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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]

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]

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]

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, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H 1 and L p norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H 1 and L p norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods. [ABSTRACT FROM AUTHOR]

The linear weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads ∂ t u − F (u) (− Δ) s / 2 u + (− Δ) s / 2 (u F (u)) = 0 , x ∈ T d , with s ∈ (0 , 1 ] . We are interested in solutions stemming from periodic positive bounded initial data. The given function F ∈ C ∞ (R +) must satisfy F ′ > 0 a.e. on (0 , + ∞) . For instance, all the functions F (u) = u n with n ∈ N ∗ are admissible non-linearities. The local theory can also be developed on the whole space, however the most complete well-posedness result requires the periodic setting. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in L ∞ . We show that any weak solution is instantaneously regularized into C ∞ . We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations, in particular (Ann. Fac. Sci. Toulouse, Math. 25(4):723–758, 2016; Ann. Fac. Sci. Toulouse, Math. 27(4):667–677, 2018). [ABSTRACT FROM AUTHOR]

This study introduces a new numerical approach named flatlet oblique multiwavelets (FOMW) to solve fractional‐order stochastic integro‐differential equation (FSI‐DE). The FOMW is used to create an operational matrix of the stochastic integral, which helps transform the FSI‐DE into a linear system of algebraic equations. This method requires only a small number of basis functions to achieve satisfactory results. The paper also includes a comprehensive discussion on error estimation and convergence analysis, which provides an upper bound of error for the proposed method. Furthermore, numerical experiments have been conducted and compared with other methods, demonstrating the superiority of FOMW in terms of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence result of weak solutions to a class fractional p(x,y)$$ p\left(x,y\right) $$‐Laplacian problems involving the nonlocal integro‐differential operator of elliptique type LKp(x)$$ {\mathcal{L}}_K^{p(x)} $$. The main tool used here is based on the topological degree theory combined with the theory fractional Sobolev spaces with variable exponent. Our results generalize and improve the existing results. [ABSTRACT FROM AUTHOR]

In this paper, an effective numerical Bernstein polynomials operational matrix is exploited to study the nonlinear forced vibration of fractional viscoelastic curved beam with viscoelastic nonlinear boundary conditions, for the first time. The Caputo fractional derivative is employed to incorporate viscoelastic material having nonlinear behavior. Based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity, the governing equation of viscoelastic curved beam which is a nonlinear integro-partial fractional differential equation is introduced. Based on the collocation method of Bernstein polynomials approximation, the generalized integer-order operational matrices of differentiation and integration as well as fractional-order operational matrix of differentiation are derived. Using Bernstein polynomials operational matrices (BPOM), the nonlinear static problem is discretized; then, it is solved via Newton's method. To compute the linear vibration mode, the linear vibration problem is discretized via BPOM and it is solved as a linear eigenvalue problem. Discretized by the Galerkin approximation, the fractional-order nonlinear integro-partial differential equation is transformed into a nonlinear fractional-order duffing-type equation. The fractional-order duffing equation is discretized using BPOM resulting nonlinear eigenvalue problem, which can be easily solved by pseudo-arc length continuation algorithm. The effects of the Caputo fractional derivative order, foundation parameters, amplitude of initial curvature, viscoelastic parameters and amplitude of excitation force on the nonlinear forced vibration of the viscoelastic beam are investigated through a detailed parametric study. The proposed procedure is supportive in the analysis and design of curved viscoelastic structure with non-classical viscoelastic boundary conditions under the dynamic mechanical loads. [ABSTRACT FROM AUTHOR]

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin–Voigt friction. [ABSTRACT FROM AUTHOR]

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin–Voigt friction. [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]

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]

This paper aims to study the Mode-I penny-shaped crack problem of an infinite body of one-dimensional hexagonal piezoelectric quasicrystal. The problem is transformed into a mixed-boundary value problem in the context of electro-elasticity of quasicrystals, and the corresponding integro-differential equations are analytically solved. Two extreme cases of electrically impermeable and permeable crack surface are considered. By virtue of the generalized potential theory method, the three-dimensional complete analytical solutions of three-dimensional crack problems under symmetric concentrated and uniform loads are expressed in terms of elementary functions. Important parameters in fracture mechanics are explicitly derived, such as crack surface displacements, the distributions of generalized stresses at the crack tip and the corresponding generalized stress intensity factors. The validity of the proposed solutions and the coupling effect of phonon-phason-electric fields are investigagted through numerical examples. [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]

