Integro-differential Equations Research Articles

Two-step crystallization in a supersaturated solution with application to protein crystal growth: Theory and analytical solutions

A theory of two-step nucleation and evolution of crystals with allowance for their diffusion in the space of particle radii is developed. The theory is based on a two-step growth law of crystals appearing in dense liquid precursors, initially evolving within them with a diffusion-limited growth law, and then leaving them and evolving with another growth law. This two-step crystal growth law influences the integrodifferential system of kinetic and balance equations for the crystal-size distribution function and liquid supersaturation. The system of integrodifferential equations for the evolution of polydisperse ensemble of crystals is solved using the integral Laplace transform and saddle-point methods for the Weber–Volmer–Frenkel–Zel’dovich and Meirs kinetic mechanisms. A complete analytical solution to this non-linear system has been constructed in a parametric form. Namely, the particle-size distribution function, liquid supersaturation and time have been found as the functions of decision variable — the maximum size of crystals. The theory is in agreement with the experimental data on the growth of beta-lactoglobulin and insulin crystals.

Overcompensation of transient and permanent death rate increases in age-structured models with cannibalistic interactions

There has been renewed interest in understanding the mathematical structure of ecological population models that lead to overcompensation, the process by which a population recovers to a higher level after suffering a permanent increase in predation or harvesting. Here, we apply a recently formulated kinetic population theory to formally construct an age-structured single-species population model that includes a cannibalistic interaction in which older individuals prey on younger ones. Depending on the age-dependent structure of this interaction, our model can exhibit transient or steady-state overcompensation of an increased death rate as well as oscillations of the total population, both phenomena that have been observed in ecological systems. Analytic and numerical analysis of our model reveals sufficient conditions for overcompensation and oscillations. We also show how our structured population partial integrodifferential equation (PIDE) model can be reduced to coupled ODE models representing piecewise constant parameter domains, providing additional mathematical insight into the emergence of overcompensation.

Superconvergent method for weakly singular Fredholm-Hammerstein integral equations with non-smooth solutions and its application

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.

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

This study investigates the initial value problem of high-order variable-order φ-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions.

Applications of The Double General Rangaig Integral Transform in Integro-Differential Equations

Applications of The Double General Rangaig Integral Transform in Integro-Differential Equations

Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation

Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation

A computational method for singularly perturbed reaction–diffusion type system of integro-differential equations with discontinuous source term

A computational method for singularly perturbed reaction–diffusion type system of integro-differential equations with discontinuous source term

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

A new fractional derivative extending classical concepts: Theory and applications

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle’s theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

Vector valued piecewise continuous almost automorphic functions and some consequences

In the present work, for X a Banach space, the notion of piecewise continuous Z-almost automorphic functions with values in finite dimensional spaces is extended to piecewise continuous Z-almost automorphic functions with values in X. Several properties of this class of functions are provided, in particular it is shown that if X is a Banach algebra, then this class of functions constitute also a Banach algebra; furthermore, using the theory of Z-almost automorphic functions, a new characterization of compact almost automorphic functions is given. As consequences, with the help of Z-almost automorphic functions, it is presented a simple proof of the characterization of almost automorphic sequences by compact almost automorphic functions; the method permits us to give explicit examples of compact almost automorphic functions which are not almost periodic. Also, using the theory developed here, it is shown that almost automorphic solutions of differential equations with piecewise constant argument are in fact compact almost automorphic. Finally, it is proved that the classical solution of the 1D heat equation with continuous Z-almost automorphic source is also continuous Z-almost automorphic; furthermore, we comment applications to the existence and uniqueness of the asymptotically continuous Z-almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions.

Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method

Abstract In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

Dynamics of chains of a large number of fully coupled oscillators, as well as oscillators with one-way and two-way delay couplings

This articles considers chains consisting of identical nonlinear equations of second order with couplings in linear elements. It is assumed that there is a large delay in the coupling elements. Moreover, we assume that the chain possesses a large number of elements. Therefore, instead of the original system with many elements, we can study a second order nonlinear integro-differential equation with periodic boundary conditions. The main study is focused on the local dynamics of chains with one-sided, two-sided couplings, as well as fully coupled chains. Also, we study the stability of equilibrium states and identify the critical cases. The condition of a sufficiently large delay helps to determine the parameters for the implementation of critical cases explicitly. Our methodology is based on the infinite-dimensional normalisation method proposed by the author, namely the method of quasinormal forms. We construct quasinormal forms for the chains under consideration that determine the asymptotics of the leading terms of the asymptotic expansions of the solutions to the original system.

