Search Results (62)

This study extends the post-Keynesian framework for cyclical economic growth, initially proposed by Goodwin in 1967, by integrating government intervention to stabilize employment amidst wage mismatches. Given the pressing challenges of unemployment and wage disparity in developing economies, particularly South Africa, this extension is necessary to better understand how policy interventions can influence labor market dynamics. Central to the study is the endogenous interaction between capital and labor, where class dynamics influence economic growth patterns. The research focuses on the competitive relationship between real wage growth and its effects on employment. Methodologically, the study measures the impact of intervention strategies using a system of coupled ordinary differential equations (Lotka–Volterra type), along with econometric techniques such as quantile regression, moving averages, and mean absolute error to measure wages mismatch. Results demonstrate the inherent contradictions of capitalism under intervention, confirming established theoretical perspectives. This work further contributes to economic optimality discussions, especially regarding the timing and persistence of economic cycles. The model provides a quantifiable approach for formulating intervention strategies to achieve long-term economic equilibrium and sustainable labor–capital coexistence. Full article

This paper presents an innovative numerical method for solving two-dimensional weakly singular Volterra integral equations, including fractional Volterra integral equations with weak singularities. Solving these equations in higher dimensions and in the presence of fractional and weak singularities is highly challenging. The proposed approach uses Euler wavelets (EWs) within an operational matrix (OM) framework combined with advanced numerical techniques, initially transforming these equations into a linear algebraic system and then solving it efficiently. This method offers very high accuracy, strong computational efficiency, and simplicity of implementation, making it suitable for a wide range of such complex problems, especially those requiring high speed and precision in the presence of intricate features. 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 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

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

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

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK $\alpha ,\gamma$-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique. Full article

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in $L_2\left[\Omega \right]\times C\left[0,T\right],T<1,$$\Omega$ is the position domain of integration and T is the time. The convergence error and the stability error are considered. Then, after using the separation technique, the MIE transforms into a system of Hammerstein integral equations (SHIEs) with time-varying coefficients. The nonlinear algebraic system (NAS) is obtained after using the degenerate method. New and special cases are derived from this work. Moreover, numerical results are computed using MATLAB R2023a software. Full article

This research relies on several kinds of Volterra-type integral differential systems and their associated stability concerns under the impulsive effects of the Volterra integral terms at certain time instants. The dynamics are defined as delay-free dynamics contriobution together with the contributions of a finite set of constant point delay dynamics, plus a Volterra integral term of either a finite length or an infinite one with intrinsic memory. The global asymptotic stability is characterized via Krasovskii–Lyapuvov functionals by incorporating the impulsive effects of the Volterra-type terms together with the effects of the point delay dynamics. Full article

The complexity of modern power grids, caused by integrating renewable energy sources, especially inverter-based resources, presents a significant challenge to grid operation and planning, since linear models are unable to capture the complex nonlinear dynamics of power systems with coupled muti-scale dynamics, and it necessitate an alternative approach utilizing more advanced and data-driven algorithms to improve modeling accuracy and system optimization. This study employs the sparse identification of nonlinear dynamics method by leveraging compressed sensing and sparse modeling principles, offering robustness and the potential for generalization, allowing for identifying key dynamical features with relatively few measurements, and providing deeper theoretical understanding in the field of power system analysis. Taking advantage of the this method in recognizing the active terms (first and high order) in the system’s governing equation, this paper also introduces the novel Volterra-based nonlinearity index to characterize system-level nonlinearity. The distinction of dynamics into first-order linearizable terms, second-order nonlinear dynamics, and third-order noise is adopted to clearly show the intricacy of power systems. The findings demonstrate a fundamental shift in system dynamics as power sources transit to inverter-based resources, revealing system-level (second-order) nonlinearity compared to module-level (first order) nonlinearity in conventional synchronous generators. The proposed index quantifies nonlinear-to-linear relationships, enriching our comprehension of power system behavior and offering a tool for distinguishing between different nonlinearities and visualizing their distinct patterns through the profile of the proposed index. Full article

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis. Full article

The paper discusses the prospect of using a combined model based on finite segments (polynomials) of the Volterra integral power series. We consider a case when the problem of identifying the Volterra kernels is solved. The predictive properties of the classic Volterra polynomial are improved by adding a linear part in the form of an equivalent continued fraction. This technique allows us to distinguish an additional parameter—the connection coefficient $\alpha$, which is effective in adapting the constructed integral model to changes in technical parameters at the input of a dynamic system. In addition, this technique allows us to take into account the case of perturbing the kernel of the linear term of the Volterra polynomial in the metric $C[0,T]$ by a given value $\delta$, implying the ideas of Volterra regularizing procedures. The problem of choosing the connection coefficient is solved using a special extremal problem. The developed algorithms are used to solve the problem of identifying input signals of test dynamic systems, among which, in addition to mathematical ones, thermal power engineering devices are used. Full article

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article