Nonlinear partial integro-differential equations (PIDEs) are applied to present the various practical phenomena in a multitude of sectors of modern science and engineering, especially in optic fiber, quantum electronics, modern physics, and the special field of nonlinear wave motion. Basically, our research demonstrates a way to generate a significant quantity of solutions to these types of two PIDEs. In this research, we have used an efficient mathematical tool namely the generalized G ′ / G -expansion method to acquire the closed form soliton solutions for the (2 + 1)-dimensional first integro-differential Kadomtsev–Petviashivili (KP) hierarchy equation and the (2 + 1)-dimensional second integro-differential KP hierarchy equation utilizing a code likely Mathematica. The explicit closed form soliton solutions of these two PIDEs are found in the pattern of trigonometric, hyperbolic, and rational functions which are compared to all the well-known results that are yielded in the paper. We attain solutions that are graphically displayed in addition to physically described in 3D structure, contour, and 2D, such as a bell-shaped soliton, a singular bell-shaped soliton, and some kink-shaped solitons. As far as the authors' wisdom, the outcomes of these problems gained by the offered expansion method are renewed closed form solitary wave and investigated here for the first time. The analysis of obtained results will be able to provide a constructive explanation of the physical phenomena in optical physics and engineering. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method for solving the fractional order Fredholm integrodifferential equations via the Caputo-Liouville derivative is presented. The method uses the wellknown shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson's 1/3, and Simpson's 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton's method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program. [ABSTRACT FROM AUTHOR]

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]

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, a high-order compact exponential (HOE) scheme is constructed for solving time fractional integro-differential convection-diffusion equations with initial singularity. For spatial discretization, the convection and diffusion terms are approximated by a compact exponential difference formula. For temporal discretization, we use the L1 formula and trapezoidal PI rule with nonuniform meshes to approximate the Caputo derivative and the Riemann-Liouville integral, respectively. For this fully discrete implicit scheme, the local truncation error is obtained. The stability and convergence of the proposed scheme are analyzed based on the discrete fractional Gronwall inequality and the energy method. Numerical experiences are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

This paper uses the resolvent operator technique to investigate second-order non-autonomous neutral integrodifferential equations with impulsive conditions in a Banach space. We study the existence of a mild solution and the system's approximate controllability. The semigroup and resolvent operator theory, graph norm, and Krasnoselskii's fixed point theorem are used to demonstrate the results. Finally, we present our findings with an example. [ABSTRACT FROM AUTHOR]

This study focuses on integrodifferential equations involving fractal–fractional differential operators characterized by exponential decay, power law, and generalized Mittag–Leffler kernels. Utilizing linear growth and Lipschitz conditions, we investigate the existence and uniqueness of solutions, as well as the Hyers–Ulam stability of the proposed equations. For every instance, a numerical method is utilized to derive a numerical solution for the specified equation. The paper includes illustrations of fractal–fractional integrodifferential equations, with their precise solutions determined and subsequently compared with the numerical outcomes. This methodology can be applied to demonstrate convergence, and graphical presentations are included in relevant examples to illustrate our proposed approach. [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 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 paper, we are concerned with the convergence rates of an implicit-explicit (IMEX) difference scheme for solving a two-dimensional partial integro-differential equation (PIDE) with an inexact boundary which arises in option pricing with stochastic intensity jumps. First, the IMEX scheme is proposed to solve the two-dimensional PIDE and its inexact boundary governed by a one-dimensional PIDE. Then the second-order convergence rates of the IMEX scheme for the main PIDE are proved for both time and space based on the second-order convergence analysis in the discrete H 1 -norm of the IMEX scheme for the boundary PIDE. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we model, classify and investigate the solutions of (normalized) second order ode's with non-constant continuous coefficients. We introduce a generalized frequency function as the solution of a nonlinear integro-differential equation , show its existence and then derive a representation formula for (all) solutions of (normalized) second order ode's with non-constant continuous coefficients. Because this formula specifies the interplay between the coefficients of the ode, the relaxation function ("strongly" decreasing positive function) and the frequency function of the oscillation, it can be applied to design models of dissipative oscillations. As an application, we present and discuss some oscillation models that stop within a finite time period. Via two examples about the Airy function and the Bessel functions, we show that the formula might be useful in analysing certain special functions. Moreover, we demonstrate that a large class of oscillations can be used to design and analyze dissipative waves. In particular, it is easy to model dissipative waves that cause in each point of space an oscillation that stops after a finite time period. Finally, we derive an alternate form of the dissipative wave equation (for a large class of dissipative waves), which is uniquely defined by the sound speed c 0 and the parameter functions in the oscillation equation. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two-sided a priori estimates for the solution of a homogeneous third-order Volterra integro-differential equation with a power-law nonlinearity and difference kernel. It is shown that the lower a priori estimate, which plays the role of a weight function when constructing a metric in the cone of the space of continuous functions, is sharp. Using these estimates, by the weighted metric method (an analog of A. Bielecki's method), we prove a global theorem on the existence and uniqueness of a nontrivial solution of the initial value problem for this integro-differential equation in the class of nonnegative continuous functions on the positive half-line and on the method for finding this solution. It is shown that the solution can be found by the successive approximation method, and an estimate of the rate of convergence of the approximations to the exact solution is obtained. Examples are given to illustrate the results obtained. [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]

