System Of Nonlinear Integro-differential Equations Research Articles

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.

Well-posedness and exponential stability of nonlinear Maxwell equations for dispersive materials with interface

In this paper we consider an abstract Cauchy problem for a Maxwell system modeling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.

APPROXIMATE AND STABILITY SOLUTION FOR NON-LINEAR SYSTEM OF INTEGRODIFFERENTIAL EQUATIONS OF VOLTERRA TYPE WITH BOUNDARY CONDITIONS

In this paper, we investigate the approximation and stability solutions of non-linear systems of integro-differential equations of Volterra type with boundary conditions, by using the numerical-analytic method which were introduced by Samoilenko. The study of such integro-differential equations leads to extend the results obtained by Butris for changing the system of non-linear integro- differential equations of Volterra type to the system of non-linear integro-differential equations of the Volterra type with boundary conditions. Theorems on a solutions are established under some necessary and sufficient conditions on compact spaces.

Absolute Stability and Synchronization in Second-Order Neural Fields with Conduction Delays

We consider a class of neural field models represented by a second-order nonlinear system of integro-differential equations with space-dependent delays. Such a system models interaction of populations of neurons, each with a continuum description. We justify the existence and uniqueness of solution for the system in a suitable function space. Global existence and boundedness of solutions for the system are confirmed. Two methodologies, the comparison argument and sequential contracting, are developed to establish sufficient conditions for absolute stability and synchronization among different populations of the system. Finally, we present some numerical examples to demonstrate the theoretical results.

New algorithm to solve nonlinear functional equations applying linearization then double discretization scheme (L.D.D)

Solving functional nonlinear equations leads to the question: which to begin with, linearization or discretization? Recent papers confirmed that linearizing then discretizing (L.D) is better. In this paper, we develop a new numerical scheme that begins with the linearization phase and then, a double discretizations process (L.D.D). This new method gives a different theoretical framework where the convergence process is satisfied under some hypotheses. Many examples are offered to show the effectiveness of our new scheme. Starting with a comparison of the obtained numerical results with the results of a recently published research, moreover, applications to the system of nonlinear integro-differential equations. Obtained numerical results show that our (L.D.D) method is more efficient for solving nonlinear functional equations.

Forced vibrations of hereditary deformable shell elements of aircraft structures in gas flow under influence of atmospheric turbulence

The problem of vibrations of thin, hereditarily deformable shell elements moving in a gas under the action of atmospheric turbulence is considered. The work aims to study the flutter phenomenon of aircraft elements in a gas flow under the action of loads caused by atmospheric turbulence. Assuming that the relationship between stresses and strains for the shell material is linear-hereditary, a thin shell is used, which obeys the Kirchhoff-Love hypothesis. The aerodynamic force is written according to the linearized piston theory. A system of nonlinear integro-differential equations in partial derivatives is obtained to describe nonlinear oscillations of a thin isotropic viscoelastic shell. The system of nonlinear integro-differential equations is solved numerically by the method proposed by F. Badalov, which is based on the Bubnov-Galerkin methods, finite differences, and power series. When using an exponential kernel, the flutter rate increases to approximately 1.5%. Therefore, when using an exponential kernel, the flutter velocity of a viscoelastic shell practically coincides with the critical flutter velocity for ideally elastic plates. It was also found that the critical flutter velocity increases with an increase in the number of pinched sides of the shell.

Nonlinear vibrations of earth structures

This article offers a detailed analysis of the current state of plane structure dynamics, addressing the complexities posed by non-homogeneous materials exhibiting nonlinear elastic and viscoelastic properties during real-life operations. The study proposes a comprehensive mathematical model and algorithm to investigate the dynamic behavior of such structures, employing the non-linear hereditary Boltzmann-Volterra theory to describe viscoelastic material properties accurately. Nonlinear oscillatory systems are analyzed using Lagrange equations based on the d’Alembert principle. The problem is approached through a multi-step process. Initially, the linear elastic problem of the structure’s natural oscillations is solved to determine its natural frequencies and modes of oscillations. Subsequently, these eigenmodes are employed as coordinate functions to address forced nonlinear oscillations in viscoelastic non-homogeneous systems. The complexity of the problem necessitates solving a Cauchy problem comprising a system of nonlinear integrodifferential equations. To illustrate the methodology, the study examines the Gissarak earth dam, considering real operational conditions and nonlinear, viscoelastic material properties near resonant modes of vibrations. Utilizing numerical methods, the dynamic behavior of the structure is analyzed, assessing displacements and stress components at different time points under non-stationary kinematic action. Stress concentration regions within the structure are identified for resonant vibrations, allowing the evaluation of its strength. Furthermore, the impact of nonlinear elasticity and viscoelasticity on the structural dynamics is quantified. This research provides valuable insights into the behavior of plane non-homogeneous structures, considering real-world scenarios and material complexities, ultimately contributing to an improved understanding of structural dynamics and facilitating the identification and mitigation of potential structural challenges.

