Search Results (486)

In this study, we investigate a class of mixed fractional partial integro-differential equations (FrPI-DE) involving symmetric singular kernels. The considered model problem involves Caputo fractional derivatives and integral operators that describe spatial interactions in a bounded domain. For the purpose of analysis, the original problem is reformulated in the form of a nonlinear Volterra–Fredholm integral equation (NV-FIE). The existence and uniqueness of the solution are established by the Banach fixed point theorem. To compute numerical solutions, a modified Toeplitz matrix method (TMM) is proposed to handle the singular kernel efficiently. The method transforms the integral equation to a system of nonlinear algebraic equations, which can be solved numerically. The convergence properties of the resulting numerical scheme are analyzed and illustrate the effectiveness of the method by providing numerical examples involving logarithmic, Cauchy-type, and weakly singular kernels. Numerical results indicate that the proposed method provides highly accurate approximations and exhibits stable convergence behavior for different parameter values. Furthermore, these results confirm the effectiveness and reliability of the proposed method for solving fractional integro-differential equations that include symmetric singular kernels. Full article

A symbolic method is presented for examining the solvability and constructing the exact solution to boundary value problems for systems of linear ordinary loaded differential equations and loaded integro-differential equations with nonlocal boundary conditions. The method uses the inverse of the differential operator involved in the system of loaded differential or integro-differential equations. A solvability criterion based on the determinant of a matrix and an exact analytical matrix-form solution formula are presented. For the implementation of the method into computer algebra system software, two algorithms are provided. The effectiveness of the method is demonstrated by solving several problems. The theoretical and practical results obtained complement the existing literature on the subject. Full article

Integral-form nonlocal elasticity provides a mechanically meaningful approach to describing size effects, yet it leads to Volterra-type integro-differential equations that are difficult to solve analytically and numerically challenging for boundary layers and high-order modes. In this work, we developed a symplectic numerical integration framework for Eringen’s two-phase (local/nonlocal mixture) integral model by embedding the constitutive operator into a Hamiltonian formulation and discretizing the influence domain in a belt-wise manner. A step-increase strategy was incorporated to allow flexible spatial marching while preserving the geometric (symplectic) structure of the transfer operation. In addition, a symmetry-explicit, element-level stiffness representation was derived for the discretized integral operator; it exposes a mirrored long-range coupling pattern and enables symmetric, energy-consistent assembly. The resulting kernel-agnostic algorithm accommodates both smooth and finite-range kernels. Static benchmarks and longitudinal vibrations are investigated for exponential, Gaussian, and triangular kernels over representative length ratios and mixture parameters. Comparisons with available analytical and asymptotic solutions show good agreement within their validity ranges, and the method yields stable higher-order eigenfrequencies when asymptotic expansions may be unreliable. The current study is limited to a linear one-dimensional rod setting, and validation is restricted to published analytical/asymptotic solutions rather than experimental calibration. Full article

In this paper, we investigate the spectrum distribution and asymptotic behavior of an M/$G^B$/1 bulk service queueing system. In this system, the server processes customers in batches of a fixed maximum capacity B, and the time required to serve a batch is governed by a general distribution with a service rate function $\eta (·)$, which determines the instantaneous probability of service completion. The system dynamics are described by an infinite set of partial integro-differential equations. First, by introducing the probability generating function and employing Greiner’s boundary perturbation method, we establish that the time-dependent solution (TDS) of the system converges strongly to its steady-state solution (SSS) in the natural Banach state space. To this end, when the service rate $\eta (·)$ is a bounded function, we prove that zero is an eigenvalue of both the system operator and its adjoint operator, with geometric multiplicity one. Moreover, we show that every point on the imaginary axis except zero belongs to the resolvent set of the system operator. Second, we analyze the spectrum of the system operator on the left real axis. When the service rate $\eta (·)$ is constant and the fixed maximum capacity B equals 2, we apply Jury’s stability criterion for cubic equations to demonstrate that the system operator possesses an uncountably infinite number of eigenvalues located on the negative real axis. Additionally, we prove that an open interval near zero on the left real axis is not part of the point spectrum of the system operator. Consequently, these results imply that the semigroup generated by the system operator is not compact, eventually compact, quasi-compact, or essentially compact. Full article