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 examine the synthesis problem for a distributed thermal control system in the case where a Fredholm integral operator is involved in the boundary-value problem. We use the method proposed by A. I. Egorov and develop it based on the Bellman scheme. Using the notions of a generalized solution of a boundary-value problem and the Fréchet differential for the Bellman functional, we obtain a partial integro-differential equation. For synthesizing a distributed optimal control, we propose a numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a second‐order accurate scheme is considered for the temporal discretization of the abstract Volterra integrodifferential equation with a weakly singular kernel. The time discrete scheme is constructed by the Crank–Nicolson method for approximating the time derivative and product integration (PI) rule for approximating the integral term. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at t=0$$ t=0 $$. Under the suitable assumptions, the stability and convergence are established by the energy argument, and the error is of order k2$$ {k}^2 $$, where k$$ k $$ is the parameter for the time grids. Numerical experiments validate the theoretical estimate. [ABSTRACT FROM AUTHOR]

The Levin‐Nohel integro‐differential equation is one of the famous equations in various fields of science and engineering. It was first studied by Volterra in connection with biological applications. This paper deals with the nonlinear neutral Levin‐Nohel integro‐differential equation with Caputo fractional derivative and variable delays. The Ulam‐Hyers‐Rassias stability, Ulam‐Hyers stability, semi‐Ulam‐Hyers‐Rassias stability are studied. The existence and uniqueness of solutions are established by using Krasnoselskii's fixed point theorem and contraction mapping principle. An illustrative example is also given and that is studied by the help of the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

In this paper, we study a time‐fractional initial‐boundary value problem of Kirchhoff type involving memory term for non‐homogeneous materials. As a consequence of energy argument, we derive L∞0,T;H01(Ω)$$ {L}^{\infty}\left(0,T;{H}_0^1\left(\Omega \right)\right) $$ bound as well as L2(0,T;H2(Ω))$$ {L}^2\left(0,T;{H}^2\left(\Omega \right)\right) $$ bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz‐Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of O(h+k2−α)$$ O\left(h+{k}^{2-\alpha}\right) $$, where α(0<α<1)$$ \alpha \kern0.1em \left(0<\alpha <1\right) $$ is the fractional derivative exponent and h$$ h $$ and k$$ k $$ are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2‐1 σ$$ {}_{\sigma } $$ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims. [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, 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]

The paper presents the results of solving a boundary value problem for a system of two integro-differential equations that simulate the action of an external electric field on a plasma layer. This system is an implication of the Boltzmann–Maxwell equations, and the physical meaning of the sought functions is the strength of a self-consistent electric field and perturbation of the electron distribution density. The solution of the problem is constructed using the theories of Fourier transform of generalized functions and singular integral equations with the Cauchy kernel. The dependence of the solution on the frequency of the external field is studied. [ABSTRACT FROM AUTHOR]

In this present paper, we first obtained some estimates involving parts of ε -regular mild solutions of the fractional integro-differential equation. In this sense, through these preliminary results, we investigate the main results of this paper, i.e., the existence, regularity and continuous dependence of ε -regular mild solutions for fractional abstract integro-differential equations in Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we consider recurrent behavior of bounded solutions for a functional integro-differential equation arising from heat conduction in materials with memory. Prior to the main results, we give a new version of composite theorem on measure pseudo almost automorphic functions involved in delay. Based on recently obtained results on the uniform exponential stability as well as contraction mapping principle, we prove some existence and uniqueness theorems on the recurrence of bounded mild solutions for the addressed equations with infinite delay. Finally, we finish this paper with an example on partial integro-differential equation which frequently comes to light in the study of heat conduction. [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]