Search Results (487)

This study aims to demonstrate how mathematics, especially calculus concepts, can be expanded to include semi-entities and how these can be applied to sampling activities. Here, the multivalued logic uses pseudo-complemented lattices, instead of Boolean algebras. Truth values can express the intensity of a property: for example, the property of being heavy intensifies as weight increases. They can also express the state-of-the-art knowledge of an individual about a certain thing. To express that a number $x$ approaches $a$ is to say that the statement “$x=b$” is not fully true but approaches the full-true value as $b-a$ approaches zero. This approach generalizes the concept of a limit and the concepts derived from it, such as differentiation and integration. A Monte Carlo algorithm replaces one function with another with finite domain, preferably its finite part, by sampling the domain and calculating its map. The discussion extends to integration over an unbounded interval, integral transforms, and differential equations. This study then covers strategies for producing Monte Carlo estimates of respective problems and determining their crucial truth values. In the discussion, a topic related to axiomatizing set theory is also suggested. Full article

This paper presents a combined theoretical, numerical, and experimental analysis to investigate the lateral plastic crushing behavior and energy absorption of circular and non-circular thin-walled rings between two rigid plates. Theoretical solutions incorporating both linear material hardening and power-law material hardening models are solved via numerical shooting methods. The theoretically predicted force-denting displacement relations agree excellently with both FEA and experimental results. The FEA simulation clearly reveals the coexistence of an upper moving plastic region and a fixed bottom plastic region. A robust automatic extraction method of the fully plastic region at the bottom from FEA is proposed. A modified criterion considering the unloading effect based on the resultant moment of cross-section is proposed to allow accurate theoretical estimation of the fully plastic region length. The detailed study implies an abrupt and almost linear drop of the fully plastic region length after the maximum value by the proposed modified criterion, while the conventional fully plastic criterion leads to significant over-estimation of the length. Evolution patterns of the upper and lower plastic regions in FEA are clearly illustrated. Furthermore, the distribution of plastic energy dissipation is compared in the bottom and upper regions through FEA and theoretical results. Purely analytical solutions are formulated for linear hardening material case by elliptical integrals. A simple algebraic function solution is derived without necessity of solving differential equations for general power-law hardening material case by adopting a constant curvature assumption. Parametric analyses indicate the significant effect of ovality and hardening on plastic region evolution and crushing force. This paper should enhance the understanding of the crushing behavior of circular and non-circular rings applicable to the structural engineering and impact of the absorption domain. Full article

Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such equations. In this approach, the Riemann–Liouville fractional integral operator of variable order is evaluated in closed form via a regularized incomplete Beta function, enabling the transformation of the governing equation into a system of algebraic equations. This wavelet-based spectral scheme attains extremely high accuracy, yielding significantly lower errors than existing numerical techniques. In particular, numerical results show that the proposed method achieves notably improved accuracy compared to existing methods under the same number of basis functions. Its strong convergence properties allow high precision to be achieved with relatively few wavelet basis functions, leading to efficient computations. The method’s accuracy and efficiency are demonstrated on several practical diffusion–wave examples, indicating its suitability for real-world applications. Furthermore, it readily applies to a wide class of fractional partial differential equations (FPDEs) with spatially or temporally varying order, demonstrating versatility for diverse applications. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics. Full article

This paper is about a problem at the intersection of differential geometry, spectral analysis and the theory of manifolds. The study of finite-type subvarieties was initiated by Chen in the 1970s, with the aim of obtaining improved estimates for the mean total curvature of compact subvarieties in Euclidean space. The concept of a finite-type subvariety naturally extends that of a minimal subvariety or surface, the latter being closely related to variational calculus. In this work, we classify factorable surfaces in the Lorentz–Heisenberg space ${\mathbb{H}}_3$, equipped with a flat metric satisfying ${\Delta }^I{r}_i={\lambda }_i{r}_i$, which satisfies algebraic equations involving coordinate functions and the Laplacian operator with respect to the surface’s first fundamental form. Full article

We present a formal framework for modeling neural network dynamics using Category Theory, specifically through Markov categories. In this setting, neural states are represented as objects and state transitions as Markov kernels, i.e., morphisms in the category. This categorical perspective offers an algebraic alternative to traditional approaches based on stochastic differential equations, enabling a rigorous and structured approach to studying neural dynamics as a stochastic process with topological insights. By abstracting neural states as submeasurable spaces and transitions as kernels, our framework bridges biological complexity with formal mathematical structure, providing a foundation for analyzing emergent behavior. As part of this approach, we incorporate concepts from Interacting Particle Systems and employ mean-field approximations to construct Markov kernels, which are then used to simulate neural dynamics via the Ising model. Our simulations reveal a shift from unimodal to multimodal transition distributions near critical temperatures, reinforcing the connection between emergent behavior and abrupt changes in system dynamics. Full article

