An improved dung beetle optimization algorithm using hybrid multi- strategy is proposed. Firstly, the cubic chaotic mapping approach is used to initialize the population to improve the diversity, expand the search range of the solution space, and enhance the global optimization ability. Secondly, the cooperative search algorithm is utilized to strength communication between individual dung beetles and dung beetle groups in foraging stage to expand the search range of the solution space and enhance the global optimization ability. Thirdly, T-distribution mutation and differential evolutionary variation strategies are introduced to provide perturbation to enhance the diversity of the population and avoid falling into local optimization. Fourthly, the proposed algorithm(named as SSTDBO) is compared with other optimization algorithms, including GODBO, QHDBO, DBO, KOA, NOA, WOA and HHO, by 29 benchmark test functions in CEC2017. The results show that the proposed algorithm has stronger robustness and optimization ability, and algorithm’s performance has substantially enhanced. Finally, the proposed algorithm is applied to solve the real-world robot path planning engineering cases, to demonstrate its effectiveness in dealing with real optimization engineering cases, which further verified how noteworthy the enhanced strategy’s efficacy and the enhanced algorithm’s superiority are in addressing real-world engineering cases.

A Multi-Objective Black-Winged Kite Algorithm for Multi-UAV Cooperative Path Planning

In UAV path-planning research, it is often difficult to achieve optimal performance for conflicting objectives. Therefore, the more promising approach is to find a balanced solution that mitigates the effects of subjective weighting, utilizing a multi-objective optimization algorithm to address the complex planning issues that involve multiple machines. Here, we introduce an advanced mathematical model for cooperative path planning among multiple UAVs in urban logistics scenarios, employing the non-dominated sorting black-winged kite algorithm (NSBKA) to address this multi-objective optimization challenge. To evaluate the efficacy of NSBKA, it was benchmarked against other algorithms using the Zitzler, Deb, and Thiele (ZDT) test problems, Deb, Thiele, Laumanns, and Zitzler (DTLZ) test problems, and test functions from the conference on evolutionary computation 2009 (CEC2009) for three types of multi-objective problems. Comparative analyses and statistical results indicate that the proposed algorithm outperforms on all 22 test functions. To verify the capability of NSBKA in addressing the multi-UAV cooperative problem model, the algorithm is applied to solve the problem. Simulation experiments for three UAVs and five UAVs show that the proposed algorithm can obtain a more reasonable collaborative path solution set for UAVs. Moreover, path planning based on NSBKA is generally superior to other algorithms in terms of energy saving, safety, and computing efficiency during planning. This affirms the effectiveness of the meta-heuristic algorithm in dealing with multiple objective multi-UAV cooperation problems and further enhances the robustness and competitiveness of NSBKA.

Tornado optimizer with Coriolis force: a novel bio-inspired meta-heuristic algorithm for solving engineering problems

This paper proposes a new meta-heuristic algorithm named tornado optimizer with Coriolis force (TOC) which is applied to solve global optimization and constrained engineering problems in continuous search spaces. The fundamental concepts and ideas beyond the proposed TOC Optimizer are inspired by nature based on the observation of the cycle process of tornadoes and how thunderstorms and windstorms evolve into tornadoes using Coriolis force. The Coriolis force is applied to windstorms that directly evolve to form tornadoes based on the developed optimization method. The proposed TOC algorithm mathematically models and implements the behavioral steps of tornado formation by windstorms and thunderstorms and then dissipation of tornadoes on the ground. These steps ultimately lead to feasible solutions when applied to solve optimization problems. These behavioral steps are mathematically represented along with the Coriolis force to allow for a proper balance between exploration and exploitation during the optimization process, as well as to allow search agents to explore and exploit every possible area of the search space. The performance of the proposed TOC optimizer was thoroughly examined on a simple benchmark set of 23 test functions, and a set of 29 well-known benchmark functions from the CEC-2017 test for a variety of dimensions. A comparative study of the computational and convergence analysis results was carried out to clarify the efficacy and stability levels of the proposed TOC optimizer compared to other well-known optimizers. The TOC optimizer outperformed other comparative algorithms using the mean ranks of Friedman’s test by 20.75%, 27.248%, and 25.85% on the 10-, 30-, and 50-dimensional CEC 2017 test set, respectively. The reliability and appropriateness of the TOC optimizer were examined by solving real-world problems including eight engineering design problems and one industrial process. The proposed optimizer divulged satisfactory performance over other competing optimizers regarding solution quality and global optimality as per statistical test methods.

