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The purpose of this paper is to study the controllability in the α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach space. To do this, we give sufficient conditions ensuring the controllability by assuming that the undelayed part admits a resolvent operator in the sense of Grimmer and that the delayed part is continuous with respect to the fractional power of the generator. The results are obtained by using the Schauder fixed-point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo S-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration. [ABSTRACT FROM AUTHOR]

This work presents a novel approach by considering the hyperbolic integro-differential equation as time fractional and multi-term, employing the generalized Caputo derivative in a two-dimensional domain for the first time. The key contribution of this study lies in the development of an explicit formula for the operator matrices of ordinary and fractional derivatives, as well as the weakly singular kernel, using shifted Vieta-Pell-Lucas polynomials. These polynomials are proven to converge in the new space L p , γ , υ , which is defined based on the generalized Caputo derivative. By utilizing the derived operator matrices and applying the collocation technique, we successfully transform the hyperbolic multi-term time fractional integro-differential equation into a system of algebraic equations. This transformation allows us to calculate the approximate solution of the equation efficiently. To demonstrate the effectiveness of the proposed method, several examples are presented and analyzed. The accuracy of the method is evaluated through these examples, showcasing its reliability in solving the hyperbolic integro-differential equations with time fractional and multi-term characteristics. The obtained results highlight the novelty and potential of our approach in addressing such complex equations. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of the Mehler–Fock convolution operator. We also analyse the Banach algebraic structure on the space of integrable functions $ L_1(1,\infty) $ L 1 (1 , ∞) with the multiplication being the Mehler–Fock convolution. The Titchmarsh-type theorem for this convolution operator is also obtained. As applications, we apply these properties of the convolution operator to solve some classes of Fredholm integral and integro-differential equations and prove some priori estimations under the given conditions. [ABSTRACT FROM AUTHOR]

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In this work, we study a new class of integro‐differential equations with delay, where the informations from the past are represented as an average of the state over state‐dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro‐differential equations arising in the theory of population dynamics. [ABSTRACT FROM AUTHOR]

At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann) steady state. Here we consider scale-invariant power-law potentials V (x) ∼ | x | m , where m > 0, and employ the optimal fluctuation method (OFM) to determine the large- | x | tails of the steady-state probability distribution P (x) of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large- | x | tails of ln ⁡ P (x) up to a dimensionless factor α (H , m) , where 0 < H < 1 is the Hurst exponent. We determine α (H , m) analytically (i) in the limits of H → 0 and H → 1, and (ii) for m = 2 and arbitrary H, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact P (x) and autocovariance. The form of the tails of P (x) yields exact conditions, in terms of H and m, for the particle confinement in the potential. For H ≠ 1 / 2 , the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for m = 2. To compute the optimal paths and the factor α (H , m) for arbitrary permissible H and m, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which accounts analytically for an intrinsic cusp singularity of the optimal paths for H < 1 / 2 . [ABSTRACT FROM AUTHOR]

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example. [ABSTRACT FROM AUTHOR]

In this work, we study the coupled system of fractional integro-differential equations, which includes the fractional derivatives of the Riemann–Liouville type and the fractional q-integral of the Riemann–Liouville type. We focus on the utilization of two significant fixed-point theorems, namely the Schauder fixed theorem and the Banach contraction principle. These mathematical tools play a crucial role in investigating the existence and uniqueness of a solution for a coupled system of fractional q-integro-differential equations. Our analysis specifically incorporates the fractional derivative and integral of the Riemann–Liouville type. To illustrate the implications of our findings, we present two examples that demonstrate the practical applications of our results. These examples serve as tangible scenarios where the aforementioned theorems can effectively address real-world problems and elucidate the underlying mathematical principles. By leveraging the power of the Schauder fixed theorem and the Banach contraction principle, our work contributes to a deeper understanding of the solutions to coupled systems of fractional q-integro-differential equations. Furthermore, it highlights the potential practical significance of these mathematical tools in various fields where such equations arise, offering a valuable framework for addressing complex problems. [ABSTRACT FROM AUTHOR]