On Fractional Order Model of Tumor Growth with Cancer Stem Cell

This paper generalizes the integer-order model of the tumour growth into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the cellular response. This model describes the dynamics of cancer stem cells and non-stem (ordinary) cancer cells using a coupled system of nonlinear integro-differential equations. Our analysis focuses on the existence and boundedness of the solution in correlation with the properties of Mittag-Leffler functions and the fixed point theory elucidating the proof. Some numerical examples with different fractional orders are shown using the finite difference scheme, which is easily implemented and reliably accurate. Finally, numerical simulations are employed to investigate the influence of system parameters on cancer progression and to confirm the evidence of tumour growth paradox in the presence of cancer stem cells.

On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order

In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovskiĭ functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new.

Existence results for a nonlinear coupled radiative–conductive heat transfer system

In this paper, we present existence results for a one-dimensional coupled system of nonlinear integro-differential equations on Lp-spaces with 1≤p<∞. This system describes the steady-state combined radiative–conductive heat transfer. We reformulate the full coupled system as a fixed point problem. This formulation is applied to prove existence results for the radiation intensity and the temperature fields. We treat separately the cases p=1 and p>1. The analysis for p>1 uses the Krasnosel’skii fixed point theorem, while the existence results in the case p=1 are obtained via a new variant of that theorem for the weak topology established in Latrach et al. (2006).

On solvability of boundary value problem for a nonlinear Fredholm integro-differential equation

The paper proposes a constructive method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation. Using D.S. Dzhumabaev parametrization method, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integrodifferential equations with parameters on the subintervals. When applying the parametrization method to a nonlinear Fredholm integro-differential equation, the intermediate problem is a special Cauchy problem for a system of nonlinear integro-differential equations with parameters. By substitution the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The iterative methods are used to solve both the constructed system of algebraic equations in parameters and the special Cauchy problem. An algorithm for solving boundary value problem under consideration is provided.

ASYMPTOTIC ANALYSIS OF AN INTEGRO-DIFFERENTIAL SYSTEM MODELING THE BLOW UP OF CANCER CELLS UNDER THE IMMUNE RESPONSE

<abstract> In this paper, we derive and analyze a phenomenological model at the cellular level of the immune response to cancer evolution based on the kinetic theory of active particles. The model consists of a system of nonlinear integro-differential equations describing the binary interactions between epithelial, tumor, naive immune cells, and activated immune cells. It also takes into account the phenotypic mutations in the epithelial and immune cells, which are known to result in the uncontrolled growth of tumor cells. We prove the well-posedness of the related Cauchy problem and the non-negativity of the solution. We give sufficient conditions for which the solution may exist globally in time. A detailed asymptotic analysis has been developed with the aim of predicting the effect of mutation events on the tumor-immune dynamics. The analysis shows that under some critical values of the model's parameters and initial conditions, we can specify some biological states of the blow up of tumor cells. Indeed, the analysis gives useful indications to be properly explored toward the design of therapeutical actions. </abstract>

Local asymptotic stability of a system of integro-differential equations describing clonal evolution of a self-renewing cell population under mutation.

In this paper we consider a system of non-linear integro-differential equations (IDEs) describing evolution of a clonally heterogeneous population of malignant white blood cells (leukemic cells) undergoing mutation and clonal selection. We prove existence and uniqueness of non-trivial steady states and study their asymptotic stability. The results are compared to those of the system without mutation. Existence of equilibria is proved by formulating the steady state problem as an eigenvalue problem and applying a version of the Krein-Rutmann theorem for Banach lattices. The stability at equilibrium is analysed using linearisation and the Weinstein-Aronszajn determinant which allows to conclude local asymptotic stability.

