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This article studies the kinetic dynamics of the rock-paper-scissors binary game in a measure setting given by a non local and non linear integrodifferential equation. After proving the wellposedness of the equation, we provide a precise description of the asymptotic behavior in large time. To do so we adopt a duality approach, which is well suited both as a first step to construct a measure solution by mean of semigroups and to obtain an explicit expression of the asymptotic measure. Even thought the equation is non linear, this measure depends linearly on the initial condition. This result is completed by a decay in total variation norm, which happens to be subgeometric due to the nonlinearity of the equation. This relies on an unusual use of a confining condition that is needed to apply a Harris-type theorem, taken from a recent paper [2] that also provides a way to compute explicitly the constants involved in the aforementioned decay in norm. [ABSTRACT FROM AUTHOR]

This paper presents numerical methods for solving the one-dimensional fractional reaction-diffusion equation with the fractional Caputo derivative. The proposed methods are based on transformation of the fractional differential equation to an equivalent form of a integro-differential equation. The paper proposes an improvement of the existing implicit method, and a new explicit method. Stability and convergence tests of the methods were also conducted. [ABSTRACT FROM AUTHOR]

This paper is concerned with the mathematical modeling of complex living systems whose element microscopic state contains variables which can attain discrete values. Specifically, the main mathematical frameworks of the discrete thermostatted kinetic theory for active particles are reviewed and generalized. In the generalized thermostatted frameworks, which are based on nonlinear ordinary or partial differential equations, the elements of the system are viewed as active particles that are able to perform certain strategies modeled by introducing a functional-state variable called activity. Interactions, which are responsible of the evolution of the system, are modeled using the fundamentals of stochastic game theory and may be influenced by the action of an external force field coupled to a Gaussian-type thermostat. In particular, the interaction domain is modeled by introducing a weighted function and different non-homogeneous discrete frameworks are proposed and coupled with a specific thermostat. Two recent models derived within this approach are reviewed and refer to vehicular and pedestrian dynamics. Future research perspectives are discussed in the whole paper from theoretical and modeling viewpoints. [ABSTRACT FROM AUTHOR]

The work of Jia et al [[25]] discusses the stable control of one class of chaotic systems, and a control method based on the accurate exponential solution of a differential equation is used. Many problems in the broad spectrum of science require the solution of a partial differential equation. In [[23]], Karmran et al. propose a localized transform-based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo's sense with a weakly singular kernel. 10.3390/math8111854 25 Jia H., Guo C. The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System. [Extracted from the article]

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order βξ instead of β1; for details, see Lemma 2.2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature. [ABSTRACT FROM AUTHOR]

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples. [ABSTRACT FROM AUTHOR]

In this article, we present as a comparative result of Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Variational Iteration Method (VIM). These methods used for developed to find the analytical approximate solution of linear Fredholm integro-differential equations. The main purpose of this paper was to show a better method for Numerical equations which does not give easily analytical solution. So, in this paper, we find approximate solutions of linear Fredholm integro-differential equations. We explain the convergence of ADM, MADM and VIM by using examples of a deterministic model by graphs and tables. All the calculations performed by the help of MATLAB (2018) Version 9.4. [ABSTRACT FROM AUTHOR]

This document is a preface to a special issue of the journal Mathematics, focusing on abstract fractional integro-differential equations and fixed point theory with applications. The preface explains that fractional calculus is used to model complex systems in various disciplines, and abstract fractional integro-differential equations arise from different scientific research areas. The special issue includes 12 research papers selected from 36 submissions, covering topics such as almost periodic solutions, stability analysis, and convex analysis. The authors express their appreciation to the contributors and reviewers, and hope that the papers will inspire future research in these areas. [Extracted from the article]

Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls-Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on their initial position or orientation. In more detail, the evolutive Peierls-Nabarro model which we consider describes the evolution of a phase-field function vε(t, x) which represents the atom deformation in a crystal. The model is a reaction-diffusion equation of Allen-Cahn type with the half Laplacian. The small parameter ε is the ratio between the atomic distance and the typical distance between phase transitions in vε. The position of a phase transition determines the position of a dislocation, and the sign of the transition (up or down) determines the orientation. The goal of this paper is to derive the asymptotic behavior of the function vε as ε → 0 up to arbitrary end time T; in particular beyond collisions. We prove that vε converges to a piecewise constant function v, whose jump points in the spatial variable satisfy the ODE system which represents discrete dislocation dynamics with annihilation. Our proof method is to explicitly construct and patch together several sub- and supersolutions of vε, and to show that they converge to the same limit v. [ABSTRACT FROM AUTHOR]