We consider a Mikusiński-type convolution algebra C α , including functions with power-type singularities at the origin as well as all functions continuous on [ 0 , ∞) . Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed. [ABSTRACT FROM AUTHOR]

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell's equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography. [ABSTRACT FROM AUTHOR]

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed. [ABSTRACT FROM AUTHOR]

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence, uniqueness, and analysis of two types of k$$ k $$‐Mittag–Leffler–Ulam stabilities in a Volterra integro‐differential fractional differential equation that involves the (k,ϱ)$$ \left(k,\varrho \right) $$‐Hilfer operator. We utilize the Banach fixed‐point theorem to establish the existence and uniqueness of solutions. We examine the stability properties, including the k$$ k $$‐Mittag–Leffler–Ulam–Hyers k$$ k $$‐ MLUH$$ \mathcal{MLUH} $$ and k‐Mittag–Leffler–Ulam–Hyers–Rassias k$$ k $$‐ MLUHR$$ \mathcal{MLUHR} $$ stabilities, by employing the Grönwall–Bellman inequality. Additionally, we provide an example to confirm our findings. [ABSTRACT FROM AUTHOR]

The Basset–Boussinesq force is often perfunctorily neglected when studying small inertial particles in turbulence. This force arises from the diffusion of vorticity from the particles and, since it depends on the particles' history, complicates the dynamics by transforming their equations of motion into integrodifferential equations. However, this force is of the same order as other viscous forces acting on the particles, and beyond convenience, the reasons for neglecting it are unclear. This study addresses the following question: Under what conditions can the Basset–Boussinesq force be neglected in light particles in geophysical flows? We derive strict bounds for the magnitude of the Basset–Boussinesq force in stably stratified flows, in contexts of interest for geophysical turbulence. The bounds are validated by direct numerical simulations. The Basset–Boussinesq force is negligible when a buoyancy Stokes number Sb = N τ p is small, where N is the flow Brunt–Väisälä frequency and τ p is the particle's Stokes time. Interestingly, for most oceanic particles this force may be negligible. Only for very strong stratification, or for particles with very large inertia, this force must be considered in the dynamics. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of mild solutions for a nonlocal ξ-Caputo fractional non-instantaneous impulses semilinear integro-differential equation in a Banach space. The proofs are based on some fixed point theorems for condensing maps. As an application, an example is given to illustrate our theoretical results [ABSTRACT FROM AUTHOR]

The methodology based on the so‐called global relation, introduced by the first author, has recently led to the derivation of a novel nonlinear integral‐differential equation characterizing the classical problem of the Saffman–Taylor fingers with nonzero surface tension. In the particular case of zero surface tension, this equation is satisfied by the explicit solution obtained by Saffman and Taylor. Here, first, for the case of zero surface tension, we present a new nonlinear integrodifferential equation characterizing the Saffman–Taylor fingers. Then, by using the explicit Saffman–Taylor solution valid for the particular case of zero surface tension, we show that the above equations give rise to sets of remarkable integral trigonometric identities. [ABSTRACT FROM AUTHOR]

In this manuscript, we introduced the radial basis function based three implicit-explicit (IMEX) finite difference techniques for pricing European and American options in an extended Markovian regime-switching jump-diffusion (RSJD) economy. A partial integrodifferential equation (PIDE) yields the values of the European option, which is one of the financial options, and a linear complementary problem (LCP) yields the prices of the American option. To solve the LCP for American option pricing, we combine the suggested techniques with the operator splitting methods. The suggested methods are designed to prevent the use of any fixed-point repetition approaches at each economic stage and time increment. We analyzed the stability of the proposed time discretization methods. We performed numerical experiments and illustrated the second-order convergence and efficiency of the three IMEX numerical techniques (BDF2, CNAB, CNLF) under the extended RSJD model. [ABSTRACT FROM AUTHOR]