On the absence of weak solutions for a sequential time-fractional Klein–Gordon equation with quadratic nonlinearity

The Klein–Gordon equation with quadratic nonlinearity is frequently used to model various problems of physics. On the other hand, it was shown that in many situations, the use of fractional derivatives instead of ordinary derivatives provides more realistic models. In this work, a one-dimensional sequential time-fractional Klein–Gordon equation posed on a bounded interval, subject to certain initial and boundary conditions, is investigated. The fractional derivatives are considered in the Caputo sense and a weight function [Formula: see text] is allowed in front of the quadratic nonlinearity. Namely, we first establish general sufficient conditions for the nonexistence of solutions. Next, some particular cases of weight functions [Formula: see text] are discussed. In our approach, we make use of the test function method. This method requires an appropriate choice of a family of test functions. In this paper, we make a judicious choice of test functions taking into consideration the nonlocal property of the Caputo fractional derivative, the initial and boundary conditions, and the geometry of the domain.

Structurally Modulated NiV-LDH with CdMoSe-Quantum Dots: Unlocking the Active Centers at S-Scheme Heterojunctions for Stimulating Photocatalytic H2O2 Production and H2 Evolution.

Designing and accumulating quantum dots (QD) onto layered double hydroxide (LDH) for the photocatalytic production of H2 and H2O2 is a formidable task. Here, we intended the synthesis procedure of CdMoSe-QD (CMS)-incorporated NiV-LDH (NV) through a facile in situ reflux method and explored the photocatalytic activities of the CMS/NV (CNV) heterostructure. CNV-1 exhibits a large interface contact area and assures excellent interfacial charge transfer ability. Moreover, CNV-1 exhibits outstanding H2 and H2O2 production rates, i.e., 6.4 and 2.5 times higher than that of pristine NV, respectively, due to formation of an S-scheme heterojunction between NV and CMS. Both NV and CMS function as n-type semiconductors and extend photoresponse to visible regions. The CNV-1 composite achieves 1.67% SCC for photocatalytic H2O2 generation and 7.36% ACE for photocatalytic H2 evolution. The excellent activity is ascribed to higher anodic photocurrent, the quantum confinement effect of CMS, large surface-active sites, and delayed recombination of excitons as supported by PL and EIS measurements. Further, the S-scheme mechanism was authenticated through a radical scavenging test and work function, evaluated by UPS measurement. Altogether, this study exemplifies the concepts of designing a CNV heterostructure, which operates via an n-n-based S-scheme mechanism and aims to enhance photocatalytic H2 and H2O2 production.

Legendre wavelet collocation method for singularly‐perturbed problems using operational matrix and Laplace transformation

AbstractIn this paper, a collocation method based on the Legendre wavelets has been developed for the numerical solution of singularly perturbed problems (SPPs). The integral operators of Legendre scaling functions have been derived using two methods: the operational matrix method and the Laplace transformation method. The operational matrix method gives the approximated value of the integral operator of the Legendre scaling functions, whereas the Laplace transformation gives the exact solution for the integral operators. The Legendre wavelet collocation method together with its properties has been utilized to convert the SPP into the system of algebraic equations with the unknown coefficients. Moreover, the ‐error estimate in the function approximation with Legendre scaling functions has been derived, and it has been verified by taking a test function. The stability analysis of the proposed method has been discussed in detail. In the last, illustrative linear and non‐linear examples have been considered to demonstrate the applicability of the proposed method and the results have been compared with the other existing numerical methods in the recent literature.

BCA: Besiege and Conquer Algorithm

This paper introduces a bio-inspired meta-heuristic algorithm, the Besiege and Conquer Algorithm (BCA), developed to tackle complex and high-dimensional optimization problems. Drawing inspiration from the concept of symmetry and guerrilla warfare strategies, the BCA incorporates four core components: besiege, conquer, balance, and feedback. The besiege strategy strengthens exploration, while the conquer strategy enhances exploitation. Balance and feedback mechanisms maintain a dynamic equilibrium between these capabilities, ensuring robust optimization performance. The algorithm’s effectiveness is validated through benchmark test functions, demonstrating superior results in comparison with existing methods, supported by Friedman rankings and Wilcoxon signed-rank tests. Beyond theoretical and experimental validation, the BCA showcases its real-world relevance through applications in engineering design and classification problems, addressing practical challenges. These results underline the algorithm’s strong exploration, exploitation, and convergence capabilities and its potential to contribute meaningfully to diverse real-world domains.