In recent years, fractional calculus (FC) has filled in a hole in traditional calculus in terms of the effect of memory, which lets us know things about the past and present and guess what will happen in the future. It is very important to have this function, especially when studying biological models and integral equations. This paper introduces developed mathematical strategies for understanding a direct arrangement of fractional integro-differential equations (FIDEs). We have presented the least squares procedure and the Legendre strategy for discussing FIDEs. We have given the form of the Caputo concept fractional order operator and the properties. We have presented the properties of the shifted Legendre polynomials. We have shown the steps of the technique to display the solution. Some test examples are given to exhibit the precision and relevance of the introduced strategies. Mathematical outcomes show that this methodology is a comparison between the exact solution and the methods suggested. To show the theoretical results gained, the simulation of suggested strategies is given in eye-catching figures and tables. Program Mathematica 12 was used to get all of the results from the techniques that were shown. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for a class of integrodifferential Kirchhoff equations. These equations involve a nonlocal operator with a measurable kernel that satisfies "structural properties" that are more general than the standard kernel of the fractional Laplacian operator. Additionally, the potential can be periodic or asymptotically periodic, and the nonlinear term exhibits critical exponential growth in the sense of Trudinger–Moser inequality. To guarantee the existence of solutions, we employ variational methods, specifically the mountain-pass theorem. In this context, it is important to emphasize that we have additional difficulties due to the lack of compactness in our problem, because we deal with critical growth nonlinearities in unbounded domains. Moreover, the Kirchhoff term adds complexity to the problem, as it requires suitable calculations for control the estimate the minimax level, representing the main challenge in this work. Finally, we consider two different approaches to estimate the minimax level. The first approach is based on a hypothesis proposed by D. M. Cao, while the second one involves a slightly weaker assumption addressed by Adimurthi and Miyagaki. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fourth-order compact discretization method for solving a secondorder boundary value problem governed by the nonlinear Fredholm integro-differential equations. Using an efficient approximate polynomial, a compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations. It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some numerical examples are presented to demonstrate its approximation accuracy and computational efficiency, as well as to compare the derived results with those obtained in the literature. [ABSTRACT FROM AUTHOR]

This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii's fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results [ABSTRACT FROM AUTHOR]

In this paper, we have studied the behaviour of two symmetric mode-III collinear cracks in a functionally graded material (FGM). The fundamental goal of this paper is to provide insight on the interaction of two cracks in FGMs with the strain gradient effect. To assess the influence of gradient elasticity, we have considered two key parameters ℓ and ℓ ′ , which describe the size scale effect caused by the underlying microstructure and are related to volumetric and surface strain energy, respectively. The crack boundary value problem have been solved by the approach involving Fourier transforms and the innovative hyper-singular integro-differential equation method, where the integral equation contains the two terms in integrals for the both cracks. A system of equations has been constructed by employing the Chebyshev polynomial expansion and then by choosing the suitable collocation points the system of equation have been solved. Our investigation involves the determination of stress intensity factors at both crack tips. These factors are vital for understanding the material's fracture behavior and structural integrity. Furthermore, we explore the variations in the displacement profile when the distance between the cracks is reduced to close proximity. This particular scenario is of significant interest as it provides insights into how the interaction between the cracks impacts the overall structural response. [ABSTRACT FROM AUTHOR]

This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon. [ABSTRACT FROM AUTHOR]

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order F. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods. [ABSTRACT FROM AUTHOR]

This paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing random variations in the economic and financial environments. We prove a sufficient condition on the distribution of jumps of the business process ensuring the smoothness of the ruin probability as a function of the initial capital and obtain for this function an integro-differential equation. [ABSTRACT FROM AUTHOR]

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy- α -stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method. [ABSTRACT FROM AUTHOR]