Eringen’s integral constitutive relation is more general than its differential counterpart for modeling small-scale effects in micro- and nanostructures; however, it leads to integro-differential governing equations that are difficult to solve, which has limited the practical use of integral formulations. To directly address this gap, this paper introduces a novel symplectic-based numerical method that efficiently and accurately analyzes the free vibration of small-scale Kirchhoff plates governed by Eringen’s integral nonlocal model. The method discretizes the nonlocal integral operator by introducing inter-belt elements for long-range interactions and adopting a truncated influence domain, while balancing computational efficiency and accuracy. The effects of the nonlocal parameter, two-phase mixture parameter, mode numbers, kernel types, and geometric parameters on the natural frequencies are systematically investigated. The results indicate stiffness softening. For a simply supported square nanoplate with side length $a=10 \mathrm{n}\mathrm{m}$, the first-order frequency parameter decreases by approximately 25% as the nonlocal parameter increases from 0 to 4 nm, and higher-order modes exhibit substantially greater sensitivity to nonlocal effects. Convergence and accuracy are validated against published continuum-level solutions and molecular dynamics simulations; relative deviations are below 2% in most cases, and the local limit $\left(l_a=0\right)$ yields errors on the order of ${10}^{-3}$. Full article

In the classical non-life insurance models, the capital reserve of an insurance company increases at a constant rate and decreases by downward jumps. We consider a generalization of this model by supposing that a fixed portion of the capital reserve is continuously invested in a risky asset whose price follows a geometric Brownian motion, while the complementary part is placed in a bank account with a constant rate of return. The quantity of interest is the ruin probability on the infinite time horizon as a function of the initial capital. In the present note, we assume only the continuity of the distribution of claims together with a standard moment restriction called “light tails.” Our main contribution is that we reveal, under such “minimalistic” hypotheses, that the ruin probability is smooth and satisfies a second-order integro-differential equation in the classical sense. We obtain the exact asymptotics for large values of the initial capital with “computable” constants and present results of numerical experiments. In contrast with other methods used in the theory, we rely upon only standard mathematics, allowing implementation in lecture courses for master’s students. Full article

Natural waves are widely used in seismology and seismic exploration as tools for nondestructive testing of the surface layer. The study examines longitudinal and transverse vibrations of a polymer pipeline transporting petroleum products, which is modeled as a viscoelastic cylindrical shell filled with a viscous fluid. This work examines the longitudinal–transverse vibrations of a viscoelastic cylindrical shell filled with a viscous fluid, considering the viscous properties of both the fluid and the cylindrical shell during longitudinal–transverse oscillations. The differential equations governing the longitudinal–transverse vibrations of a cylindrical shell in contact with a viscous fluid are derived based on thin-shell equations satisfying the Kirchhoff–Love hypotheses, while the motion of the viscous fluid obeys the Navier–Stokes equations. The viscoelastic properties of the shell are described using the Boltzmann–Volterra hereditary integral. After applying the “freezing method” to the system of integro-differential equations, we obtain ordinary differential equations with complex coefficients, which are subsequently solved by the method of separation of variables and Godunov’s orthogonal sweep combined with Müller’s and Gauss’s methods in complex arithmetic. It is established that for small viscosity, the frequencies of both modes are close to each other in the low-frequency region, while at high frequencies, the phase velocity of the first mode tends toward the velocity of the dry shell. Full article

We study the Cauchy problem for a loaded fractional integro-differential equation with a time-dependent diffusion coefficient. By reducing the problem to an equivalent Volterra integral equation of the second kind, we derive explicit analytical representations of solutions under appropriate regularity assumptions. The construction of the associated resolvent kernel allows us to establish existence and uniqueness results and to investigate the role of the fractional order and the loading term in the solution structure. Two illustrative examples are presented to demonstrate the applicability of the proposed approach. Full article

Variant emergence continues to pose a threat to public health, despite the widespread use of vaccination. To quantify how vaccine strain compositions shape evolutionary and epidemiological outcomes, we extend a previous genotype-structured transmission model with vaccination and study the impact of different vaccination formulations on variant emergence. It consists of a set of partial differential equations coupled with an integro-differential one. We begin by showing that the model reproduces variant emergence followed by a period of co-circulation in the absence of vaccination. Then, we introduce vaccination and show important trade-offs shaped by the breadth and cross-protection of vaccine-induced immunity. In our simulations, narrow-spectrum vaccines substantially reduce the immediate infection burden but inadvertently promote the emergence of non-targeted variants. After that, we study the effects of more complex shapes such as triangular and M-shaped configurations. We show that M-triangular distributions outperform triangular ones by limiting secondary variant expansion for vaccines with narrow cross-protection. In contrast, triangular compositions are more protective when considering broader cross-protection. We also show that targeting the genetic area between co-circulating variants is more beneficial than focusing on specific variants when using vaccines with a broad cross-protection. Together, these results highlight how vaccine breadth and antigenic targeting influence both epidemic size and the trajectory of variant emergence, offering quantitative guidance for monovalent and multivalent vaccine design. Full article