An alternating direction implicit (ADI) scheme is proposed to study the numerical solution of a three-dimensional integrodifferential equation (IDE) with multi-term tempered singular kernels. Firstly, we employ the Crank-Nicolson method and the product integral (PI) rule on a uniform grid to approximate the temporal derivative and the multi-term tempered-type integral terms, thus establishing a second-order temporal discrete scheme. Then, a second-order finite difference method is used for spatial discretization and combined with the ADI technique to improve computational efficiency. Based on regularity conditions, the stability and convergence analysis of the ADI scheme is given by the energy argument. Finally, numerical examples confirm the results of the theoretical analysis and show that the method is effective. [ABSTRACT FROM AUTHOR]

In this work, we deal with a stochastic fractional integrodifferential equations associated to a Poisson random measure. We first prove existence and uniqueness of solution in the case of Lipschitz coefficients but also in the non Lipschitz case. In the second part, we show an averaging principle in the sense of Khasminskii approach for a class of this equation with non Lipschitz coefficients and weak averaging conditions. [ABSTRACT FROM AUTHOR]

The work introduces a novel numerical method for solving the Fredholm Integro-Differential Equations (FIDEs) and system of Fredholm Integro-Differential Equations (SFIDEs) by employing the fourth-order compact finite difference methods in conjunction with Simpson's quadrature rule. The accuracy of the proposed scheme is rigorously evaluated using [Math Processing Error] l 2 and [Math Processing Error] l ∞ norms, while the computational efficiency is measured by assessing the CPU-time values, demonstrating a notable reduction in computational cost compared to standard finite difference schemes. The significance of this approach lies in its ability to maintain high levels of accuracy, addressing a common challenge in traditional methods. The methods presented exhibit fourth accuracy in space, as evidenced by numerical experiments. The mentioned work signifies a notable progress in tackling problems related to FIDEs and SFIDEs. It introduces a robust and efficient numerical methodology that proves particularly effective in situations where obtaining exact solutions poses challenges. This advancement is crucial as it addresses a common difficulty faced in the solution of FIDEs and SFIDEs problems, offering a reliable numerical approach that can handle complex scenarios and contribute to more accurate and practical solutions in various fields of study. [ABSTRACT FROM AUTHOR]

Analog computing based on miniaturized surfaces has gained attention for its high‐speed and low‐power mathematical operations. Building on recent advances, an anisotropic space‐time digital metasurface for parallel and programmable wave‐based mathematical operations is proposed. Using frequency conversions, our metasurface performs 1st‐order and 2nd‐order spatial differentiations, integrodifferential equations, and sharp edge detection in spatially encoded images. The anisotropic nature of the meta‐particle enables independent and simultaneous operations for two orthogonal polarizations. Reconfigurability is achieved through tunable gate biasing of an indium tin oxide layer. Illustrative examples demonstrate that the metasurface's output signals and transfer functions closely match ideal transfer functions, confirming its versatility and effectiveness. Unlike other wave‐based signal processors, the design handles wide spatial frequency bandwidths, even with high spatial frequency inputs. [ABSTRACT FROM AUTHOR]

Quantum many-body systems are of great interest for many research areas, including physics, biology, and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning, in general, are a way to face this challenge. For instance, methods like tensor networks and neural quantum states are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schrödinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploit automatic differentiation to solve integro-differential equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an artificial intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states [ABSTRACT FROM AUTHOR]

Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far-field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen's extension of Noll's theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips. [ABSTRACT FROM AUTHOR]

This article investigates the finite-approximate controllability properties for semi-linear integrodifferential systems involving higher-order fractional derivatives in Riemann-Liouville sense excluding Lipschitz assumptions of nonlinear operators. We discuss the existence of mild solutions by utilizing the Schaefer fixed point principle and compactness condition on the fractional resolvent. Then we show that one can steer the system in an arbitrary neighbourhood of any given target state simultaneously obeying the finitely many constraints. Lastly, an illustrative example is presented to validate the proposed methodology. [ABSTRACT FROM AUTHOR]

