Search Results (51)

A geodesic or geodetic line on a sphere is called the orthodrome. Research has shown that the orthodrome can be defined in a large number of ways. This article provides an overview of various definitions of the orthodrome. We recall the definitions of the orthodrome according to the greats of geodesy, such as Bessel and Helmert. We derive the equation of the orthodrome in the geographic coordinate system and in the Cartesian spatial coordinate system. A geodesic on a surface is a curve for which the geodetic curvature is zero at every point. Equivalent expressions of this statement are that at every point of this curve, the principal normal vector is collinear with the normal to the surface, i.e., it is a curve whose binormal at every point is perpendicular to the normal to the surface, and that it is a curve whose osculation plane contains the normal to the surface at every point. In this case, the well-known Clairaut equation of the geodesic in geodesy appears naturally. It is found that this equation can be written in several different forms. Although differential equations for geodesics can be found in the literature, they are solved in this article, first, by taking the sphere as a special case of any surface, and then as a special case of a surface of rotation. At the end of this article, we apply calculus of variations to determine the equation of the orthodrome on the sphere, first in the Bessel way, and then by applying the Euler–Lagrange equation. Overall, this paper elaborates a dozen different approaches to orthodrome definitions. Full article

This paper introduces a theoretical large-scale radio channel model for the downlink in cellular systems, aimed at estimating variations in received signal power at the user terminal as a function of device mobility. This enables applications such as direction-of-arrival (DoA) estimation, estimating power at subsequent points based on received power, and detection of coverage anomalies. The model is validated using real-world measurements from urban and suburban environments, achieving a maximum estimation error of 7.6%. In contrast to conventional models like Okumura–Hata, COST-231, Third Generation Partnership Project (3GPP) stochastic models, or ray-tracing techniques, which estimate average power under static conditions, the proposed model captures power fluctuations induced by terminal movement, a factor often neglected. Although advanced techniques such as wave-domain processing with intelligent metasurfaces can also estimate DoA, this model provides a simpler, geometry-driven approach based on empirical traces. While it does not incorporate infrastructure-specific characteristics or inter-cell interference, it remains a practical solution for scenarios with limited information or computational resources. Full article

To address the limitations of traditional tumor diagnostic methods in image feature extraction and model generalization, this study innovatively proposes a synergistic diagnostic model that integrates fractional Fourier transform (FrFT) and error back-propagation (BP) neural networks. The model leverages the time–frequency analysis capability of FrFT and incorporates the fractal characteristics observed during tumor proliferation, effectively enhancing multi-scale feature extraction and representation. Experimental results show that the proposed model achieves an accuracy of 93.177% in classifying benign and malignant tumors, outperforming the support vector machine (SVM) method. The integration of FrFT improves feature distinguishability and reduces dependence on manual extraction. This study not only represents a breakthrough in tumor diagnostic technology but also paves new avenues for the application of fractional calculus and fractal geometry in medical image analysis. The findings show great potential for clinical application and future development. Full article

This article presents the Transfer Matrix Method as a mathematical approach for the calculus of different structures that can be discretized into elements using an iterative calculus for future applications in the vehicle industry. Plate calculus is important in construction, medicine, orthodontics, and many other fields. This work is original due to the mathematical apparatus used in the calculus of long rectangular plates embedded in both long borders and required by a uniformly distributed force on a line parallel to the long borders. The plate is discretized along its length in unitary beams, which have the width of the rectangular plate. The unitary beam can also be discretized into parts. As applications, the long rectangular plates embedded on the two long borders and charged with a vertical uniform load that acts on a line parallel to the long borders are studied. A state vector is associated with each side. For each of the four cases studied, a matrix relationship was written for each side, based on a transfer matrix, the state vector corresponding to the origin side, and the vector due to the action of external forces acting on the considered side. After, it is possible to calculate all the state vectors for all sides of the unity beam. Now, the efforts, deformations, and stress can be calculated in any section of the beam, respectively, for the long rectangular plate. This calculus will serve as a calculus of resistance for different pieces of the components of vehicles. Full article

This paper presents a generalization of the cross product to N dimensions, extending the classical operation beyond its traditional confines in three-dimensional space. By redefining the cross product to accommodate $N-1$ arguments in N dimensions, a framework has been established that retains the core properties of orthogonality, magnitude, and anticommutativity. The proposed method leverages the determinant approach and introduces the polar sine function to calculate the magnitude of the cross product, linking it directly to the volume of an N-dimensional parallelotope. This generalization not only enriches the theoretical foundation of vector calculus but also opens up new applications in high-dimensional data analysis, machine learning, and multivariate time series. The results suggest that this extension of the cross product could serve as a powerful tool for modeling complex interactions in multi-dimensional spaces, with potential implications across various scientific and engineering disciplines. Full article