Research on building energy-saving based on GA-BP coupled improved multi-objective whale optimization algorithm

High building energy consumption, significant greenhouse gas emissions, and high investment are current issues in building renovation. To address this issue, an optimization model based on the improved multi-objective whale optimization algorithm (IMOWOA) coupled with GA-BP is proposed. Firstly, an energy consumption simulation model for buildings was established, and the simulated data was trained using GA-BP. Next, three improvement strategies were introduced: optimal Latin hypercube sampling, cosine control factor, and archimedes spiral. Enhance the search efficiency and convergence time of the IMOWOA algorithm. To test the optimization performance of the IMOWOA algorithm, it was evaluated using DTLZ and UF series test functions. Finally, the Entropy Weight-TOPSIS method was utilized to further screen the optimization results, thereby determining the best renovation strategy. The results show that the R2 of GA-BP is close to 1, indicating good predictive performance. The GA-BP training results can serve as the fitness function for the IMOWOA algorithm. The optimal renovation strategy results in a building energy consumption of 49.98 kW•h /m2, an environmental benefit of ¥1.67 × 105, and a net present value of ¥1.21 × 105. It is evident that the proposed method can achieve a trade-off renovation strategy among multiple conflicting objectives. The feasibility of combining GA-BP with the IMOWOA algorithm for multi-objective building energy-saving renovation was verified, which provides a theoretical reference for future similar studies.

Research on Mobile Robot Path Planning Based on MSIAR-GWO Algorithm

Path planning is of great research significance as it is key to affecting the efficiency and safety of mobile robot autonomous navigation task execution. The traditional gray wolf optimization algorithm is widely used in the field of path planning due to its simple structure, few parameters, and easy implementation, but the algorithm still suffers from the disadvantages of slow convergence, ease of falling into the local optimum, and difficulty in effectively balancing exploration and exploitation in practical applications. For this reason, this paper proposes a multi-strategy improved gray wolf optimization algorithm (MSIAR-GWO) based on reinforcement learning. First, a nonlinear convergence factor is introduced, and intelligent parameter configuration is performed based on reinforcement learning to solve the problem of high randomness and over-reliance on empirical values in the parameter selection process to more effectively coordinate the balance between local and global search capabilities. Secondly, an adaptive position-update strategy based on detour foraging and dynamic weights is introduced to adjust the weights according to changes in the adaptability of the leadership roles, increasing the guiding role of the dominant individual and accelerating the overall convergence speed of the algorithm. Furthermore, an artificial rabbit optimization algorithm bypass foraging strategy, by adding Brownian motion and Levy flight perturbation, improves the convergence accuracy and global optimization-seeking ability of the algorithm when dealing with complex problems. Finally, the elimination and relocation strategy based on stochastic center-of-gravity dynamic reverse learning is introduced for the inferior individuals in the population, which effectively maintains the diversity of the population and improves the convergence speed of the algorithm while avoiding falling into the local optimal solution effectively. In order to verify the effectiveness of the MSIAR-GWO algorithm, it is compared with a variety of commonly used swarm intelligence optimization algorithms in benchmark test functions and raster maps of different complexities in comparison experiments, and the results show that the MSIAR-GWO shows excellent stability, higher solution accuracy, and faster convergence speed in the majority of the benchmark-test-function solving. In the path planning experiments, the MSIAR-GWO algorithm is able to plan shorter and smoother paths, which further proves that the algorithm has excellent optimization-seeking ability and robustness.

Mendelian randomization analysis of female reproductive factors on osteoarthritis.

