Pub Date : 2022-08-31DOI: 10.15377/2409-5761.2022.09.9
Jun-Sheng Duan, Ji-Lian Wang
We propose the conjecture that for any positive integers r and n with n > 2, there do not exist 2r + 1 consecutive positive integers in natural order such that the sum of n-th powers of the first r + 1 integers equals the sum of n-th powers of the subsequent r integers, i.e., there are no positive integers r, m and n, where r < m and n > 2, satisfying (m – r)n + (m – r + 1)n + … + mn = (m + 1)n + (m + 2)n + … + (m + r)n. We prove that the conjecture is true for the cases n = 3 and n = 4. We also verified by using Mathematica that the conjecture is true for the cases 3 < n < 10 and m < 5000.
我们propose任何积极的conjecture那integers r和n与n > 2,有do不存在2r + 1 consecutive积极integers在自然秩序如此那sum of鲍尔n-th》第一r + 1 integers equals the sum of鲍尔n-th of the subsequent r integers,神盾局,没有积极integers是r, m和n, r < m和n > 2,在令人满意(m + r) n (r + 1) n + ... mn = (m + 1) n (m + 2) + ... + ( m + r) n。我们证明了对形势的影响是真实的。我们还通过使用Mathematica来验证，这种联系是真实的3 < n < 10和m < 5000。
{"title":"A Conjecture Congenetic with Fermat’s Last Theorem","authors":"Jun-Sheng Duan, Ji-Lian Wang","doi":"10.15377/2409-5761.2022.09.9","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.9","url":null,"abstract":"We propose the conjecture that for any positive integers r and n with n > 2, there do not exist 2r + 1 consecutive positive integers in natural order such that the sum of n-th powers of the first r + 1 integers equals the sum of n-th powers of the subsequent r integers, i.e., there are no positive integers r, m and n, where r < m and n > 2, satisfying (m – r)n + (m – r + 1)n + … + mn = (m + 1)n + (m + 2)n + … + (m + r)n. We prove that the conjecture is true for the cases n = 3 and n = 4. We also verified by using Mathematica that the conjecture is true for the cases 3 < n < 10 and m < 5000.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132590381","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}
Pub Date : 2022-05-31DOI: 10.15377/2409-5761.2022.09.8
Dimitrios Katsikis, A. Muradova, G. Stavroulakis
A modern approach to solving mathematical models involving differential equations, the so-called Physics-Informed Neural Network (PINN), is based on the techniques which include the use of artificial neural networks and the method of fitting the governing differential equations at collocation points. In this paper, training of the PINN with an application of optimization techniques is performed on simple one-dimensional mechanical problems of elasticity, namely rods and beams. Different boundary conditions are considered. Required computer algorithms are implemented using Python programming packages with the intention of creating neural networks. Numerical results are presented, and the efficiency of the proposed technique is investigated through numerical experiments with different numbers of epochs, batches, hidden layers, neurons, and collocation points. The paper provides useful skills for using a PINN for different problems of solid mechanics. The proposed methodology is a continuation of our intention of using PINNs for problems of the theory of elasticity. The objectives are to present simply the main steps of constructing PINN and an implementation of it. A detailed explanation of the Python programming code, based on the scientific software Tensorflow, built in the Keras library and optimizers, may help compose an effective code for complicated models in mechanics. PINNs are proposed in many recent publications to solve complicated direct and inverse problems. It seems to be a promising method that will play a central role in the development of computational mechanics in the near future. Nevertheless, the lack of educational material does not help new users to enter this scientific area. The present contribution describes the method for the solution of elementary rod and beam problems and gives computer codes that may help the reader to understand the method and to apply it to other problems.