For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained by this model. We use the improved $(G^{\prime }/G)$ expansion technique and a modified extended direct algebraic technique to obtain these solutions. Results for trigonometry, hyperbolic, and rational functions are obtained. The impact of the fractional-order derivative is also covered. We use Mathematica software to verify our findings. Furthermore, we use contour graphs in two and three dimensions to illustrate some wave solitons that are obtained. The results obtained have applications in ocean engineering, fluid dynamics, and other fields. The stability analysis of the considered equation is also performed. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. Furthermore, the used methods are useful for other nonlinear fractional partial differential equations in different areas of applied science and engineering. Full article

Bilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists, for the induced system on projective space. We use the system semigroup by considering piecewise constant controls and use spectral properties to extend the result to bilinear systems in ${\mathbb{R}}^d$. The contribution of this paper highlights the relationship between all the existent control sets. We show that the controllability property of a bilinear system is equivalent to the existence and uniqueness of a control set of the projective system. Full article

This work systematically addresses the dual challenges of non-inertial dynamic coupling and kinematic constraint redundancy encountered in dynamic modeling of serial–parallel–serial hybrid robotic mechanisms, and proposes an improved Newton–Euler modeling method with constraint compensation. Taking the Skiing Simulation Platform with 6-DOF as the research mechanism, the inverse kinematic model of the closed-chain mechanism is established through $G_F$ set theory, with explicit analytical expressions derived for the motion parameters of limb mass centers. Introducing a principal inertial coordinate system into the dynamics equations, a recursive algorithm incorporating force/moment coupling terms is developed. Numerical simulations reveal a 9.25% periodic deviation in joint moments using conventional methods. Through analysis of the mechanism’s intrinsic properties, it is identified that the lack of angular momentum conservation constraints on the end-effector in non-inertial frames leads to system controllability degradation. Accordingly, a constraint compensation strategy is proposed: establishing linearly independent differential algebraic equations supplemented with momentum/angular momentum balance equations for the end platform. Co-Simulation results demonstrate that the optimized model reduces the maximum relative error of actuator joint moments to 0.98%, and maintains numerical stability across the entire configuration space. The constraint compensation framework provides a universal solution for dynamics modeling of complex closed-chain mechanisms, validated through applications in flight simulators and automotive driving simulators. Full article

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

Material defects resulting from manufacturing and processing can significantly affect material properties. Voids and dislocations are material defects considered in this study, in which a numerical solution of the 3D stress field of a dislocation line (infinite or finite) outside a cylindrical void (either parallel to the cylinder axis or not) is developed using the collocation point method. The collocation point method is utilized to solve ordinary differential equations, partial differential equations, differential-algebraic equations, and integral equations by enforcing the solution at a set of spatial collocation points. Analytical solutions for such three-dimensional (3D) problems, e.g., a dislocation line or segment near an internal void of any shape, were not found. Therefore, a numerical solution for this problem has been constructed in this paper. The numerical solution developed is verified using an existing two-dimensional analytical solution. The numerical results and the 2D analytical solution are in perfect agreement as long as the cylindrical void is sufficiently long and the Saint-Venant’s principle is followed. Full article

This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. The scheduled second-order dynamic systems are equivalent to the stated second-order differential equations, and they are used for analysis purposes. In the first study, a control law is synthesized for a forced time-invariant nominal version of the current time-varying one so that their respective two-point boundary values are coincident. Afterward, the parameter that fixes the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying parameter that is a control function to be synthesized without performing, in this case, any comparison with a nominal time-invariant version of the system. Such a control law is designed in such a way that, for given arbitrary and finite initial conditions of the differential system, prescribed final conditions along a time interval of finite length are matched by the state trajectory solution. As a result, the solution of the dynamic system, and thus that of its differential equation counterpart, is subject to prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. Also, some algebraic constraints between the eigenvalues of the Sturm–Liouville system and their evolution operators are formulated later on. Those constraints are based on the fact that the solutions corresponding to each of the eigenvalues match the same two-point boundary values. Full article

This article proposes a method to solve nonlinear optimal control problems with arbitrary performance indices and terminal constraints, which is based on the neighboring extremal method and Gauss pseudospectral collocation. Firstly, a quadratic performance index is formulated, which minimizes the second-order variation of the nonlinear performance index and fully considers the deviations in initial states and terminal constraints. Secondly, the first-order necessary conditions are applied to derive the perturbation differential equations involving deviations in state and costate variables. Therefore, a quadratic optimal control problem is formulated, which is subject to such perturbation differential equations. Thirdly, the Gauss pseudospectral collocation is used to transform the differential and integral operators into algebraic operations. Therefore, an analytical solution of the control correction can be successfully derived in the polynomial space, which comes close to the optimal solution. This method has a fast computation speed and low computational complexity due to the discretization at orthogonal points, making it suitable for online applications. Finally, some simulations and comparisons with the optimal solution and other typical methods have been carried out to evaluate the performance of the method. Results show that it not only performs well in computational efficiency and accuracy but also has great adaptability and optimality. Moreover, Monte Carlo simulations have been conducted. The results demonstrate that it has strong robustness and excellent performance even in highly dispersed environments. Full article

This paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations. Full article