Epidemiology shows women have a higher incidence of osteoarthritis (OA) than men. However, there is not enough evidence to suggest a direct correlation between female reproductive factors and OA. Therefore, this study will employ Mendelian randomization (MR) analysis to investigate whether there is a causal relationship between the 2. This study used a 2-sample MR analysis with single nucleotide polymorphisms significantly associated with female reproductive factors as instrumental variables (IV). We used inverse variance weighted (IVW), MR-Egger regression, weighted median method to infer a causal relationship between female reproductive factors and OA, Cochran Q heterogeneity test by IVW and MR-Egger method, MR PRESSO method and IVW-radial method to detect outliers, MR_pleiotropy_test function and MR PRESSO method for multivariate validity test, and calculation of F-value was used to assess the presence of weak IVs. Finally, the stability of the findings was assessed using the leave-one-out method. Our research shows that there is no reliable causal relationship between an increase in Age at menarche (years) (AAM) and Age at menopause (years) (AM) and OA, that an increase in Age first had sexual intercourse (years) (AFS) is associated with a decreased risk of knee OA and/or hip OA and hand OA, that an increase in Age at first live birth (years) (AFB) is associated with a decreased risk of knee OA and/or hip OA and knee OA, and that an increase in Number of live births (NOB) is associated with an increased risk of knee OA and/or hip OA. This study provides genetic support for an increase in AFS as a reduced knee OA and/or hip OA and hand OA risk factor, an increase in AFB as a reduced knee OA and/or hip OA and knee OA risk factor, and an increase in NOB as an increased knee OA and/or hip OA risk factor. Further studies are needed to elucidate the potential mechanisms underlying the causal associations between AFS, AFB, and NOB and site-specific OA.

On a Rayleigh-type quotient involving a variable exponent which depends on test functions

Abstract Let $$p:{{\mathbb {R}}}\rightarrow (1,\infty )$$ p : R → ( 1 , ∞ ) be a bounded and continuous function. In this paper, we are concerned with the study of the positivity of the infimum $$\inf \limits _{u\in C_0^\infty (\Omega ){\setminus }\{0\}}\frac{\displaystyle {\int _\Omega }|\nabla u(x)|^{p(u(x))}\;dx}{\displaystyle {\int _\Omega }|u(x)|^{p(u(x))}\;dx}\,$$ inf u ∈ C 0 ∞ ( Ω ) \ { 0 } ∫ Ω | ∇ u ( x ) | p ( u ( x ) ) d x ∫ Ω | u ( x ) | p ( u ( x ) ) d x for all open and bounded domains $$\Omega \subset {{\mathbb {R}}}^N$$ Ω ⊂ R N ( $$N\ge 1$$ N ≥ 1 ). In particular, we give some sufficient conditions on the function p in order to get the positivity of the above infimum and we provide examples of functions p for which the infimum vanishes.

A Two-Stage Adaptive Differential Evolution Algorithm with Accompanying Populations

Stochastic simulations are often used to determine the crossover rates and step size of evolutionary algorithms to avoid the tuning process, but they cannot fully utilize information from the evolutionary process. A two-stage adaptive differential evolution algorithm (APDE) is proposed in this article based on an accompanying population, and it has unique mutation strategies and adaptive parameters that conform to the search characteristics. The global exploration capability can be enhanced by the accompanying population to achieve a balance between global exploration and local search. This algorithm has proven to be convergent with a probability of 1 using the theory of Markov chains. In numerical experiments, the APDE is statistically compared with nine comparison algorithms using the CEC2005 and CEC2017 standard set of test functions, and the results show that the APDE is statistically superior to the comparison methods.

Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations

Abstract: Hydropower is a vital strategic component of China’s clean energy development. Its construction and optimized water resource allocation are crucial for addressing global energy challenges, promoting socio-economic development, and achieving sustainable development. However, the optimization scheduling of cascade hydropower stations is a large-scale, multi-constrained, and nonlinear problem. Traditional optimization methods suffer from low computational efficiency, while conventional intelligent algorithms still face issues like premature convergence and local optima, which severely hinder the full utilization of water resources. This study proposed an improved whale optimization algorithm, the Black-winged Differential-variant Whale Optimization Algorithm (BDWOA), which enhanced population diversity through a Logistic-Sine-Cosine combination chaotic map, improved algorithm flexibility with an adaptive adjustment strategy, and introduced the migration mechanism of the black-winged kite algorithm along with a differential mutation strategy to enhance the global search ability and convergence capacity. The BDWOA algorithm was tested using test functions with randomly generated simulated data, with its performance compared against five related optimization algorithms. Results indicate that the BDWOA achieved the optimal value with the fewest iterations, effectively overcoming the limitations of the original whale optimization algorithm. Further validation using actual runoff data for the cascade hydropower station optimization scheduling model showed that the BDWOA effectively enhanced power generation efficiency. In high-flow years, the average power generation increased by 8.3%, 6.5%, 6.8%, 4.1%, and 8.2% compared to the five algorithms while achieving the shortest computation time. Significant improvements in power generation were also observed in normal-flow and low-flow years. The scheduling solutions generated by the BDWOA can adapt to varying inflow conditions, offering an innovative approach to solving complex hydropower station optimization scheduling problems. This contributes to the sustainable utilization of water resources and supports the long-term development of renewable energy.