{"title":"A Gentle Introduction to Physics-Informed Neural Networks, with Applications in Static Rod and Beam Problems","authors":"Dimitrios Katsikis, A. Muradova, G. Stavroulakis","doi":"10.15377/2409-5761.2022.09.8","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.8","url":null,"abstract":"A modern approach to solving mathematical models involving differential equations, the so-called Physics-Informed Neural Network (PINN), is based on the techniques which include the use of artificial neural networks and the method of fitting the governing differential equations at collocation points. In this paper, training of the PINN with an application of optimization techniques is performed on simple one-dimensional mechanical problems of elasticity, namely rods and beams. Different boundary conditions are considered.\u0000Required computer algorithms are implemented using Python programming packages with the intention of creating neural networks. Numerical results are presented, and the efficiency of the proposed technique is investigated through numerical experiments with different numbers of epochs, batches, hidden layers, neurons, and collocation points.\u0000The paper provides useful skills for using a PINN for different problems of solid mechanics. The proposed methodology is a continuation of our intention of using PINNs for problems of the theory of elasticity. The objectives are to present simply the main steps of constructing PINN and an implementation of it. A detailed explanation of the Python programming code, based on the scientific software Tensorflow, built in the Keras library and optimizers, may help compose an effective code for complicated models in mechanics.\u0000PINNs are proposed in many recent publications to solve complicated direct and inverse problems. It seems to be a promising method that will play a central role in the development of computational mechanics in the near future. Nevertheless, the lack of educational material does not help new users to enter this scientific area. The present contribution describes the method for the solution of elementary rod and beam problems and gives computer codes that may help the reader to understand the method and to apply it to other problems.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"249 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114552785","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}
Pub Date : 2022-05-24DOI: 10.15377/2409-5761.2022.09.3
Tat’y Mwata-Velu, J. Ruiz-Pinales, J. Aviña-Cervantes, J. González-Barbosa, J. Contreras-Hernandez
Brain-Computer Interface (BCI) paradigms based on Motor Imagery Electroencephalogram (MI-EEG) signals have been developed because the related signals can be generated voluntarily to control further applications. Researches using strong and stout limbs MI-EEG signals reported performing significant classification rates for BCI applied systems. However, MI-EEG signals produced by imagined movements of small limbs present a real classification challenge to be effectively used in BCI systems. It is due to a reduced signal level and increased noisy distorted effects. This study aims to decode individual right-hand fingers’ imagined movements for BCI applications, using MI-EEG signals from C3, Cz, P3, and Pz channels. For this purpose, the Empirical Mode Decomposition (EMD) preprocesses the non-stationary and non-linear EEG signals to finally use a Bidirectional Long Short-Term Memory (BiLSTM) to classify corresponding feature sequences. An average accuracy of 98.8 % was achieved for ring-finger movements decoding using k-fold cross-validation on a public dataset (Scientific-Data). The obtained results support that the proposed framework can be used for BCI control applications.
{"title":"Empirical Mode Decomposition and a Bidirectional LSTM Architecture Used to Decode Individual Finger MI-EEG Signals","authors":"Tat’y Mwata-Velu, J. Ruiz-Pinales, J. Aviña-Cervantes, J. González-Barbosa, J. Contreras-Hernandez","doi":"10.15377/2409-5761.2022.09.3","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.3","url":null,"abstract":"Brain-Computer Interface (BCI) paradigms based on Motor Imagery Electroencephalogram (MI-EEG) signals have been developed because the related signals can be generated voluntarily to control further applications. Researches using strong and stout limbs MI-EEG signals reported performing significant classification rates for BCI applied systems. However, MI-EEG signals produced by imagined movements of small limbs present a real classification challenge to be effectively used in BCI systems. It is due to a reduced signal level and increased noisy distorted effects. This study aims to decode individual right-hand fingers’ imagined movements for BCI applications, using MI-EEG signals from C3, Cz, P3, and Pz channels. For this purpose, the Empirical Mode Decomposition (EMD) preprocesses the non-stationary and non-linear EEG signals to finally use a Bidirectional Long Short-Term Memory (BiLSTM) to classify corresponding feature sequences. An average accuracy of 98.8 % was achieved for ring-finger movements decoding using k-fold cross-validation on a public dataset (Scientific-Data). The obtained results support that the proposed framework can be used for BCI control applications.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115589757","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}
Accurate semantic segmentation of each coronary artery using invasive coronary angiography (ICA) is important for stenosis assessment and coronary artery disease (CAD) diagnosis. In this paper, we propose a multi-step semantic segmentation algorithm based on analyzing arterial segments extracted from ICAs. The proposed algorithm firstly extracts the entire arterial binary mask (binary vascular tree) using a deep learning-based method. Then we extract the centerline of the binary vascular tree and separate it into different arterial segments. Finally, by extracting the underlying arterial topology, position, and pixel features, we construct a powerful coronary artery segment classifier based on a support vector machine. Each arterial segment is classified into the left coronary artery (LCA), left anterior descending (LAD), and other types of arterial segments. The proposed method was tested on a dataset with 225 ICAs and achieved a mean accuracy of 70.33% for the multi-class artery classification and a mean intersection over union of 0.6868 for semantic segmentation of arteries. The experimental results show the effectiveness of the proposed algorithm, which provides impressive performance for analyzing the individual arteries in ICAs.
