Search Results (169)

This paper deals with a new coupled system of integro-differential equations involving the generalized proportional Caputo derivatives equipped with nonlocal four-point boundary conditions. Sufficient criteria for the existence and uniqueness of solutions for the studied system are derived based on Krasnoselskii’s and Banach fixed-point theorems, respectively. Applications are constructed with three different cases to illustrate the main results. Full article

The stability of energy demand–supply systems is often affected by delayed feedback caused by regulatory inertia, communication lags, and heterogeneous agent responses. Conventional models typically assume discrete delays, which may oversimplify real dynamics and reduce controller effectiveness. This work addresses this limitation by introducing a novel class of nonlinear energy models with distributed delay feedback governed by gamma-distributed memory kernels. Specifically, we consider both weak (exponential) and strong (Erlang-type) kernels to capture a spectrum of memory effects. Using the linear chain trick, we reformulate the resulting integro-differential model into a higher-dimensional system of ordinary differential equations. Analytical conditions for local asymptotic stability and Hopf bifurcation are derived, complemented by Lyapunov-based global stability criteria. The related numerical analysis confirms the theoretical findings and reveals a distinct stabilization regime. Compared to fixed-delay approaches, the proposed framework offers improved flexibility and robustness, with implications for delay-aware energy control and infrastructure design. Full article

We develop and analyze a distributed-delay model for nutrient–fish–mussel dynamics in multitrophic aquaculture systems. Extending the classical discrete-delay framework, we incorporate gamma-distributed kernels to capture the time-distributed nature of nutrient assimilation, yielding a more realistic and analytically tractable representation. These kernels introduce a form of temporal symmetry in the system’s memory, where past nutrient levels influence present dynamics in a balanced and structured way. Using the linear chain trick, we reformulate the integro-differential equations into ordinary differential systems for both weak and strong memory scenarios. We derive conditions for local stability and Hopf bifurcation, and establish global stability using Lyapunov-based methods. Numerical simulations confirm that increased delay can destabilize the system, leading to oscillations, while stronger memory mitigates this effect and enhances resilience. Bifurcation diagrams, time series, and phase portraits illustrate how memory strength governs the system’s dynamic response. This work highlights how symmetry in memory structures contributes to system robustness, offering theoretical insights and practical implications for the design and management of ecologically stable aquaculture systems. Full article

This paper investigates a robust optimal reinsurance and investment problem for an insurance company operating in a Markov-modulated financial market. The insurer’s surplus process is modeled by a diffusion process with jumps, which is correlated with financial risky assets through a common shock structure. The economic regime switches according to a continuous-time Markov chain. To address model uncertainty concerning both diffusion and jump components, we formulate the problem within a robust optimal control framework. By applying the Girsanov theorem for semimartingales, we derive the dynamics of the wealth process under an equivalent martingale measure. We then establish the associated Hamilton–Jacobi–Bellman (HJB) equation, which constitutes a coupled system of nonlinear second-order integro-differential equations. An explicit form of the relative entropy penalty function is provided to quantify the cost of deviating from the reference model. The theoretical results furnish a foundation for numerical solutions using actor–critic reinforcement learning algorithms. Full article

We develop a new mathematical framework for describing impurity diffusion in multiphase, stochastically inhomogeneous media with internal deterministic mass sources. The main contribution of the paper is the structural preservation of the original multiphase problem while reducing it to a single integro-differential diffusion equation for the entire body. Using a Feynman diagram technique, we obtain a Dyson-type equation for the averaged concentration field; its kernel (mass operator) summarizes the cumulative effect of random phase interfaces and internal sources. This diagrammatic formulation offers clear advantages: it systematically organizes the contributions of complex interphase interactions and source terms, ensures convergence of the Neumann-series solution, and facilitates extensions to more intricate source distributions. The approach allows us to analyze the behavior of the averaged impurity concentration under various temporally or spatially distributed internal sources and provides a foundation for further refinement of transport models in complex multiphase systems. Full article