An elastic-constant stress cohesive zone model for an orthotropic piezoelectric actuator adhesively bonded to a substrate

One of the common failure mechanisms observed in electromechanical systems utilizing actuators, sensors, or flexible electronics is the occurrence of interfacial debonding. In this paper, an elastic-constant stress cohesive zone model is constructed to analyze the interface behavior of piezoelectric actuators used in electromechanical systems when imperfectly bonded to a host medium. The analytical formulation treats the piezoelectric actuator as an orthotropic thin film, employing the membrane assumption. It is assumed that the cohesive zone extends from a point, initially unknown, to the edge of the film. The governing integro-differential equation is solved for various bonding scenarios, including perfect bonding with cohesive zones at the edges, an elastic case with a delaminated zone at the center, and an elastic case with both a delaminated zone at the center and cohesive zones at the edges of the film. Detailed examinations are conducted to explore the effects of stiffness ratio parameters and critical actuating voltage on the stresses within the film and substrate as well as the cohesive zone length. The results indicate that the cohesive zone length covers 45% of the film length when the actuating voltage has a fivefold increase over the critical threshold. For the imperfect bonding, the surface in-plane stress of the substrate exhibits a tensile peak near the boundary of the cohesive zone in addition to compressive singularities at the edges of the film. The shear stress singularity factor has an asymptotic behavior as the actuating voltage goes beyond the critical threshold. The critical actuating voltage reduces by a factor of 5% only if the delamination length does not exceed 70% of the film length.

Approximation Solution for Initial Value Problem of Singularly Perturbed Integro-Differential Equation

Adomian decomposition method (ADM) is used to approximately solve the initial value problem of the singularly perturbed Volterra and Fredholm differential equation. With this method, the desired accurate results are obtained in only a few terms. The approach is simple and effective. An example application is made to demonstrate the effectiveness of ADM. The result obtained is compared with the exact solution. Convergence analysis of the method was performed.

Weak convergence of the split-step backward Euler method for stochastic delay integro-differential equations

In this paper, our primary objective is to discuss the weak convergence of the split-step backward Euler (SSBE) method, renowned for its exceptional stability when used to solve a class of stochastic delay integro-differential equations (SDIDEs) characterized by global Lipschitz coefficients. Traditional weak convergence analysis techniques are not directly applicable to SDIDEs due to the absence of a Kolmogorov equation. To bridge this gap, we employ modified equations to establish an equivalence between the SSBE method used for solving the original SDIDEs and the Euler–Maruyama method applied to modified equations. By demonstrating first-order strong convergence between the solutions of SDIDEs and the modified equations, we establish the first-order weak convergence of the SSBE method for SDIDEs. Finally, we present numerical simulations to validate our theoretical findings.

Exploring New Traveling Wave Solutions to the Nonlinear Integro-Partial Differential Equations with Stability and Modulation Instability in Industrial Engineering

In this research article, we demonstrate the generalized expansion method to investigate nonlinear integro-partial differential equations via an efficient mathematical method for generating abundant exact solutions for two types of applicable nonlinear models. Moreover, stability analysis and modulation instability are also studied for two types of nonlinear models in this present investigation. These analyses have several applications including analyzing control systems, engineering, biomedical engineering, neural networks, optical fiber communications, signal processing, nonlinear imaging techniques, oceanography, and astrophysical phenomena. To study nonlinear PDEs analytically, exact traveling wave solutions are in high demand. In this paper, the (1 + 1)-dimensional integro-differential Ito equation (IDIE), relevant in various branches of physics, statistical mechanics, condensed matter physics, quantum field theory, the dynamics of complex systems, etc., and also the (2 + 1)-dimensional integro-differential Sawda–Kotera equation (IDSKE), providing insights into the several physical fields, especially quantum gravity field theory, conformal field theory, neural networks, signal processing, control systems, etc., are investigated to obtain a variety of wave solutions in modern physics by using the mentioned method. Since abundant exact wave solutions give us vast information about the physical phenomena of the mentioned models, our analysis aims to determine various types of traveling wave solutions via a different integrable ordinary differential equation. Furthermore, the characteristics of the obtained new exact solutions have been illustrated by some figures. The method used here is candid, convenient, proficient, and overwhelming compared to other existing computational techniques in solving other current world physical problems. This article provides an exemplary practice of finding new types of analytical equations.

Analytical and Computational Results for Neutral Impulsive Fractional System on Time Scales

This paper delves into the existence and uniqueness of neutral fractional integro-differential impulsive dynamic equations across various time scales, enriched by nonlocal initial conditions using the Caputo-Nabla derivative. By leveraging the refined fixed point theorem, the study provides a robust framework for establishing existence. The theoretical findings are elegantly illustrated through detailed graphical representations, enhancing the comprehension and appeal of the results.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.