{"title":"Semantic Segmentation to Extract Coronary Arteries in Invasive Coronary Angiograms","authors":"Chen Zhao, Robert Bober, Haipeng Tang, Jinshan Tang, Minghao Dong, Chaoyang Zhang, Zhuo He, Michele Esposito, Zhihui Xu, Weihua Zhou","doi":"10.15377/2409-5761.2022.09.6","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.6","url":null,"abstract":"Accurate semantic segmentation of each coronary artery using invasive coronary angiography (ICA) is important for stenosis assessment and coronary artery disease (CAD) diagnosis. In this paper, we propose a multi-step semantic segmentation algorithm based on analyzing arterial segments extracted from ICAs. The proposed algorithm firstly extracts the entire arterial binary mask (binary vascular tree) using a deep learning-based method. Then we extract the centerline of the binary vascular tree and separate it into different arterial segments. Finally, by extracting the underlying arterial topology, position, and pixel features, we construct a powerful coronary artery segment classifier based on a support vector machine. Each arterial segment is classified into the left coronary artery (LCA), left anterior descending (LAD), and other types of arterial segments. The proposed method was tested on a dataset with 225 ICAs and achieved a mean accuracy of 70.33% for the multi-class artery classification and a mean intersection over union of 0.6868 for semantic segmentation of arteries. The experimental results show the effectiveness of the proposed algorithm, which provides impressive performance for analyzing the individual arteries in ICAs.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"21 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120874369","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}
Pub Date : 2022-05-24DOI: 10.15377/2409-5761.2022.09.2
Zitao Chen, Quanbin Deng, Kairui Chen
Both adaptive dynamic programming and other intelligent algorithms can solve the economic dispatch problem in the microgrid. Adaptive dynamic programming can reduce the computational burden, which the intelligent algorithms suffer from, by using function approximation structure to approximate performance index function. In recent years, it has been also widely used in economic dispatch in the microgrid. In this article, we introduce some recent research trends within the field of adaptive dynamic programming based economic dispatch. Adaptive dynamic programming is firstly reviewed. Then, the current research works about adaptive dynamic programming based economic dispatch are summarized and compared. Furthermore, we point out some topics for future studies.
{"title":"Adaptive Dynamic Programming and Its Application to Economic Dispatch in Microgrid: A Brief Overview","authors":"Zitao Chen, Quanbin Deng, Kairui Chen","doi":"10.15377/2409-5761.2022.09.2","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.2","url":null,"abstract":"Both adaptive dynamic programming and other intelligent algorithms can solve the economic dispatch problem in the microgrid. Adaptive dynamic programming can reduce the computational burden, which the intelligent algorithms suffer from, by using function approximation structure to approximate performance index function. In recent years, it has been also widely used in economic dispatch in the microgrid. In this article, we introduce some recent research trends within the field of adaptive dynamic programming based economic dispatch. Adaptive dynamic programming is firstly reviewed. Then, the current research works about adaptive dynamic programming based economic dispatch are summarized and compared. Furthermore, we point out some topics for future studies.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122129630","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}
Pub Date : 2022-05-24DOI: 10.15377/2409-5761.2022.09.7
S. Yang, Jiapeng Wang, Hengfei Yang
The planetary gearbox is an important part of the wind turbine. There are many random uncertain factors in the process of design, production, installation, and use, and these uncertain factors greatly influence the service life and reliability of the planetary gearbox. Therefore, the influence of uncertain factors needs to be considered in the design process to reduce the risk of failure. In this paper, an uncertainty design optimization method based on evidence theory is proposed, which can consider both interval variables and random variables in the optimization process. Then the megawatt wind turbine planetary gearbox is taken as the research object to analyze its uncertainty sources. Finally, the planetary gearbox is optimized by the proposed method. By comparing the results, the design scheme obtained by the method proposed in this paper is more reliable.