In the mathematical modeling of a far-from-equilibrium complex system an important target is the understanding of the large time behavioun This paper focuses on a continuous-homogeneousconservative mathematical framework coming from the thermostatted kinetic theory recently proposed for the modeling of complex living systems. Specifically by introducing a scaling parameter and letting this parameter going towards zero, the large time behavior of the system is reached, which consists in a nonequilibrium stationary state. The formal proof is obtained in the Lebesgue space L. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear perturbed system of neutral delay integro-differential equations (NDIDEs). We prove two new theorems, Theorems 1 and 2, such that these theorems include sufficient conditions and are related to asymptotical stability of zero solution of the perturbed system of NDIDEs. The technique of the proofs depends upon the definitions of two new and more suitable Lyapunov-Krasovskiĭ functionals (LKFs). When we compared the results of this paper with those found in the literature, our results improve and extend some classical results, and contribute new contents to the topic of NDIDEs and to the profession. [ABSTRACT FROM AUTHOR]

A new mathematical model described by a Volterra integrodifferential equation (VIDE) with constant delay is examined. New agreeable conditions on the uniformly asymptotic stability, boundedness, and square integrability of solutions of the VIDE are obtained by using the Lyapunov–Razumikhin technique. The established conditions improve some former results. They can be also regarded as nonlinear generalizations of these results. Moreover, they are weaker than some available results cited in the bibliography. Two examples are presented to demonstrate possible applications of these results and the introduced concepts. The application of the Lyapunov–Razumikhin technique leads to a significant difference and gives certain advantages over the related methods used in the books and papers cited in the bibliography. [ABSTRACT FROM AUTHOR]

There is growing evidence that broad-line regions (BLRs) in active galactic nuclei (AGNs) have regular substructures, such as spiral arms. This is supported by the fact that the radii of BLRs measured by reverberation mapping (RM) observations are generally consistent with the self-gravitating regions of accretion disks. We showed in Paper I that the spiral arms excited by the gravitational instabilities in these regions may exist in some disk-like BLRs. Here, in the second paper of the series, we investigate the loosely wound spiral arms excited by gravitational instabilities in disk-like BLRs and present their observational characteristics. We solve the governing integro-differential equation by a matrix scheme. The emission-line profiles, velocity-delay maps, and velocity-resolved lags of the BLR spiral arms are calculated. We find that the spiral arms can explain some of the phenomena seen in observations: (1) different asymmetries in the emission-line profiles in the mean and rms spectra; (2) complex subfeatures (incomplete ellipse) in some velocity-delay maps, for example that of NGC 5548; and (3) the short timescales of the asymmetry changes in emission-line profiles (rms spectra). These features are attractive for modeling the observed line profiles and the properties of reverberation, and for revealing the details of the BLR geometry and kinematics. [ABSTRACT FROM AUTHOR]

Transport of vesicles and organelles inside cells consists of constant-speed bidirectional movement along cytoskeletal filaments interspersed by periods of idling. This transport shows many features of anomalous diffusion. In this paper, we develop a non-Markovian persistent random walk model for intracellular transport that incorporates the removal rate of organelles. The model consists of two active states with different speeds and one resting state. The organelle transitions between states with switching rates that depend on the residence time the organelle spends in each state. The mesoscopic master equations that describe the average densities of intracellular transport in each of the three states are the main results of the paper. We also derive ordinary differential equations for the dynamics for the first and second moments of the organelles' position along the cell. Furthermore, we analyse models with power-law distributed random times, which reveal the prevalence of the Mittag-Leffler resting state and its contribution to subdiffusive and superdiffusive behaviour. Finally, we demonstrate a non-Markovian non-additivity effect when the switching rates and transport characteristics depend on the rate of organelles removal. The analytical calculations are in good agreement with numerical Monte Carlo simulations. Our results shed light on the dynamics of intracellular transport and emphasise the effects of rest times on the persistence of random walks in complex biological systems. [ABSTRACT FROM AUTHOR]

This work focuses on stochastic functional differential equations (SFDEs) with wide-band noise perturbation, which is a class of actual physical process and has wide application in the modeling of communication and signal system. Using functional derivatives together with martingale methods and weak convergence techniques, this paper examines the asymptotic properties of the underlying systems as the small parameter tends to zero. Based on the techniques, this paper also establishes the average principle for a class of SFDEs with two-time scales. As a special case, this paper also considers a class of integro-differential equations with wide-band noise perturbation and examines its approximate properties. As an example, a scalar stochastic Lotka–Volterra integro-differential system with wide-band noise perturbation is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of positive periodic solutions of a third-order nonlinear integrodifferential equation with distributed delays, by using the Green function and the Krasnosel'skii fixed point theorem in cones of Banach spaces, providing new results on this field. Three examples are analyzed to illustrate the effectiveness of the abstract results. [ABSTRACT FROM AUTHOR]

Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points. Design/methodology/approach: Thanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled. Findings: All computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question. Originality/value: This paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigated Caputo fractional integro-differential equations with noninstantaneous impulses and nonlocal conditions. By employing the solution operator, the Mönch fixed point theorem, and the stepwise estimation method, we eliminated the Lipschitz condition of the nonlinear term, while also dispensing with the requirement for the compressibility coefficient condition 0 < k < 1. The main results presented represented a generalization and enhancement of previous findings. Furthermore, an example was provided to verify the application of our main results. [ABSTRACT FROM AUTHOR]

The aim of this paper was to provide systematic approaches to study the existence of results for the system fractional relaxation integro-differential equations. Applied problems require definitions of fractional derivatives, allowing the utilization of physically interpretable boundary conditions. Impulsive conditions serve as basic conditions to study the dynamic processes that are subject to sudden changes in their state. In the process, we converted the given fractional differential equations into an equivalent integral equation. We constructed appropriate mappings and employed the Schaefer's fixed-point theorem and the Banach fixed-point theorem to show the existence of a unique solution. We presented an example to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach's fixed point theorem, while by applying Leray–Schauder's nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate the controllability in the α -norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system's controllability using resolvent operator theory introduced by Grimmer, fractional power operators, and fixed-point theorems associated with generalized measures of noncompactness for condensing operators in vector Banach spaces. Finally, we present an application example to validate the proposed methodology in this research. [ABSTRACT FROM AUTHOR]

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

When we transform an initial value problem into an integral equation, Volterra integral equations form. Compared to the starting value problem, the Volterra integral equation is much simpler to solve as a significant subclass of integral equations, numerous academics and mathematicians focus on Volterra integral equations to provide approximate or precise solutions. Recently, mathematics scientists have turned their attention to using the integral transform to solve issues in a variety of scientific domains, including engineering and mathematical sciences. The fundamental problem is reduced to a simpler algebraic equation using the complex transform. This paper utilizes the complex Sadik transform to track down the specific answer for the second kind of linear Volterra integro-differential equation. After that, the solution to this fundamental problem can be found by solving this equation and using the complex Sadik transform's inverse. Through some numeric examples, complex Sadik integral transform's effectiveness and capacity to deliver a precise solution with the least number of calculations are shown. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the (recently introduced by Alavrez et al.) concept of the so-called asymptotically (ω,c) periodic functions. We investigate further properties of this class of functions which complement the currently known ones. These properties allowed us to prove the existence and uniqueness of asymptotically (ω,c) - periodic mild solution of an integro-differential equation in a Banach space, a result which generalizes some recent works in the class of S-asymptotically - periodic functions by Brindle and N'Guérékata. [ABSTRACT FROM AUTHOR]

In this paper, the aim is to study the weakly singular integro-differential equations, typically its significance appears in various applications of engineering and science. The contributions of this paper have been presented through the study of existence, uniqueness, and different stability of solution of the weakly singular integro-differential equations, and solving this type of equations analytically. We propose an analytic method based on Laplace transform to solve the weakly singular integro-differential equations, whose advantages lie in simplicity application and obtaining exact solutions. Some suitable examples have been provided to better understand this work and the results of experiments exhibited the proposed method as a simple approach and superiority in accuracy and efficiency of solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the variance and volatility swaps pricing problem under the framework of double Heston jump diffusion model with approximative fractional stochastic volatility. The pricing formulas of discretely-sampled variance and volatility swaps are obtained by deriving the characteristic function and solving the governing partial integro-differential equations(PIDEs). We also obtain the limits of discretely-sampled variance and volatility swaps pricing formulas, which are the pricing formulas of continuously-sampled variance and volatility swaps. Finally, the effectiveness of the pricing formula is illustrated by comparing with some existing works, and the influence of approximation factor and Hurst parameter variation on the prices of swaps are studied. [ABSTRACT FROM AUTHOR]