Статья посвящена описанию разработанной численной схемы для моделирования эредитарной динамической системой, являющейся моделью двумодового гидромагнитного динамо. Модели включают в себя два генератора магнитного поля — крупномасштабный и турбулентный (α-эффект). Влияние магнитного поля на движения среды представлено через подавление α-эффекта функционалом от компонент поля, что вводит в модель память (эредитарность). Модель описывается интегро-дифференциальной системой уравнений.В работе представлена сама численная схема и исследован порядок точности на вложенных сетках. Численная схема состоит из двух частей, для дифференциальной части используется метод трапеций, а для интегральной квадратурная формула трапеций. В результате сопряжения схем получаем нелинейную алгебраическую систему уравнений. Для решения такой системы необходимо привлечение методов для нелинейных алгебраических систем. В работе был выбран метод Ньютона. Показано, что в случае экспоненциального ядра функционала подавления модель может быть сведена к классической системе Лоренца. Известный характер динамики системы Лоренца при различных параметрах позволил верифицировать численную схему. Показано, что численная схема позволяет решать на качественном уровне интегро-дифференциальную систему уравнений, которая является моделью космического динамо. Данная численная схема была разработана для конкретной модели, но может быть легко обобщена для других квадратично-нелинейных интегро-дифференциальных систем.

Работа посвящена классу стохастических двумодовых эредитарных моделей космического динамо. Модели включают в себя два генератора магнитного поля — крупномасштабный и турбулентный (α-эффект). Влияние магнитного поля на движения среды представлено через подавление α-эффекта функционалом от компонент поля, что вводит в модель память (эредитарность). Модель описывает динамику только крупномасштабных компонент, однако учитывает возможное воздействие мелкомасштабных мод с помощью стохастического члена. Это член моделирует влияние возможной спонтанной синхронизации мелкомасштабных мод. Так же в работе представлена численная схема для решения интегро-дифференциальных уравнений модели. Численная схема состоит из двух частей: для дифференциальной части используется метод «предиктор-корректор» Адамса четвертого порядка, а для интегральной части — метод Симпсона. Основным результатом работы является обобщенная модель динамо-системы, с аддитивным добавлением случайной поправка в α-генератор. Учет такой поправки существенно разнообразит динамические режимы в модели.

In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators ∂ t u (t , x) = L a u (t , x) + f (t , x) , t > 0 in L q (L p) spaces. Our spatial operator L a is an integro-differential operator of the form ∫ R d (u (x + y) − u (x) − ∇ u (x) ⋅ y 1 | y | ≤ 1) a (t , y) j d (| y |) d y. Here, a (t , y) is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on j d (r) which yield L q (L p) -regularity of solutions. Our assumptions on j d are general so that j d (r) may be comparable to r − d ℓ (r − 1) for a function ℓ which is slowly varying at infinity. For example, we can take ℓ (r) = log ⁡ (1 + r α) or ℓ (r) = min ⁡ { r α , 1 } (α ∈ (0 , 2)). Indeed, our result covers the operators whose Fourier multiplier ψ (ξ) does not have any scaling condition for | ξ | ≥ 1. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on ψ are considered. [ABSTRACT FROM AUTHOR]

In this work, we focus on a class of Ψ− fractional integro-differential equations (Ψ-FIDEs) involving Ψ-Caputo derivative. The objective of this paper is to derive the numerical solution of Ψ-FIDEs in the truncated Bell series. Firstly, Ψ-FIDEs by using the definition of Ψ− Caputo derivative is converted into a singular integral equation. Then, a computational procedure based on the Bell polynomials, Gauss-Legendre quadrature rule, and collocation method is developed to effectively solve the singular integral equation. The convergence of the approximation obtained in the presented strategy is investigated. Finally, the effectiveness and superiority of our method are revealed by numerical samples. The results of the suggested approach are compared with the results obtained by extended Chebyshev cardinal wavelets method (EChCWM). [ABSTRACT FROM AUTHOR]

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy.