{"title":"Evidence Theory based Uncertainty Design Optimization for Planetary Gearbox in Wind Turbine","authors":"S. Yang, Jiapeng Wang, Hengfei Yang","doi":"10.15377/2409-5761.2022.09.7","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.7","url":null,"abstract":"The planetary gearbox is an important part of the wind turbine. There are many random uncertain factors in the process of design, production, installation, and use, and these uncertain factors greatly influence the service life and reliability of the planetary gearbox. Therefore, the influence of uncertain factors needs to be considered in the design process to reduce the risk of failure. In this paper, an uncertainty design optimization method based on evidence theory is proposed, which can consider both interval variables and random variables in the optimization process. Then the megawatt wind turbine planetary gearbox is taken as the research object to analyze its uncertainty sources. Finally, the planetary gearbox is optimized by the proposed method. By comparing the results, the design scheme obtained by the method proposed in this paper is more reliable.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133662131","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}
Pub Date : 2022-05-24DOI: 10.15377/2409-5761.2022.09.5
Emmanuel Ovalle-Magallanes, Dora E. Alvarado-Carrillo, J. Aviña-Cervantes, I. Cruz-Aceves, J. Ruiz-Pinales, J. Contreras-Hernandez
Coronary stenosis results from unnatural narrowing of the heart arteries due to the accumulation of adipose depots, leading to different heart diseases and yielding top mortality worldwide. Thus far, deep learning-based methods for automatic stenosis over X-ray Coronary Angiography (XCA) have employed state-of-the-art architectures to solve the ImageNet challenge. With the advance of deep learning, contemporary architectures incorporated a variety of attention mechanisms to improve performance. Therefore, this paper presents a study of three attention mechanisms for stenosis detection in XCA images. Extensive experiments and comparisons over different Residual backbone networks are presented to verify the effectiveness of including such attention modules. An improvement of 4%, 10%, and 10% on the accuracy, recall, and F1-score was achieved using the approach, reaching mean values of 0.8787, 0.8610, and 0.8732, respectively.
{"title":"Attention Mechanisms Evaluated on Stenosis Detection using X-ray Angiography Images","authors":"Emmanuel Ovalle-Magallanes, Dora E. Alvarado-Carrillo, J. Aviña-Cervantes, I. Cruz-Aceves, J. Ruiz-Pinales, J. Contreras-Hernandez","doi":"10.15377/2409-5761.2022.09.5","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.5","url":null,"abstract":"Coronary stenosis results from unnatural narrowing of the heart arteries due to the accumulation of adipose depots, leading to different heart diseases and yielding top mortality worldwide. Thus far, deep learning-based methods for automatic stenosis over X-ray Coronary Angiography (XCA) have employed state-of-the-art architectures to solve the ImageNet challenge. With the advance of deep learning, contemporary architectures incorporated a variety of attention mechanisms to improve performance. Therefore, this paper presents a study of three attention mechanisms for stenosis detection in XCA images. Extensive experiments and comparisons over different Residual backbone networks are presented to verify the effectiveness of including such attention modules. An improvement of 4%, 10%, and 10% on the accuracy, recall, and F1-score was achieved using the approach, reaching mean values of 0.8787, 0.8610, and 0.8732, respectively.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117284727","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}
Pub Date : 2022-05-24DOI: 10.15377/2409-5761.2022.09.4
Thiago Carreta Moro, F. R. Chavarette, Roberto Outa, Igor Feliciano Merizio, Estevão Fuzaro Almeida
Large buildings and skyscrapers are vulnerable to environmental, temporal, and anthropological stresses, generating wear and tear that can lead to this social and economic impediment's collapse. The technological improvements of the fourth industrial revolution have resulted in changes in the connection between physical space and man, known as the cyber physic model, which necessitates monitoring systems to protect the structural branch and so correct this structural vulnerability. Thus, the structural health monitoring system is the exact measure of the evolution required by the cyber physic model in construction and the protection of the monumental buildings, ensuring not only their economic development but also the safety of society. Therefore, this research work presents the innovative proposal of the cyber-physical structural health monitoring system aimed at buildings and skyscrapers, based on and differentiated by intelligent computing techniques, using the negative selection algorithm to perform the analysis and monitoring of structural integrity, overcoming the existing traditional work. The cyber-physical structural health monitoring system will be applied to experimental data obtained from the shear building model that represents these imposing skyscrapers. An artificial immune system will be developed and used in the decision-making process based on the acquisition and processing of the obtained signals to perform the identification, localization, and quantification of possible structural damage. Observing the results, this work proved to be efficient, robust, and economically feasible, having high performance and overcoming the shortcomings of traditional techniques. It represents the perfect measure of cyber physics in the monitoring of large buildings and skyscrapers.