The All ORders Spectral Algorithm (AORSA) wave equation solver by Jaeger (Jaeger et al 2001 Phys. Plasmas 8 1573) solves the integro-differential wave equation relevant for the radio frequency (RF) domain and for fusion-relevant conditions in tokamaks or stellarators, retaining all finite Larmor radius corrections by substituting the continuous Fourier integrals by a sum over a discrete set of modes. Its strength is also its weakness: the simplicity of the method results in significant computational effort, a full matrix needing to be inverted to solve the associated linear system. Based on the notion that modes are gradually more independent if their eigenvalues differ, the present paper proposes a straightforward numerical method to partly alleviate this need, allowing to substitute the full system matrix by a banded one. The adopted method can be applied to a wide variety of equations. A few 1D examples—of relevance for solving the wave equation in the RF domain of frequencies—are provided: the tunneling equation is used to illustrate the potential of the method, and the all-FLR wave equation (retaining all Finite Larmor Radius corrections in the dielectric response) adopted by Jaeger is solved comparing the solutions found to those based on simpler models (a cold plasma and a 'tepid plasma' - i.e. a kinetic model truncated at zero order in Larmor radius—description). [ABSTRACT FROM AUTHOR]

In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C 0 —semigroup theory, we prove that the system has a unique positive time-dependent solution (TDS) that satisfies the probability condition. As our second result, we prove that the TDS of the system strongly converges to its steady-state solution (SSS) if the service rates of the servers are constants. For this asymptotic behavior, we analyze the spectrum of the system operator associated with the system. Additionally, the stability of the semigroup generated by the system operator is also discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a compact implicit-explicit (IMEX) difference scheme for solving a moving boundary partial integro-differential equation (PIDE) system of the regime-switching jump-diffusion Asian option pricing. First, the IMEX scheme is proposed for temporal discretization and combined with the compact difference scheme for spatial discretization. Then the unconditional stability, unique solvability, and convergence rates of second-order in time and fourth-order in space are proved in the discrete L 2 and L ∞ norms. Numerical examples are given to demonstrate the theoretical results and show the effectiveness of the proposed scheme with respect to the efficiency and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we explore the iterated Crank–Nicolson (ICN) algorithm for the one-dimensional peridynamic model. The peridynamic equation of motion is an integro-differential equation that governs structural deformations such as fractures. The ICN method was originally developed for hyperbolic advection equations. In peridynamics, we apply the ICN algorithm for temporal discretization and the midpoint quadrature method for spatial integration. Several numerical tests are carried out to evaluate the performance of the ICN method. In general, the ICN method demonstrates second-order accuracy, consistent with the Störmer–Verlet (SV) method. When the weight is 1/3, the ICN method behaves as a third-order Runge–Kutta method and maintains strong stability-preserving (SSP) properties for linear problems. Regarding energy conservation, the ICN algorithm maintains at least second-order accuracy, making it superior to the SV method, which converges linearly. Furthermore, selecting a weight of 0.25 results in fourth-order superconvergent energy variation for the ICN method. In this case, the ICN method exhibits energy variation similar to that of the fourth-order Runge–Kutta method but operates approximately 20% faster. Higher-order convergence for energy can also be achieved by increasing the number of iterations in the ICN method. [ABSTRACT FROM AUTHOR]

In this paper, our primary goal is to investigate the existence of solutions for an impulsive fractional integro-differential system (IFIDS) subject to boundary value conditions (BVC), utilizing the Caputo-Fabrizio operator (CFO). Through the application of fixed-point methodologies, we derive original insights into the existence of solutions. The final section of the study incorporates three numerical examples, serving to illustrate the practical application of the theoretical concepts introduced. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear Volterra integro-differential equation with Caputo fractional derivative, multiple kernels, and multiple constant delays is considered. The aim of this paper is to investigate qualitative properties of solutions of this equation such as uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution as well as boundedness of nonzero solutions. Here, we prove four new theorems on the mentioned properties of the solutions of the considered fractional integro-differential equation. The technique used in the proofs of these theorems includes defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method. To illustrate the obtained results, two examples are provided, one of them related to an RLC circuit, to illustrate and show applications of the given results. The obtained results are new, original, and they can be useful for applied researchers in sciences and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter. The estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices and , the mild solution of the approximation problem converges to the solution of the associated limit problem in as. The resolvent operator family and a set of kernel assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as. In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter . The estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices and , the mild solution of the approximation problem converges to the solution of the associated limit problem in as . The resolvent operator family and a set of kernel assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as . [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations. [ABSTRACT FROM AUTHOR]