In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem.

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on L p ( G ) ( 1 < p < + ∞, G is a graded Lie group). We show the explicit solution of the considered equation. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on G. Also, we provide L p ( G ) − L q ( G ) ( 1 < p ⩽ 2 ⩽ q < + ∞) norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in L 2 ( G ) that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given. [ABSTRACT FROM AUTHOR]

In this study, we present an immuno-epidemic model to understand mitigation options during an epidemic break. The model incorporates comorbidity and multiple-vaccine doses through a system of coupled integro-differential equations to analyze the epidemic rate and intensity from a knowledge of the basic reproduction number and time-distributed rate functions. Our modeling results show that the interval between vaccine doses is a key control parameter that can be tuned to significantly influence disease spread. We show that multiple doses induce a hysteresis effect in immunity levels that offers a better mitigation alternative compared to frequent vaccination which is less cost-effective while being more intrusive. Optimal dosing intervals, emphasizing the cost-effectiveness of each vaccination effort, and determined by various factors such as the level of immunity and efficacy of vaccines against different strains, appear to be crucial in disease management. The model is sufficiently generic that can be extended to accommodate specific disease forms. [ABSTRACT FROM AUTHOR]

In the article, we establish the global well‐posedness in W1,2,2(ℝ×ℝ+)$$ {W}&#x0005E;{1,2,2}\left(\mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}&#x0005E;{&#x0002B;}\right) $$ of the integro‐differential equation in the case of anomalous diffusion when the one‐dimensional negative Laplace operator is raised to a fractional power in the presence of the transport term. The model is relevant to the cell population dynamics in the mathematical biology. Our proof relies on a fixed point technique. [ABSTRACT FROM AUTHOR]

We have performed a comparative computational study of magnetic particle alignment in a strong magnetic field and ambient viscous fluid using the first-principles Navier–Stokes equation as well as numerical modeling using the finite element method (FEM). FEM solution has been compared with the solution of the unsteady rotations of a magnetic particle in a viscous fluid during the alignment process described with nonlocal integro-differential equations (IDEs) for torque and angular velocity. The assumption of nonlocality comes from the history term of acceleration torque with a non-Basset kernel function, which has its origin in the presence of vortices in the flow of a particle fluid environment due to its unsteady rotation and ambient fluid inertia and friction. The flow vortices of the ambient fluid are explicitly shown in the solution of the FEM model. Moreover, the solution of the time evolution of the alignment angle and angular velocity for the FEM model are in good agreement with an IDE model which we have recently developed. The obtained results are justification for interchangeability of the FEM and IDE models, which may have important consequences for large-scale simulations of magnetic microparticles. [ABSTRACT FROM AUTHOR]

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of "damage" increments accelerates according to the increasing number of "damages". We start from the partial differential equation satisfied by its transition density, in the case of exponentially distributed addends, and then we generalize it by introducing a space derivative of convolution type (i.e., defined in terms of the Laplace exponent of a subordinator). Then we are concerned with the solution of integro‐differential equations, under proper initial conditions, which, in a special case, reduce to a fractional one. Correspondingly, we analyze the related cumulative jump processes under a general infinitely divisible distribution of the (positive) jumps. Some special cases (such as the stable, tempered stable, gamma, and Poisson) are presented. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish an improved convergence analysis via Kulkarni method to approximate the solution of an integro‐differential equation in L2([−1,1],R). We prove the following convergence orders: Kulkarni order is n−3r, and Kulkarni iterated order is n−4r. The present study extends and improves earlier results in the literature. A numerical example illustrates the theoretical results and shows the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integro-differential equations comprising of the Ψ-Caputo fractional derivative and the Ψ-Riemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko's attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples. [ABSTRACT FROM AUTHOR]

