Search Results (180)

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

In this integrative review paper, we explore the parallels between the physical models of gyrotrons and some equations used in diverse and broad scientific fields. These include Adler’s famous equation, Van der Pol equation, the Lotka–Volterra equations and the Kuramoto model. The paper is written in the form of a pedagogical discourse and aims to provide additional insights into gyrotron physics through analogies and parallels to theoretical approaches used in other fields of research. For the first time, reachability analysis is used in the context of gyrotron physics as a modern tool for understanding the behavior of nonlinear dynamical systems. Full article

This work presents the mathematical frameworks of the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Volterra-Type” (1st-FASAM-NIDE-V) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Volterra-Type” (2nd-FASAM-NIDE-V). It is shown that the 1st-FASAM-NIDE-V methodology enables the efficient computation of exactly-determined first-order sensitivities of the decoder response with respect to the optimized NIDE-V parameters, requiring a single “large-scale” computation for solving the 1st-Level Adjoint Sensitivity System (1st-LASS), regardless of the number of weights/parameters underlying the NIE-net. The 2nd-FASAM-NIDE-V methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights involved in the NIDE-V’s decoder, hidden layers, and encoder, requiring only as many “large-scale” computations as there are non-zero first-order sensitivities with respect to the feature functions. These characteristics of the 1st-FASAM-NIDE-V and 2nd-FASAM-NIDE-V are illustrated by considering a nonlinear heat conduction model that admits analytical solutions, enabling the exact verification of the expressions obtained for the first- and second-order sensitivities of NIDE-V decoder responses with respect to the model’s functions of parameters (weights) that characterize the heat conduction model. Full article

Loaded Volterra integral equations represent a novel class of integral equations that have attracted considerable attention in recent years due to their numerous applications in various fields of science and engineering. This class of Volterra integral equations is characterized by the presence of a loading function, which complicates their theoretical and numerical analysis. In this paper, we study Volterra equations with locally loaded integral operators. The existence and uniqueness of their solutions are examined. A collocation-type method for the approximate solution of such equations is proposed, based on piecewise linear approximation of the exact solution. To confirm the convergence of the method, several numerical results for solving model problems are provided. Full article

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the $L^2$ sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the $L^2$ sense. Full article

This study investigates extremal solutions for fractional-order delayed difference equations, utilizing the Caputo nabla operator to establish mild lower and upper approximations via discrete fractional calculus. A new approach is employed to demonstrate the uniform convergence of the sequences of lower and upper approximations within the monotone iterative scheme using the summation representation of the solutions, which serves as a discrete analogue to Volterra integral equations. This research highlights practical applications through numerical simulations in discrete bidirectional associative memory neural networks. Full article

This article introduces a new fuzzy double integral transformation called fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial derivatives and fuzzy unitary convolution. All of the new results are applied to find an analytical solution to the fuzzy parabolic Volterra integro-differential equation (FPVIDE) with a suitably selected memory kernel. In addition, a numerical example is provided to illustrate how the proposed method might be helpful for solving FPVIDE utilizing symmetric triangular fuzzy numbers. Compared with other symmetric transforms, we conclude that our new approach is simpler and needs less calculations. Full article

This work presents the general mathematical frameworks of the “First and Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integral Equations of Volterra Type” designated as the 1st-FASAM-NIE-V and the 2nd-FASAM-NIE-V methodologies, respectively. Using a single large-scale (adjoint) computation, the 1st-FASAM-NIE-V enables the most efficient computation of the exact expressions of all first-order sensitivities of the decoder response to the feature functions and also with respect to the optimal values of the NIE-net’s parameters/weights after the respective NIE-Volterra-net was optimized to represent the underlying physical system. The computation of all second-order sensitivities with respect to the feature functions using the 2nd-FASAM-NIE-V requires as many large-scale computations as there are first-order sensitivities of the decoder response with respect to the feature functions. Subsequently, the second-order sensitivities of the decoder response with respect to the primary model parameters are obtained trivially by applying the “chain-rule of differentiation” to the second-order sensitivities with respect to the feature functions. The application of the 1st-FASAM-NIE-V and the 2nd-FASAM-NIE-V methodologies is illustrated by using a well-known model for neutron slowing down in a homogeneous hydrogenous medium, which yields tractable closed-form exact explicit expressions for all quantities of interest, including the various adjoint sensitivity functions and first- and second-order sensitivities of the decoder response with respect to all feature functions and also primary model parameters. Full article

This paper introduces two concepts, subcompatibility and subsequential continuity, which are, respectively, weaker than the existing concepts of occasionally weak compatibility and reciprocal continuity. These concepts are studied within the framework of neutrosophic metric spaces. Using these ideas, a common fixed point theorem is developed for a system involving four maps. Furthermore, the results are applied to solve the Volterra integral equation, demonstrating the practical use of these findings in neutrosophic metric spaces. Full article

