Integro-differential Equations Research Articles

A Numerical Scheme of a Fractional Coupled System of Volterra Integro-Differential Equations with the Caputo Fabrizio Fractional Derivative

In this paper, we have developed a new computational scheme for solving coupled systems of fractional order Volterra-type integro-differential equation (FVIDE). We construct new operational matrices, which serve as building blocks for converting the FVIDE into a Sylvester-type algebraic structure that is more easily solvable. The fractional derivatives and their inverses (integrals) are considered in the Caputo-Fabrizio sense. This computational scheme is based on Legendre polynomials. The assessment of the proposed method’s convergence involves utilizing appropriate error norms to measure the level of convergence. The algorithm is designed in such a way that it can be simulated using any computational package; we have used Matlab for simulating the proposed scheme. Graphical visualization and tabulation of data are performed to confirm convergence and to analyze errors.

New attitude on sequential Ψ-Caputo differential equations via concept of measures of noncompactness

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.

Explicit exponential Runge–Kutta methods for semilinear time-fractional integro-differential equations

In this work, we consider and analyze explicit exponential Runge–Kutta methods for solving semilinear time-fractional integro-differential equation, which involves two nonlocal terms in time. Firstly, the temporal Runge–Kutta discretizations follow the idea of exponential integrators. Subsequently, we utilize the spectral Galerkin method to introduce a fully discrete scheme. Then, we mainly focus on discussing the one-stage and two-stage methods for solving the proposed semilinear problem. Based on special abstract settings, we perform the convergence analysis for the proposed two different stage methods. In this process, we heavily use estimates about the operator family {S̃(t)}, and in combination with Lipschitz continuous condition. Finally, some numerical experiments confirm theoretical results. Meanwhile, applying this scheme to the related linear problem yields high-order convergence, highlighting the advantages of explicit exponential Runge–Kutta methods.

The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations

The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations

Stochastic fractional integrodifferential equations with jumps: application to an averaging principle

In this work we deal with a stochastic fractional integrodifferential equation 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 establish an averaging principle in the sense of Khasminskii approach for a class of this equation with non lipschitz coefficients and weak averaging conditions.

Approximate Solution of System of Multi-term Fractional Mixed Volterra–Fredholm Integro-Differential Equations by BPFs

Approximate Solution of System of Multi-term Fractional Mixed Volterra–Fredholm Integro-Differential Equations by BPFs

Bernstein computational algorithm for integro-differential equations

In this study, we introduce a computational algorithm for solving Integro-Differential Equations (IDEs) using Bernstein polynomials as basis functions. The algorithm approximates the solution by expressing it in terms of Bernstein polynomials and substituting this assumed solution into the IDE. Collocating the resulting equation at evenly spaced points yields a system of linear algebraic equations, which is solved via matrix inversion to find the Bernstein coefficients. These coefficients are then used to construct the approximate solution. Numerical examples demonstrate the method's accuracy and efficiency, highlighting its advantages in reducing computational effort.

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 < 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 < 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

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.