An Optimization Design of Energy Consumption for Aluminum Smelting Based on a Multi-Objective Artificial Vulture Algorithm

In the process of regenerative aluminum smelting, the temperature of the furnace needs to be maintained between 700 and 850 by adjusting the setting parameters of the smelting furnace. The setting parameters are usually adjusted by manual work, and inaccuracies in manual operation can lead to wasted energy as well as unstable temperatures. Energy consumption and temperature stability are two conflicting objectives, which are difficult to find optimal parameters for the aluminum smelting process. In this paper, an improved multi-objective artificial vulture algorithm (IMOAVOA) is developed to solve a multi-objective problem of energy consumption and temperature deviations in the regenerative aluminum smelting process. The dynamic switching–elimination mechanism based on crowding distance is proposed to maintain the archive, which enhances the diversity of solutions by dynamically switching the operation space for deleting redundant solutions in the archive and dynamically deleting the solution with the smallest crowding distance in the operation space. The multi-directional leader selection mechanism is developed to select better leaders. To improve the convergence of the algorithm, the bounce strategy is introduced in the IMOAVOA. The effectiveness of the proposed algorithm is verified by UF1-UF10, kursawe, Viennet2, Viennet3, ZDT1-ZDT6, DTLZ4, and DTLZ6 test functions with several multi-objective algorithms. The experimental results indicate that IMOAVOA outperforms the original algorithm and three other multi-objective algorithms in terms of the algorithm convergence, the Pareto front coverage, and the solution diversity. Finally, the proposed algorithm is tested in an application case of regenerative aluminum smelting process. The results show that the optimal parameters for the aluminum smelting process using the proposed algorithm can reduce the consumption while meeting the objective of furnace temperature.

A Comparative Analysis of African Vultures Optimization Algorithm with Current Metaheuristics

With the increasing complexity of optimization problems, new metaheuristic algorithms are being developed. These algorithms show their success by exhibiting superior performances on different problems. In this paper, the performance of 4 recently proposed metaheuristic algorithms, namely Artificial Hummingbird Algorithm (AHA), African Vultures Optimization Algorithm (AVOA), Crayfish Optimization Algorithm (COA) and Marine Predators Optimization Algorithm (MPA) on 26 test functions are compared. As a result of the comparisons, it was observed that the algorithms outperformed each other with very small differences on different functions. At the same time, the comparison results were evaluated by t-test statistical test. AVOA has shown better or comparable performance to other recent metaheuristics in evaluating the quality of solutions for several test functions. It is aimed to use AVOA on different problems in future research.

Improved Honey Badger Algorithm Based on Elite Tangent Search and Differential Mutation with Applications in Fault Diagnosis

This paper presents a critique of the Honey Badger Algorithm (HBA) with regard to its limited exploitation capabilities, susceptibility to local optima, and inadequate pre-exploration mechanisms. In order to address these issues, we propose the Improved Honey Badger Algorithm (IHBA), which integrates the Elite Tangent Search Algorithm (ETSA) and differential mutation strategies. Our approach employs cubic chaotic mapping in the initialization phase and a random value perturbation strategy in the pre-iterative stage to enhance exploration and prevent premature convergence. In the event that the optimal population value remains unaltered across three iterations, the elite tangent search with differential variation is employed to accelerate convergence and enhance precision. Comparative experiments on partial CEC2017 test functions demonstrate that the IHBA achieves faster convergence, greater accuracy, and improved robustness. Moreover, the IHBA is applied to the fault diagnosis of rolling bearings in electric motors to construct the IHBA-VMD-CNN-BiLSTM fault diagnosis model, which quickly and accurately identifies fault types. Experimental verification confirms that this method enhances the speed and accuracy of rolling bearing fault identification compared to traditional approaches.