One among the most intriguing results coming from the application of statistical mechanics to the study of the brain is the understanding that it, as a dynamical system, is inherently out of equilibrium. In the realm of non-equilibrium statistical mechanics and stochastic processes, the standard observable computed to determine whether a system is at equilibrium or not is the entropy produced along the dynamics. For this reason, we present here a detailed calculation of the entropy production in the Amari model, a coarse-grained model of the brain neural network, consisting of an integro-differential equation for the neural activity field, when stochasticity is added to the original dynamics. Since the way to add stochasticity is always to some extent arbitrary, particularly for coarse-grained models, there is no general prescription to do so. We precisely investigate the interplay between noise properties and the original model features, discussing in which cases the stationary state is in thermal equilibrium and which cases it is out of equilibrium, providing explicit and simple formulae. Following the derivation for the particular case considered, we also show how the entropy production rate is related to the variation in time of the Shannon entropy of the system. Full article

The photokinetic behavior of concomitant and independent photo- and photothermal reactions exposed to monochromatic or polychromatic irradiation, has not yet been described in photochemistry literature. The occurrence of such mixtures is reported in a wide range of fields, from living species to technologically designed devices. To address the lack of investigative tools that facilitate better understanding, quantification, and control of such parallel-reaction systems, a new holistic approach is proposed in the present study. It contributes to an effort dedicated to rationalizing photokinetics along the same criteria required for thermal kinetics. The methodology builds on a previously introduced general explicit integrated rate-law formula for single-reaction systems (whose integro-differential rate-equation is not solvable). The extension of its field of applicability to multi-component photoreactive mixtures is demonstrated in the present paper. For this purpose, a large number of combinations of both photo- and photothermal individual reactions, possessing distinctly different features, were studied in binary and ternary mixtures. The data of reactions/mixtures were generated by a fourth-order Runge–Kutta numerical integration. An excellent fitting of the species’ kinetic traces by the adapted explicit formula was obtained for all mixtures. Also, the quantification of the effects of the variation in the initial concentration of one component of the mixture, and/or the presence of inert spectator molecules in the reactor, was successfully performed. The investigative photokinetic tools proposed here are shown to be handy, efficient, and useful. The findings of the present study are also thought to expand the application possibilities of reactive photothermal systems in mixtures. Full article

This paper presents a novel class of conformable integro-differential operators designed to model systems with rapid and ultra-rapid dynamics. This class of local operators enables the design of controllers and observers that induce hyperexponential convergence and provide robustness against bounded disturbances and dynamic uncertainties. The proposed method leverages Nested Exponential Functions (NEFs) and Nested Exponential Factorial Functions (NEFFs) to capture fast dynamics effectively. Additionally, the proposed study examines the Fundamental Theorem of Calculus in the context of Nested Exponential Conformable (NEC) operators, unveiling structural properties, such as stability and robustness, that produce dynamical systems with enhanced hyperexponential convergence and faster dynamics compared to existing approaches. Stability results for NEC systems are established, and some illustrative examples based on numerical simulations are presented to demonstrate the reliability of the proposed approach. Full article

This work investigates the solutions of fractional integro-differential equations (FIDEs) using a unique kernel operator within the Caputo framework. The problem is addressed using both analytical and numerical techniques. First, the two-step Adomian decomposition method (TSADM) is applied to obtain an exact solution (if it exists). In the second part, numerical methods are used to generate approximate solutions, complementing the analytical approach based on the Adomian decomposition method (ADM), which is further extended using the Sumudu and Shehu transform techniques in cases where TSADM fails to yield an exact solution. Additionally, we establish the existence and uniqueness of the solution via fixed-point theorems. Furthermore, the Ulam–Hyers stability of the solution is analyzed. A detailed error analysis is performed to assess the precision and performance of the developed approaches. The results are demonstrated through validated examples, supported by comparative graphs and detailed error norm tables ($L_{\infty }$, $L_2$, and $L_1$). The graphical and tabular comparisons indicate that the Sumudu-Adomian decomposition method (Sumudu-ADM) and the Shehu-Adomian decomposition method (Shehu-ADM) approaches provide highly accurate approximations, with Shehu-ADM often delivering enhanced performance due to its weighted formulation. The suggested approach is simple and effective, often producing accurate estimates in a few iterations. Compared to conventional numerical and analytical techniques, the presented methods are computationally less intensive and more adaptable to a broad class of fractional-order differential equations encountered in scientific applications. The adopted methods offer high accuracy, low computational cost, and strong adaptability, with potential for extension to variable-order fractional models. They are suitable for a wide range of complex systems exhibiting evolving memory behavior. Full article

In this article, a novel type of equation, namely the $\mathsf{\Phi }$-Hilfer fractional-order integro-differential delay system ($\mathsf{\Phi }$-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes. Full article