The neutron survival probability (and related quantities including probabilities of extinction and initiation) is a central element of the broader stochastic theory of neutron populations and finds application in fields including reactor start-up, analysis of reactor power bursts and criticality accidents, and safeguards. In a full neutron transport formulation, the equation governing the single-neutron survival probability is a backward or adjoint-like integro-partial differential equation with the added complexity of being highly nonlinear. Analogous formulations of this equation exist in the context of many approximate theories of neutron transport, with the point kinetics formulation having received significant theoretical attention since the 1940s. This work continues this tradition by providing a novel analysis of the single-neutron survival probability equation using the tools of boundary layer theory. The analysis reveals that the “fully dynamic” solution of the single-neutron survival probability equation—and some key probability distributions derived from it—may be cast as a singular perturbation around the underlying quasi-static single-neutron probability of initiation. In this perturbation solution, the expansion parameter is the ratio of the neutron generation time to a macroscopic time scale characterizing the overall system evolution; this interpretation illuminates some of the fundamental structural aspects of neutron survival phenomena. Full article

Boundary value problems for systems of integrodifferential equations appear in many branches of science and engineering. Accuracy in modeling complex processes requires the specification of nonlocal boundary conditions, including multipoint and integral conditions. These kinds of problems are even harder to solve. In this paper, we present solvability criteria and a direct operator method for constructing the exact solution to systems of linear ordinary integrodifferential equations with general nonlocal boundary conditions. A symbolic algorithm is also proposed. Several examples are solved to demonstrate the effectiveness of the method. The results obtained are equally valid for nonlocal boundary value problems for systems of ordinary differential, loaded differential, and loaded integrodifferential equations. Full article

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into past, present, and future intervals and includes nonlinear mixed integral operators. Using Banach’s contraction mapping principle and Schauder’s fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions within the space of continuous functions. The study is then extended to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Ulam–Hyers–Rassias stability is also analyzed under appropriate conditions. To demonstrate the practical applicability of the theoretical framework, explicit examples with specific nonlinear functions and integral kernels are provided. Furthermore, detailed numerical simulations are conducted using MATLAB-based specialized algorithms, illustrating solution convergence and behavior in both finite-dimensional and Banach space contexts. Full article

This paper presents an analytical model for the hereditary vibrations of a coupled circular plate system interconnected by viscoelastic creep layers. The system is represented as a discrete-continuous chain of thin, isotropic plates with time-dependent material properties. Based on the theory of hereditary viscoelasticity and D’Alembert’s principle, a system of partial integro-differential equations is derived and reduced to ordinary integro-differential equations using Bernoulli’s method and Laplace transforms. Analytical expressions for natural frequencies, mode shapes, and time-dependent response functions are obtained. The results reveal the emergence of multi-frequency vibration regimes, with modal families remaining temporally uncoupled. This enables the identification of resonance conditions and dynamic absorption phenomena. The fractional parameter serves as a tunable damping factor: lower values result in prolonged oscillations, while higher values cause rapid decay. Increasing the kinetic stiffness of the coupling layers raises vibration frequencies and enhances sensitivity to hereditary effects. This interplay provides deeper insight into dynamic behavior control. The model is applicable to multilayered structures in aerospace, civil engineering, and microsystems, where long-term loading and time-dependent material behavior are critical. The proposed framework offers a powerful tool for designing systems with tailored dynamic responses and improved stability. Full article

Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed for the iterative solution of nonlinear, boundary-value problems of ordinary differential equations that makes use of modified functionals and two Lagrange multipliers, is nothing but an iterative Green’s function formulation that does not require Lagrange multipliers at all. It is also shown that the two Lagrange multipliers of GVIM are associated with the left and right Green’s functions. The convergence of iterative methods based on both the Green function and the method of variation of parameters is proven for nonlinear functions that depend on the dependent variable and is illustrated by means of two examples. Several new iterative integro-differential formulations based on Green’s functions that use a multiplicative function for convergence acceleration are also presented. Full article

In this article, we investigate a nonlinear $\psi$-Hilfer fractional order Volterra integro-delay differential equation ($\psi$-Hilfer FRVIDDE) and a nonlinear $\psi$-Hilfer fractional Volterra delay integral equation ($\psi$-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the $\psi$-Hilfer FRVIDDE and $\psi$-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article