Assessing post-COVID-19 functional recovery in healthcare workers: Insights from the 6-minute walking test and DLCO analysis.

The COVID-19 pandemic has impacted healthcare workers (HCWs) worldwide, necessitating an understanding of its effects on their health and functional capacity. This study utilized the 6-Minute Walk Test (6MWT) and pulmonary function test (PFT) to evaluate post-infection recovery of HCWs, and analyzed the results in relation to comorbidities, symptoms, and healthcare admission. HCWs who previously tested positive for SARS-CoV-2 were recruited, and the 6MWT and PFT were conducted. Clinical data, including comorbidities, symptoms, hospitalization history, intensive care unit (ICU) admission, and mechanical ventilation, were collected. Binary logistic regression analysis and Fisher's exact test were employed to examine the associations between these factors and diffusing capacity of the lungs for carbon monoxide (DLCO) and the 6MWT. The study comprised 80 HCWs, with various comorbidities and various presenting symptoms. On average, 167.24 days (± 63.83 days) post-SARS-CoV-2 infection, 34 (42.5 %) had a DLCO% < 80 % of the predicted value, while 46 (57.5 %) had a DLCO% of ≥ 80 %. The mean six-minute-walk distance (6MWD) was 400.6 ± 54.1 m. No significant associations were found between DLCO% and most of the factors examined, except for Forced Expiratory Flow 25-75 % (FEF25-75 %), although this was not statistically significant (P=0.069). The study provides significant data regarding the functional recovery of HCWs who have recovered from COVID-19 using the 6MWT. Importantly, the findings demonstrated that SARS-CoV-2 infection did not substantially impair the pulmonary functional capacity of HCWs.

The Cost of Randomness in Evolutionary Algorithms: Crossover Can Save Random Bits.

Evolutionary algorithms make countless random decisions during selection, mutation and crossover operations. These random decisions require a steady stream of random numbers. We analyze the expected number of random bits used throughout a run of an evolutionary algorithm and refer to this as the cost of randomness. We give general bounds on the cost of randomness for mutation-based evolutionary algorithms using 1-bit flips or standard mutations using either a naive or a common, more efficient implementation that uses Θ(logn) random bits per mutation. Uniform crossover is a potentially wasteful operator as the number of random bits used equals the Hamming distance of the two parents, which can be up to n. However, we show for a (2+1) Genetic Algorithm that is known to optimize the test function ONEMAX in roughly (e/2)nlnn expected evaluations, twice as fast as the fastest mutation-based evolutionary algorithms, that the total cost of randomness during all crossover operations on ONEMAX is only Θ(n). A more pronounced effect is shown for the common test function JUMPk, where there is an asymptotic decrease both in the number of evaluations and in the cost of randomness. Consequently, the use of crossover can reduce the cost of randomness below that of the fastest evolutionary algorithms that only use standard mutations.

Improved Honey Badger Algorithm and Its Application to K-Means Clustering

As big data continues to evolve, cluster analysis still has a place. Among them, the K-means algorithm is the most widely used method in the field of clustering, which can cause unstable clustering results due to the random selection of the initial clustering center of mass. In this paper, an improved honey badger optimization algorithm is proposed: (1) The population is initialized using sin chaos to make the population uniformly distributed. (2) The density factor is improved to enhance the optimization accuracy of the population. (3) A nonlinear inertia weight factor is introduced to prevent honey badger individuals from relying on the behavior of past individuals during position updating. (4) To improve the diversity of solutions, random opposition learning is performed on the optimal individuals. The improved algorithm outperforms the comparison algorithm in terms of performance through experiments on 23 benchmark test functions. Finally, in this paper, the improved algorithm is applied to K-means clustering and experiments are conducted on three data sets from the UCI data set. The results show that the improved honey badger optimized K-means algorithm improves the clustering effect over the traditional K-means algorithm.

On the generalized fractional Hankel transforms of arbitrary order in the Zemanian spaces

In this paper, we introduce the fractional Hankel transform in Zemanian spaces Hμ, the space of test functions and in its dual Hμ’, the space of distributions of slow growth, for any real value μ. Moreover, we prove some of its properties. As application, we solve some differential equations to illustrate the theoretical results and the importance of using this transform.