This work illustrates the application of the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (1st-FASAM-NIDE-F) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (2nd-FASAM-NIDE-F) to a paradigm heat transfer model. This physically based heat transfer model has been deliberately constructed so that it can be represented either by a neural integro-differential equation of a Fredholm type (NIDE-F) or by a conventional second-order “neural ordinary differential equation (NODE)” while admitting exact closed-form solutions/expressions for all quantities of interest, including state functions and first-order and second-order sensitivities. This heat transfer model enables a detailed comparison of the 1st- and 2nd-FASAM-NIDE-F versus the recently developed 1st- and 2nd-FASAM-NODE methodologies, highlighting the considerations underlying the optimal choice for cases where the neural net of interest is amenable to using either of these methodologies for its sensitivity analysis. It is shown that the 1st-FASAM-NIDE-F methodology enables the most efficient computation of exactly determined first-order sensitivities of the decoder response with respect to the optimized NIDE-F 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 NIDE-F decoder, hidden layers, and encoder. The 2nd-FASAM-NIDE-F methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights. Full article

This work presents the “First-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (1st-FASAM-NIDE-F) and the “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Integro-Differential Equations of Fredholm-Type” (2nd-FASAM-NIDE-F). It is shown that the 1st-FASAM-NIDE-F methodology enables the most efficient computation of exactly-determined first-order sensitivities of decoder response with respect to the optimized NIDE-F 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 NIDE-F decoder, hidden layers, and encoder. The 2nd-FASAM-NIDE-F methodology enables the computation, with unparalleled efficiency, of the second-order sensitivities of decoder responses with respect to the optimized/trained weights, requiring only as many large-scale computations for solving the 2nd-Level Adjoint Sensitivity System (2nd-LASS) as there are non-zero feature functions of parameters. The application of both the 1st-FASAM-NIDE-F and the 2nd-FASAM-NIDE-F methodologies is illustrated in an accompanying work (Part II) by considering a paradigm heat transfer model. 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

The magnetic field of the Sun is formed by the mechanism of hydromagnetic dynamo. In this mechanism, the flow of the conducting medium (plasma) of the convective zone generates a magnetic field, and this field corrects the flow using the Lorentz force, creating feedback. An important role in dynamo is played by memory (hereditary), when a change in the current state of a physical system depends on its states in the past. Taking these effects into account may provide a more accurate description of the generation of the Sun’s magnetic field. This paper generalizes classical dynamo models by including hereditary feedback effects. The feedback parameters such as the presence or absence of delay, delay duration, and memory duration are additional degrees of freedom. This can provide more diverse dynamic modes compared to classical memoryless models. The proposed model is based on the kinematic dynamo problem, where the large-scale velocity field is predetermined. The field in the model is represented as a linear combination of two stationary predetermined modes with time-dependent amplitudes. For these amplitudes, equations are obtained based on the kinematic dynamo equations. The model includes two generators of a large-scale magnetic field. In the first, the field is generated due to large-scale flow of the medium. The second generator has a turbulent nature; in it, generation occurs due to the nonlinear interaction of small-scale pulsations of the magnetic field and velocity. Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by an integral term of the type of convolution of a quadratic form of phase variables with a kernel of a fairly general form. The quadratic form models the influence of the Lorentz force. This integral term describes the turbulent generator quenching. Mathematically, this model is written with a system of integro-differential equations for amplitudes of modes. The model was applied to a real space object, namely, the solar dynamo. The model representation of the Sun’s velocity field was constructed based on helioseismological data. Free field decay modes were chosen as components of the magnetic field. The work considered cases when hereditary feedback with the system arose instantly or with a delay. The simulation results showed that the model under study reproduces dynamic modes characteristic of the solar dynamo, if there is a delay in the feedback. Full article

In this paper, we obtain unique solution and stability results for coupled fractional differential equations with p-Laplacian operator and Riemann–Stieltjes integral conditions that expand and improve the works of some of the literature. In order to obtain the existence and uniqueness of solutions for coupled systems, several fixed point theorems for operators in ordered product spaces are given without requiring the existence conditions of upper–lower solutions or the compactness and continuity of operators. By applying the conclusions of the operator theorem studied, sufficient conditions for the unique solution of coupled fractional integro-differential equations and approximate iterative sequences for uniformly approximating unique solutions were obtained. In addition, the Hyers–Ulam stability of the coupled system is discussed. As applications, the corresponding results obtained are well demonstrated through some concrete examples. Full article