Nonlinear Integro-differential Equations Research Articles

Three finite difference schemes for generalized nonlinear integro-differential equations with tempered singular kernel

This paper presents and investigates three different finite difference schemes for solving generalized nonlinear integro-differential equations with tempered singular kernel. For the temporal derivative, the backward Euler (BE), Crank–Nicolson (CN), and second-order backward differentiation formula (BDF2) schemes are employed. The corresponding convolution quadrature rules are utilized for the integral term involving tempered fractional kernel. In order to ensure second-order accuracy in the spatial direction, the standard central difference formula is applied, leading to fully discrete difference schemes. The convergence and stability of these three schemes are proved, by using energy method and cut-off method. Furthermore, we apply a fixed point iterative algorithm to calculate the proposed schemes, and the numerical results are consistent with the theoretical analysis.

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

Existence and Stability Results of Nonlinear Random Impulsive Integro-Differential Evolution Equations with Time-Varying Delays

This study examines the existence, uniqueness, and stability of nonlinear random impulsive integrodifferential equations with time-varying delays under sufficient conditions. Our study is based on the Leray-Schauder alternative fixed point theorem, Pachpatte’s inequality, and the Banach contraction principle. Besides, we generalize, extend, and develop some results for Kasinathan et al.

A Unified Design of Energy Stable Schemes with Variable Steps for Fractional Gradient Flows and Nonlinear Integro-differential Equations

A Unified Design of Energy Stable Schemes with Variable Steps for Fractional Gradient Flows and Nonlinear Integro-differential Equations

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.

Global Existence and Uniqueness of Solutions of Integral Equations with Multiple Variable Delays and Integro Differential Equations: Progressive Contractions

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature.

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

<abstract><p>The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.</p></abstract>

Нелинейные колебания токонесущей струны в магнитном поле

The nonlinear properties of a string must be taken into account when creating string systems that are used in transducers and musical instruments. The forced oscillations of an electrically conductive string in a magnetic field are described by a nonlinear integro-differential equation. The amplitude-frequency characteristics of nonlinear resonances of the first and second orders are obtained. The damping factor of the first-mode oscillations is determined from the optimal coincidence of the experimental and theoretical amplitude-frequency characteristic graphs. The experimental amplitude-frequency characteristics for the first-order resonance is obtained using a laboratory setup with a copper string 0.4 mm in diameter and 0.57 m long, having a certain initial tension. The cumulative damping factor of the first mode is obtained, which takes into account all the dissipative factors of a real string system. Low-frequency thermoparametric oscillations of the second mode are observed in experiments with a string. It follows from the analysis of the Mathieu equation that parametric resonance arises even at very low amplitudes of the thermal tension of a string, and it must be taken into account in the frequency analysis of oscillations of string systems. The authors are grateful to Tomilin V.A. for his help in performing the calculations and graphic work. The authors declare that they have no conflict of interests.

Investigation on integro-differential equations with fractional boundary conditions by Atangana-Baleanu-Caputo derivative.

We establish, the existence and uniqueness of solutions to a class of Atangana-Baleanu (AB) derivative-based nonlinear fractional integro-differential equations with fractional boundary conditions by using special type of operators over general Banach and Hilbert spaces with bounded approximation numbers. The Leray-Schauder alternative theorem guarantees the existence solution and the Banach contraction principle is used to derive uniqueness solutions. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining the numerical approximation to the solution. To illustrate our analytical and numerical findings, an example is provided and concluded in the final section.

A numerical comparative study for the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations on layer-adapted meshes

This article deals with the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes.

Ulam-Hyers and generalized Ulam-Hyers stability of fractional functional integro-differential equations

Unique solvability conditions for general fractional functional integro-differential equations with nonlocal initial conditions are established using fixed-point theory. Ulam-Hyers and generalized Ulam-Hyers stability conditions for a solution of these problems are formulated. An example of the nonlinear fractional functional integro-differential equation related to wheel shimmy models is considered as well.

Positivity and Periodicity in Nonlinear Neutral Mixed Type Levin-Nohel Integro-Differential Equations

In this work, we give sufficient conditions for the existence of periodic and positive periodic solutions for a nonlinear neutral mixed type Levin-Nohel integro-differential equation with variable delays by using Krasnoselskii’s fixed point theorem. Also, we obtain the existence of a unique periodic solution of the posed equation by means of the contraction mapping principle. As an application, we give an example to illustrate our results. Previous results are extended and generalized.