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance. Full article

In recent decades, many researchers have pointed out that derivatives and integrals of the non-integer order are well suited for describing various real-world materials, for example, polymers. It has also been shown that fractional-order mathematical models are more effective than integer-order mathematical models. Thereby, given these considerations, the investigation of qualitative properties, in particular, Ulam-type stabilities of fractional differential equations, fractional integral equations, etc., has now become a highly attractive subject for mathematicians, as this represents an important field of study due to their extensive applications in various branches of aerodynamics, biology, chemistry, the electrodynamics of complex media, polymer science, physics, rheology, and so on. Meanwhile, the qualitative concepts called Ulam–Hyers–Mittag-Leffler (U-H-M-L) stability and Ulam–Hyers–Mittag-Leffler–Rassias (U-H-M-L-R) stability are well-suited for describing the characteristics of fractional Ulam-type stabilities. The Banach contraction principle is a fundamental tool in nonlinear analysis, with numerous applications in operational equations, fractal theory, optimization theory, and various other fields. In this study, we consider a nonlinear fractional Volterra integral equation (FrVIE). The nonlinear terms in the FrVIE contain multiple variable delays. We prove the U-H-M-L stability and U-H-M-L-R stability of the FrVIE on a finite interval. Throughout this article, new sufficient conditions are obtained via six new results with regard to the U-H-M-L stability or the U-H-M-L-R stability of the FrVIE. The proofs depend on Banach’s fixed-point theorem, as well as the Chebyshev and Bielecki norms. In the particular case of the FrVIE, an example is delivered to illustrate U-H-M-L stability. Full article

This study investigates the stability behavior of nonlinear Fredholm and Volterra integral equations, as well as nonlinear integro-differential equations with Volterra integral terms, through the lens of symmetry principles in mathematical analysis. By leveraging fixed-point methods within b-metric spaces, which generalize classical metric spaces while preserving structural symmetry, we establish sufficient conditions for Hyers–Ulam–Rassias and Hyers–Ulam stability. The symmetric framework of b-metric spaces offers a unified approach to analyzing stability across a wide range of nonlinear systems. To illustrate the theoretical results, examples are provided that underscore the practical applicability and relevance of these findings to complex nonlinear systems, emphasizing their inherent symmetrical properties. Full article

Epoxy resins are extensively employed as adhesives and matrices in fibre-reinforced composites. As polymers, they possess a viscoelastic nature and are prone to creep and stress relaxation even at room temperature. This phenomenon is also responsible for time-dependent failure or creep fracture due to cumulative strain. Several constitutive equations have been used to describe the mechanical time-dependent response of polymers. These models have been proposed over the past six decades, with minimal direct and practical confrontation. Each model is associated with a specific application or research group. This work assesses the predictive performance of four distinct time-dependent constitutive models based on experimental data. The models were deemed sufficiently straightforward to be readily integrated into practical engineering analyses. A range of loading cases, encompassing constant strain rate, creep, and relaxation tests, were conducted on a commercial epoxy resin. Model parameter calibration was conducted with a minimum data set. The extrapolative predictive capacity of the models was evaluated for creep loading by extending the tests to five decades. The selected rheological models comprise two viscoelastic models based on Volterra-type integrals, as originally proposed by Schapery and Rabotnov; one viscoplastic model, as originally proposed by Norton and Bailey; and the Burger model, in which two springs and two dashpots are combined in a serial and parallel configuration. The number of model parameters does not correlate positively to superior performance, even if it is high. Overall, the models exhibited satisfactory predictive performance, displaying similar outcomes with some relevant differences during the unloading phases. Full article

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations. Full article

Integral–algebraic equations and their systems are a common description of many technical and engineering problems. Often, such models also describe certain dependencies occurring in nature (e.g., ecosystem behaviors). The integral equations occurring in this problem may have two types of domains—symmetric or asymmetric. Depending on whether such symmetry exists in the system describing a given problem, we must choose the appropriate method to solve this system. In this task, the absence of symmetry is more advantageous, but the presented examples demonstrate how one can approach cases where symmetry is present. In this paper, we present the application of two methods for solving such tasks: the analytical Differential Transform Method (DTM) and Physics-informed Neural Networks (PINNs). We consider a wide class of these types of equation systems, including Volterra and Fredholm integrals (which are also in a single model). We demonstrate that despite the complex nature of the problem, both methods are capable of handling such tasks, and thus, they can be successfully applied to the issues discussed in this article. Full article