This study presents a novel Fractional Whale Optimization Algorithm-Enhanced Support Vector Regression (FWOA-SVR) framework for solar energy forecasting, addressing the limitations of traditional SVR in modeling complex relationships within data. The proposed framework incorporates fractional calculus in the Whale Optimization Algorithm (WOA) to improve the balance between exploration and exploitation during hyperparameter tuning. The FWOA-SVR model is comprehensively evaluated against traditional SVR, Long Short-Term Memory (LSTM), and Backpropagation Neural Network (BPNN) models using training, validation, and testing datasets. Experimental results show that FWOA-SVR achieves superior performance with the lowest MSE values (0.036311, 0.03942, and 0.03825), RMSE values (0.19213, 0.19856, and 0.19577), and the highest R² values (0.96392, 0.96104, and 0.96192) for training, validation, and testing, respectively. These results highlight the significant improvements of FWOA-SVR in prediction accuracy and efficiency, surpassing benchmark models in capturing complex patterns within the data. The findings highlight the effectiveness of integrating fractional optimization techniques into machine learning frameworks for advancing solar energy forecasting solutions. Full article

Electric vehicles demand efficient and robust motor control to maximize range and performance. This paper presents an innovative adaptive fractional-order sliding mode (FO-SM) control approach tailored for Direct Torque Control with Space Vector Modulation (DTC-SVM) applied to induction motor drives. This approach tackles the challenges of parameter variations inherent in real-world applications, such as temperature changes and load fluctuations. By leveraging the inherent robustness of FO-SM and the fast dynamic response of DTC-SVM, our proposed control strategy achieves superior performance, significantly reduced torque ripple, and improved efficiency. The adaptive nature of the control system allows for real-time adjustments based on system conditions, ensuring reliable operation even in the presence of uncertainties. This research presents a significant advancement in electric vehicle propulsion systems, offering a powerful and adaptable control solution for induction motor drives. Our findings demonstrate the potential of this innovative approach to enhance the robustness and performance of electric vehicles, paving the way for a more sustainable and efficient future of transportation. In fact, the paper proposes using an adaptive approach to control the electric vehicle’s speed based on the fractional calculus of sliding mode control. The adaptive algorithm converges to the actual values of all system parameters. Moreover, the obtained performance results are reached without precise system modeling. Full article

Vector-borne infections pose serious public health challenges due to the complex interplay of biological, environmental, and social factors. Therefore, comprehensive approaches are essential to mitigate the burden of vector-borne infections and minimize their impact on public health. In this research, an epidemic model for the vector-borne disease malaria is structured with a saturated incidence rate via fractional calculus and preventive measures. The essential results and concepts are introduced to examine the proposed model. The solution of the system is examined for some necessary results, and the threshold parameter of the model, indicated by ${\mathcal{R}}_0$, is calculated. In this paper, the proposed malaria model is analyzed both quantitatively and qualitatively. The fixed-point theorems of Banach and Schaefer are utilized to examine the uniqueness and existence of the solution dynamics. Furthermore, the necessary conditions for the stability of the model have been determined. A numerical approach is offered to visualize the solution pathways of the system and identify its key factors. Through the results, the most influential factors for the control and management of the disease are highlighted. Full article

Support vector machine (SVM) models apply the Karush–Kuhn–Tucker (KKT-OC) optimality conditions in the ordinary derivative to the primal optimisation problem, which has a major influence on the weights associated with the dissimilarity between the selected support vectors and subsequently on the quality of the model’s predictions. Recognising the capacity of fractional derivatives to provide machine learning models with more memory through more microscopic differentiations, in this paper we generalise KKT-OC based on ordinary derivatives to KKT-OC using fractional derivatives (Frac-KKT-OC). To mitigate the impact of noise and identify support vectors from noise, we apply the Frac-KKT-OC method to the fuzzy intuitionistic version of SVM (IFSVM). The fractional fuzzy intuitionistic SVM model (Frac-IFSVM) is then evaluated on six sets of data from the UCI and used to predict the sentiments embedded in tweets posted by people with diabetes. Taking into account four performance measures (sensitivity, specificity, F-measure, and G-mean), the Frac-IFSVM version outperforms SVM, FSVM, IFSVM, Frac-SVM, and Frac-FSVM. Full article

In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a $2\alpha$-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for $2\alpha$-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for $2\alpha$-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation. Full article

The famous hypercontractive estimate discovered independently by Gross, Bonami and Beckner has had a great impact on combinatorics and theoretical computer science since it was first used in this setting in a seminal paper by Kahn, Kalai and Linial. The usual proofs of this inequality begin with two-point space, where some elementary calculus is used and then generalised immediately by introducing another dimension using submultiplicativity (Minkowski’s integral inequality). In this paper, we prove this inequality using information theory. We compare the entropy of a pair of correlated vectors in ${\{0,1\}}^n$ to their separate entropies, analysing them bit by bit (not as a figure of speech, but as the bits are revealed) using the chain rule of entropy. Full article

This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks. Full article

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications. Full article

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. Full article

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method’s compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method’s performance in practice. Full article