A Qualitative Study on Fractional Logistic Integro-differential Equations in an Arbitrary Time Scale

This manuscript deals with the investigation related to uniqueness and existence of solution of fractional order nonlinear pantograph integro-differential equation in arbitrary time scale. The fractional derivatives are defined in Riemann-Liouville sense, the primary tools are taken as Banach contraction principle and Schauder’s fixed point theory to establish the theoretical outcomes. Finally, we give examples to show the efficiency of our results.

Coupled Nonlocal Boundary Value Problems for Fractional Integro-differential Langevin System via Variable Coefficient

In this paper, we aim to study a new coupled system of nonlinear fractional integro-differential Langevin equations with coupled multipoint boundary conditions. The existence and uniqueness of solution are investigated by using the Banach’s and Krasnoselskii’s fixed point theorems. The Ulam-Hyers stability of the mentioned equation is provided by applying the classical technique of functional analysis. Two examples are presented to verify our analysis.

Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three

Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three

On Hadamard-Caputo Implicit Fractional Integro-Differential Equations With Boundary Fractional Conditions

The purpose of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional implicit integro-differential equations of Hadamard-Caputo type with fractional boundary conditions. The reasoning is inspired by diverse classical fixed point theory, such as the Schauder and Banach fixed point theorems. The theoretical findings are illustrated through an example.

Исследование динамики свободной поверхности, ограничивающей конечную массу самогравитирующего газа

The work researches the evolution of the free surface which limits the polytropic, isentropic and self-gravitating ideal gas moving in vacuum. Gas currents are described with the mathematical model of gas dynamics, constructed due to the system of nonlinear integro-differential equations, expressed in Euler coordinate system. The transformation of the given system to the Lagrangian coordinates lets reduce it to the equivalent system consisting of integral Voltair’s equations and the equation of continuity in the Lagrangian form, and also it lets break down the unknown boundaries. Free boundary is determined as a great number of points of the system of integral equations solution achieved when depicting the boundary points of the initial area within the point of the moving area. For the system there is the proved theorem of the existence and uniqueness of the solution of the Cauchy problem in the space of the infinitely differentiated functions. Methods of qualitative research of the system of gas dynamics were applied in the work for the research of the evolution of the free surface.

Properties of the solution of some multipoint problems for integro-differential equation of Barbashin type

The nonlocal boundary value problems for the nonlinear integro-differential Barbashin equation with a partial derivative of the second order is studied. The solution of the problem is understood in the classical sense. The problem reduces to the equivalent Volterra integral equation with partial integrals. The conditions for the existence and uniqueness of the solution of this equation are obtained.

Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

The aim of this paper is to investigate some types of stability such as generalized Hyers-Ulam- Rassias stability(G-H-U-R-stabile) and the relation with Hyers-Ulam(H-U-stabile) stable and Hyers-Ulam-Rassias stable (H-U-R-stabile) and generalized Hyers-Ulam stable (G-H-U- stable) to obtain which one guarantee to satisfy stability of equations included a nonlinear function some of them contains a delay time of solution and the other contain a vector of different order of derivatives for the solution to n-time and vector of fractional order of integrals with different fractional orders and that was the for using a claculse of fractional calculus to satisfies the issue of this techniques. Moreover, the nonlocal initial values for the proposal equation of nonlinear impulsive higher order differential – fractional integral delay time equations which are adding more interesting for nonlinear analytic object of nonlinear higher order integro – fractional order impulsive classes, and the impulsive difference of the equation has some necessary conditions to prove the results of solution to be stable with certain type has related with other types. The necessary and sufficient conditions which assumed on this nonlinear higher order integro-differential impulsive equation have been achieved the stability with interesting certain estimates obtain through the proving technique. Also the uniqueness of solution has been studied with same conditions was presented for stability and used for that issue a contraction fixed point theorem.

Learning nonlinear integral operators via recurrent neural networks and its application in solving integro-differential equations

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the need to perform a numerical integration in each numerical time evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can be reduced to O(nT) from O(nT2) if a nT-step trajectory is to be computed. We illustrate the efficiency and robustness of this LSTM-RNN-based numerical IDE solver with a model problem. Additionally, we highlight the generalizability of the learned integral operator by applying it to IDEs driven by different external forces. As a practical application, we show how this methodology can effectively solve the Dyson’s equation for quantum many-body systems.