{"title":"Detection, Location and Quantification of Structural Faults in a Two-Story Building Using the Artificial Immunological System","authors":"Thiago Carreta Moro, F. R. Chavarette, Roberto Outa, Igor Feliciano Merizio, Estevão Fuzaro Almeida","doi":"10.15377/2409-5761.2022.09.4","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.4","url":null,"abstract":"Large buildings and skyscrapers are vulnerable to environmental, temporal, and anthropological stresses, generating wear and tear that can lead to this social and economic impediment's collapse. The technological improvements of the fourth industrial revolution have resulted in changes in the connection between physical space and man, known as the cyber physic model, which necessitates monitoring systems to protect the structural branch and so correct this structural vulnerability. Thus, the structural health monitoring system is the exact measure of the evolution required by the cyber physic model in construction and the protection of the monumental buildings, ensuring not only their economic development but also the safety of society. Therefore, this research work presents the innovative proposal of the cyber-physical structural health monitoring system aimed at buildings and skyscrapers, based on and differentiated by intelligent computing techniques, using the negative selection algorithm to perform the analysis and monitoring of structural integrity, overcoming the existing traditional work. The cyber-physical structural health monitoring system will be applied to experimental data obtained from the shear building model that represents these imposing skyscrapers. An artificial immune system will be developed and used in the decision-making process based on the acquisition and processing of the obtained signals to perform the identification, localization, and quantification of possible structural damage. Observing the results, this work proved to be efficient, robust, and economically feasible, having high performance and overcoming the shortcomings of traditional techniques. It represents the perfect measure of cyber physics in the monitoring of large buildings and skyscrapers.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127443833","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}
Pub Date : 2022-04-28DOI: 10.15377/2409-5761.2022.09.1
Mohammed S Abdo
In this research paper, we develop and extend some qualitative analyses of a class of a nonlinear fractional integro-differential equation involving ψ-Caputo fractional derivative (ψ-CFD) and ψ-Riemann-Liouville fractional integral (ψ-RLFI). The existence and uniqueness theorems are obtained in Banach spaces via an equivalent fractional integral equation with the help of Banach’s fixed point theorem (B’sFPT) and Schaefer’s fixed point theorem (S’sFPT). An example explaining the main results is also constructed.
{"title":"Qualitative Analyses of ψ-Caputo Type Fractional Integrodifferential Equations in Banach Spaces","authors":"Mohammed S Abdo","doi":"10.15377/2409-5761.2022.09.1","DOIUrl":"https://doi.org/10.15377/2409-5761.2022.09.1","url":null,"abstract":"In this research paper, we develop and extend some qualitative analyses of a class of a nonlinear fractional integro-differential equation involving ψ-Caputo fractional derivative (ψ-CFD) and ψ-Riemann-Liouville fractional integral (ψ-RLFI). The existence and uniqueness theorems are obtained in Banach spaces via an equivalent fractional integral equation with the help of Banach’s fixed point theorem (B’sFPT) and Schaefer’s fixed point theorem (S’sFPT). An example explaining the main results is also constructed.\u0000 \u0000 ","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114941379","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}
Pub Date : 2021-12-31DOI: 10.15377/2409-5761.2021.08.4
G. Anastassiou
We present a detailed great variety of Hardy type fractional inequalities under convexity and Lp norm in the setting of generalized Prabhakar and Hilfer fractional calculi of left and right integrals and derivatives. The radial multivariate case of the above over a spherical shell is developed in detail to all directions. Many inequalities are of vectorial splitting rational Lp type or of separating rational Lp type, others involve ratios of functions and of fractional integral operators.
{"title":"Vectorial Prabhakar Hardy Type Generalized Fractional Inequalities under Convexity","authors":"G. Anastassiou","doi":"10.15377/2409-5761.2021.08.4","DOIUrl":"https://doi.org/10.15377/2409-5761.2021.08.4","url":null,"abstract":"We present a detailed great variety of Hardy type fractional inequalities under convexity and Lp norm in the setting of generalized Prabhakar and Hilfer fractional calculi of left and right integrals and derivatives. The radial multivariate case of the above over a spherical shell is developed in detail to all directions. Many inequalities are of vectorial splitting rational Lp type or of separating rational Lp type, others involve ratios of functions and of fractional integral operators.","PeriodicalId":335387,"journal":{"name":"Journal of Advances in Applied & Computational Mathematics","volume":"135 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124260257","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}
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