The main focus of this manuscript is to study an impulsive fractional integro‐differential equation with delay and Caputo fractional derivative. The existence solution of such a class of fractional differential equations is discussed for linear and nonlinear case with the help of direct integral method. Moreover, Banach's fixed point theorem and Schaefer's fixed point theorem are use to discuss the uniqueness and at least one solution of the said fractional differential equations, respectively. Some hypothesis and inequalities are utilize to present four different types of Hyers–Ulam stability of the mentioned impulsive integro‐differential equation. Example is provide for the illustration of main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following fractional Kirchhoff‐type pseudo parabolic equation driven by a nonlocal integro‐differential operator ℒK$$ {\mathcal{L}}_K $$: ut|x|2s+M([u]s2)ℒKu+ℒKut=|u|p−2ulog|u|,$$ \frac{u_t}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{2s}}&#x0002B;M\left({\left[u\right]}_s&#x0005E;2\right){\mathcal{L}}_Ku&#x0002B;{\mathcal{L}}_K{u}_t&#x0003D;{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\log \mid u\mid, $$ where [u]s$$ {\left[u\right]}_s $$ represents the Gagliardo seminorm of u$$ u $$. Instead of imposing specific assumptions on the Kirchhoff function, we introduce a more general sense to establish the local existence of weak solutions. Moreover, via the sharp fractional Hardy inequality, the decay estimates for global weak solutions, the blow‐up criterion, blow‐up rate, and the upper and lower bounds of the blow‐up time are derived. Lastly, we discuss the global existence and finite time blow‐up results with high initial energy. [ABSTRACT FROM AUTHOR]

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton-Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density. [ABSTRACT FROM AUTHOR]

This article investigates the proper existence conditions and uniqueness results for a class of fuzzy fractional Caputo Volterra-Fredholm integro-differential equations (FFCV-FIDE) with initial conditions. The findings are based on Banach's contraction principle and Schaefer's fixed point theorem. Furthermore, the solution to the posed problem is found using the Adomian decomposition technique (ADT). We support the concept with several examples. The relationship between the upper and lower reduced approximation of the fuzzy solutions was demonstrated numerically and graphically using MATLAB. [ABSTRACT FROM AUTHOR]

The influence of the deposition of nanoparticles of an aerosol, moving in a continuous reactor at a constant velocity, on the channel walls of the reactor and of the coagulation of these particles as a result of their Brownian diffusion on the nanoaerosol parameters was investigated. A simple one-dimensional model, adequately defining the diffusion, coagulation, and deposition of nanoaerosol particles in a wide range of change in its Knudsen number, has been constructed. It is shown analytically that, as the distance from the inlet of the reactor increases, the volume fraction of particles in the nanoaerosol decreases by the exponential law, and the radius of the clusters formed in the nanoaerosol as a result of the coagulation of its particles increases and tends to a limiting (maximum) value. A nonlinear integro-differential equation for the radius of such clusters has been obtained, and, from it, compact formulas for approximate calculation of its limiting radius have been derived. The characteristic distributions of the radii of clusters and of their concentrations along the reactor channel, calculated by the numerical method, are presented. It was established that, if this channel has a fairly large length, the clusters moving in it increase to their limiting size. The influence of the determining parameters of the flow of a nanoaerosol at the inlet to the continuous reactor on the distribution of its dispersion characteristics along the reactor channel, on the limiting size of the clysters in it, and on the characteristic length of the channel, at which the processes of deposition and coagulation of nanoaerosol particles are completed, is discussed. A comparison of the results of calculations of the parameters of a nanoaerosol moving in a continuous reactor by the one- and two-dimensional models has shown that the simple one-dimensional model defines the behavior of these parameters quite adequately. [ABSTRACT FROM AUTHOR]

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ ∈ [ 0 , 1 ] . The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. [ABSTRACT FROM AUTHOR]