Increasing power of generator on nonlinear magnetic nanostructure

Background: One of the most promising areas of development of modern electronics is the creation of spintronic devices, which should replace the traditional semiconductor elements. The use of electron spin as a carrier of information in magnetic nanostructures can radically change modern life. Objectives: The aim of this work is to find ways to increase the power of the generator on the magnetic nanostructure by changing its electrical circuit and more optimal external electromagnetic parameters that affect the state of electrons in the studied layered structure. Materials and methods: The solution of this problem is carried out by numerical simulation of the magnetic nanostructure using a specially created micromagnetic simulator, which implements an algorithm for the simultaneous solution of the system of Maxwell and Landau-Lifshitz-Hilbert equations. The solution of such a complex problem is accelerated by the use of a quasi-static approximation in solving the system of Maxwell's equations, which is justified by the small size of the calculation area compared to the depth of the skin layer. Further calculations of the electrodynamic system are performed using the finite element method. To obtain the best frequency and energy parameters of the generator, it is proposed to introduce a resonant circuit to the schematic diagram of the studied generator, which is excited by short nanosecond pulses. Results: A scheme of a generator on a magnetic nanostructure containing a resonator with concentrated parameters is proposed, and a system of nonlinear integro-differential equations with respect to electric currents is obtained in general. Numerical calculation of this system includes, in addition to the calculation of the scheme, also the modeling of a nonlinear electrodynamic structure by the finite element method. The energy and spectral characteristics of the studied generator are obtained. The search for the optimal values of the geometric parameters of the nanostructure and the magnitude of the external longitudinal magnetization is carried out. Conclusions: Due to the complex nature of nonlinear processes in the magnetic nanostructure, the use of an external resonator, which could improve the spectral parameters of the generated current, did not give a noticeable improvement. The influence of the value of the external magnetization on the output power of the generator is complex and nonlinear, but, in general, a decrease in the level of magnetization leads to a significant decrease in power. It is established that the thickness of the magnetic layer of 6 nm is optimal for improving the energy characteristics of the generator.

Normal Impact of A Rigid Cone-shaped against A Viscoelastic Plate on Viscoelastic Foundation

This paper considers the problem of a low-velocity normal impact of a rigid cone-shaped upon a viscoelastic plate. The contact force is defined by the modified Hertz's contact law. Approximate solutions of the system of nonlinear integro-differential equations for the contact force and local indentation have been obtained.&#x0D;

Existence and uniqueness results for the first-order non-linear impulsive integro-differential equations with two-point boundary conditions

The article discusses the existence and uniqueness of solutions for a system of nonlinear integro-differential equations of the first order with two-point boundary conditions. The Green function is constructed, and the problem under consideration is reduced to equivalent integral equation. Existence and uniqueness of a solution to this problem is analyzed using the Banach contraction mapping principle. Schaefer’s fixed point theorem is used to prove the existence of solutions.

Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

The research object of this work is an orthotropic viscoelastic plate with an arbitrarily varying thickness. The plate was subjected to dynamic periodic load. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. The Bubnov–Galerkin method, based on a polynomial approximation of the deflection and displacement, was used. The problem was reduced to solving systems of nonlinear integrodifferential equations. The solution of the system was obtained for an arbitrarily varying thickness of the plate. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The plate’s dynamic stability was investigated depending on the plate’s geometric parameters and viscoelastic and inhomogeneous material properties. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov–Rzhanitsyn kernel, the differences between elastic and viscoelastic problems are significant and amount to more than 40%. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.

On solvability of one class of third order differential equations

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

A method of solving a nonlinear boundary value problem for the Fredholm integro-differential equation

We propose a method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation on a finite interval. By introducing additional parameters chosen as the values of the solution at the left-end points of the partition subintervals, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integro-differential equations with parameters on the subintervals. For fixed parameters, we obtain a special Cauchy problem for this system, which is represented as a nonlinear operator equation and solved by an iterative method. By substitution of the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The algorithm for solving the special Cauchy problem includes two auxiliary problems: the Cauchy problems for ordinary differential equations and the evaluation of definite integrals. The accuracy of the method that we propose to solve the boundary value problem depends on the accuracy of methods applied to the auxiliary problems and does not depend on the number of the partition subintervals. Since iterative methods are used to solve both the constructed system of algebraic equations and the special Cauchy problem, we offer an approach to find an initial guess for the solutions to these problems.

Регуляризованные асимптотические решения интегродифференциальных уравнений с быстрыми и медленными переменными

The article considers a nonlinear integro-differential system of equations with fast and slow variables. Such systems were not considered previously from the point of view of constructing regularized (according to Lomov) asymptotic solutions. The known studies were mainly devoted to construction of the asymptotics of the Butuzov-Vasil'eva boundary layer type, which, as is known, can be applied only if the spectrum of the first variation matrix (on the degenerate solution) is located strictly in the open left-half plane of a complex variable. If the spectrum of this matrix falls on the imaginary axis, the S.A. Lomov regularization method is commonly used. However, this method was mainly developed for singularly perturbed differential systems that do not contain integral terms, or for integro-differential problems without slow variables. In this article, the regularization method is generalized for two-dimensional integro-differential equations with fast and slow variables.