Nonlinear Integro-differential Equations Research Articles

Stability of complement value problems for p-Lévy operators

We set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear nonlocal integro-differential p-Lévy operators. A prototypical example of integro-differential p-Lévy operators is the well-known fractional p-Laplace operator. Our main focus is on nonlinear integro-differential equations in the presence of Dirichlet, Neumann and Robin conditions and we show well-posedness results. Several results are new even for the fractional p-Laplace operator but we develop the approach for general translation-invariant nonlocal operators. We also bridge the gap from nonlocal to local, by showing that solutions to the local Dirichlet and Neumann boundary value problems associated with p-Laplacian are strong limits of the nonlocal ones.

Stability and boundedness of solutions to a certain third-order neutral integro-differential equation with constant delay

In this work, we examine a third-order nonlinear neutral integro-differential equation with constant delay. By building a Lyapunov functional, we obtain some sufficient criteria that ensure the asymptotic stability and boundedness of solutions for an analyzed equation. We present two examples to demonstrate the applicability of our conclusions, which extend and improve several well-known results in the literature.

Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations

Abstract 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 $V(x)\sim |x|^m$, where $m>0$, and employ the optimal fluctuation method (OFM) to determine the large-$|x|$ tails of the steady-state probability distribution $\mathcal{P}(x)$ 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 $\ln \mathcal{P}(x)$ up to a dimensionless factor $\alpha(H,m)$, where $0<H<1$ is the Hurst exponent. We determine $\alpha(H,m)$ analytically (i) in the limits of $H\to 0$ and $H\to 1$, 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 $\mathcal{P}(x)$ and autocovariance. The form of the tails of $\mathcal{P}(x)$ yields exact conditions, in terms of $H$ and $m$, for the particle confinement in the potential.
For $H\neq 1/2$, 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 $\alpha(H,m)$ 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 $H<1/2$.


Numerical solutions of stochastic delay integro-differential equations by block pulse functions

This paper presents an efficient numerical method for solving nonlinear stochastic delay integro-differential equations based on block pulse functions. Firstly, the equation is transformed into an algebraic system by the integral delay operator matrixes of block pulse functions. Then, error analysis is conducted on the method. Finally, some numerical examples are provided to validate the method. This work provides numerical solutions for the stochastic delay integro-differential equations by global approximation method. This method has the advantages of simple calculation and higher error accuracy.

Existence and uniqueness result for reaction-diffusion model of diffusive population dynamics

The present work investigates a convolution type nonlinear integrodifferential equation with diffusion. This type of equations represents not only pure mathematical interest, but also is widely used in various applications, especially in wide range of problems on population dynamics arising in biology. The existence of a parametric family of travelling wave solutions as well as the uniqueness of the solution in certain class of bounded continuous functions on R \mathbb {R} is proved. The study investigates also some important properties as well as asymptotic behaviour of constructed solutions. These results are then used to derive a new uniform estimate for the deviation between successive iterations, which provides us with a strong tool to control the number of iterations on our way of computing the desired numerical approximation of the exact solution. Finally, we apply our theoretical results to two well-known population problems modelled by delayed reaction-diffusion equation: Diffusion model for spatial-temporal spread of epidemics and stage structured population model.

A finite volume approximations for one nonlinear and nonlocal integrodifferential equations

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the L∞(L2(Ω)) and L2(H01(Ω)) norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

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.

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.

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.

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.

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar–Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar–Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilizing ATDM.

Euler wavelets method for optimal control problems of fractional integro-differential equations

This research introduces an efficient direct method for finding the numerical solution to optimal control problems involving linear and nonlinear fractional integro-differential equations using orthonormal Euler wavelets. Using orthonormal Euler wavelets, fractional integral, and product operational matrices have been obtained. By using these fractional operational matrices and collocation points, optimal control problems of fractional Volterra integro-differential equations have been reduced to a system of linear or non-linear equations. We establish the convergence analysis and error bound of the proposed method. Orthonormal Euler wavelets have been utilized to solve numerical examples and determine the approximate value of the cost function by approximating state and control functions. This has been done in order to verify the effectiveness of the proposed numerical technique.

Existence analysis on multi-derivative nonlinear fractional neutral impulsive integro-differential equations

This article utilizes the Atangana–Baleanu (AB) fractional derivative to examine the behavior of multi-derivative fractional neutral impulsive integro-differential equations under non-local conditions. To demonstrate the existence, uniqueness, and controllability of the solutions, we utilize fixed point theory as our primary analytical tool. Furthermore, we include a detailed example to illustrate and validate the theoretical results obtained from our study.

Controllability of impulsive nonlinear [formula omitted]-Hilfer fractional integro-differential equations

Sufficient conditions for controllability of impulsive nonlinear integro-differential equations with ψ-Hilfer fractional derivative are established. The result are obtained by using fractional calculus and Schaefer’s fixed point theorem. Finally, a numerical examples are then provided to demonstrate the outcomes.

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.

E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations

Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems.

Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

We introduce a novel, time-efficient adaptive Runge-Kutta computational scheme tailored for systematically solving linear and nonlinear Volterra-type Integro-Differential Equations (VTIDEs). This scheme is particularly effective for equations featuring a specific class of convolution memory, K(t,s)=aκ(t−s) within convolution integrals, where a,κ∈R. Such equations frequently arise in fields like viscoelasticity, population dynamics, epidemiology, control systems, financial mathematics, and neuroscience, where current states are influenced by historical data, especially in dynamical systems within physics. Our proposed scheme demonstrates remarkable computational efficiency, achieving a cost-effective computation time of O(Nt) for Nt iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of O(Ntlog⁡Nt) and O(Nt(log⁡Nt))2. The key feature of our scheme lies in its ability to elegantly and efficiently handle convoluted integrals of varying complexity such as those including nonlinearities and derivatives. Specifically, our approach eliminates the need for backstage integration, allowing the main Runge-Kutta scheme to operate efficiently with only a few data points at each time step. This not only saves time but also prevents potential data storage issues while rendering numerical implementation straightforward and tractable. Moreover, this article introduces flexible discretizations for integration and differentiation, accommodating unequally spaced data points. We implement a fast RK45-Fehlberg algorithm (TE-RK45-Fehlberg), which we use to solve three examples of nonlinear VTIDE presented in this study. We further demonstrate the effectiveness of our approach by using an example, where we employ a novel parabolic step size generator function to efficiently generate step sizes for a computationally fast RK4 scheme (Flex-RK4). For all examples tackled using TE-RK45-Fehlberg and Flex-RK4, we generate plots that compare numerical data with exact solutions, providing insight into the accuracy and reliability of our approach. We also show how the scheme provides straightforward ways of transforming it into implicit forms and obtaining stability curves.

Finite Difference with Quintic B-Splines for Solving a system of nonlinear Volterra Integro-Differential Equations of integer order

This study presents a novel numerical approach for solving systems of nonlinear Volterra Integro-Differential Equations (VIDEs) of integer order by integrating the Finite Difference Method (FDM) with Quintic B-Splines for. The proposed method aims to address the challenges posed by the nonlinearity and integral terms inherent in VIDEs while providing enhanced accuracy and stability in the numerical solution. An error estimate of the approximate solution is proved. Finally, Some example are presented to illustrate the simplicity and the effectiveness of the propose method