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.

Modeling pull-in instability of CNT nanotweezers under electrostatic and van der Waals attractions based on the nonlocal theory of elasticity

This work investigates the electromechanical response and pull-in instability of an electrostatically-actuated CNT tweezer taking into consideration a TPNL constitutive behavior of the CNTs as well as the intermolecular forces, both of which provide a significant contribution at the nanoscale. The nonlocal response of the material introduces two additional parameters in the formulation, which are effective in capturing the size effects observed at the nanoscale. The problem is governed by a nonlinear integrodifferential equation, which can be reduced to a sixth-order nonlinear ODE with two additional boundary conditions accounting for the nonlocal effects near to the CNT edges. A simplified model of the device is proposed based on the assumption of a linear or parabolic distribution of the loading acting on the CNTs. This assumption allows us to formulate the problem in terms of a linear ODE subject to two-point boundary conditions, which can be solved analytically. The results are interesting for MEMS and NEMS design. They show that strong coupling occurs between the intermolecular forces and the characteristic material lengths as smaller structure sizes are considered. Considering the influence of the nonlocal constitutive behavior and intermolecular forces in CNT tweezers will equip these devices with reliability and functional sensitivity, as required for modern engineering applications.

A higher-order collocation method based on Haar wavelets for integro-differential equations with two-point integral condition

In this article, the higher-order Haar wavelet collocation method (HCMHW) is investigated to solve linear and nonlinear integro-differential equations (IDEs) with two types of conditions: simple initial condition and the point integral condition. We reproduce and compare the numerical results of the conventional Haar wavelet collocation method (CMHW) with those of HCMHW, demonstrating the superior performance of HCMHW across various conditions. Both methods effectively handle different types of given conditions. However, numerical results reveal that HCMHW exhibits a faster convergence rate than CMHW. To address nonlinear IDEs, we employ the quasi-linearization technique. The computational stability of both methods is evaluated through various experiments. Additionally, the article provides examples to illustrate the overall performance and accuracy of HCMHW compared to CMHW for both linear and nonlinear IDEs.

İNTEGRO-DİFERANSİYEL DENKLEMLERİN SAYISAL ÇÖZÜMÜNE UYGULANAN KOLLOKASYON YÖNTEMİ

Integro-differential equations are encountered in such fields of study as mechanics, physics, chemistry, biophysics, astronomy, economic theory, and population dynamics. In rare cases the solution methods for differential and/or integral equations can be generalized to integro-differential equations; but in general, numerical methods have to be applied. Recent years have seen the development of a large number of methods applicable to integro-differential equations. The present study aims to compare these newer methods with the classical method of point collocation, which is one of the weighted residual methods. The method was applied to test problems chosen from the literature, both linear and nonlinear integro-differential equations, and was seen to give good results.

Some New Results on Hadamard Neutral Fractional Nonlinear Volterra-Fredholm Integro-Differential Equations

Some New Results on Hadamard Neutral Fractional Nonlinear Volterra-Fredholm Integro-Differential Equations

SOLUTIONS OF NONLINEAR BOUNDARY SYSTEM WITH COUPLED INTEGRAL BOUNDARY CONDITIONS

This article presented some theorems on a novel non-linear multiple Integro-differential equations of boundary T-system. It has been studied the numerical-analytic method and Banach fixed point theorem for the existence and approximation of the solutions over considered boundary system in compact space. In this work, we demonstrate that the mention methods can be discussed and analyzed for the existence and uniqueness, of a solution for the vector system. The paper satisfies the Hölder condition.

Pattern Formation in Mesic Savannas

We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels.

A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations

We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp.

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.

Existence of solutions for fractional integro-differential equations with integral boundary conditions

"The aim of this study is to prove the existence of solutions for Caputo boundary value problems of nonlinear fractional integro-differential equations with integral boundary conditions, by using the measure of non compactness combined with Mönch's fixed point theorem. Two examples are offered to demonstrate our outcomes."

An efficient parametric finite difference and orthogonal spline approximation for solving the weakly singular nonlinear time-fractional partial integro-differential equation

An efficient parametric finite difference and orthogonal spline approximation for solving the weakly singular nonlinear time-fractional partial integro-differential equation

A block-by-block approach for nonlinear fractional integro-differential equations

In this paper, a block-by-block scheme is proposed for a class of nonlinear fractional integro-differential equations. This method is based on the Gauss–Lobatto numerical integration method, which shows the high accuracy at all time intervals. Also, the method convergence for this type of equations is proved and it is shown that the order of convergence is at least eight. Finally, the high accuracy, fast calculations and good performance of the method are investigated by solving some numerical examples.

A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.

A new wavelet method for fractional integro-differential equations with [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative is successfully used to define a new class of nonlinear fractional integro-differential equations. The extended Chebyshev cardinal wavelets (CCWs), as a proper set of wavelet functions, are employed to establish a computational procedure for such equations. To do this, a new operational matrix for the ψ-Riemann–Liouville fractional integral of the extended CCWs is extracted with the help of the block-pulse functions. The established approach obtains the problem solution by solving an algebraic system of nonlinear equations that is created from expanding the solution by the extended CCWs (with some unknown coefficients) and utilizing the expressed fractional operational matrix. In fact, by solving this system, the unknown coefficients and, subsequently the solution of the original problem are obtained. The convergence analysis of the proposed method is investigated. The accuracy and correctness of the scheme are illustrated by solving three examples.

Comparison of several numerical solvers for a discretized nonlinear diffusion model with source terms

Abstract The numerical solution of the nonlinear system of equations resulting from a real engineering problem is discussed. We use the approximate solution of a system of two nonlinear integrodifferential equations to build the nonlinear system of equations. This system can be solved by Newton’s method if the solution is differentiable, or using some derivative-free methods, such as Steffensen’s method. Here we show that Steffensen’s method does not always converge and secant method requires more iterations than Traub’s method and Newton’s method. We recommend Traub’s method in case the solution is not differentiable.

Pontryagin’s maximum principle for a fractional integro-differential Lagrange problem

In this paper, we study an optimal control problem of Lagrange type in which a control system is described by a nonlinear and singular integro-differential equation of Volterra type with a Caputo derivative. The necessary first-order optimality conditions for a local in (y,u,v) solution (in the form of a maximum principle) for the considered problem are derived. Our approach to deriving these conditions is based on an extremum principle for an abstract optimal control problem obtained in Idczak and Walczak (2020), where the main assumption is smoothness in (y,u,v) of a cost and an operator describing constraints.

Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function

Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.

Approximating local solution of IVPs of nonlinear first order ordinary hybrid integrodifferential equations

In this paper, we prove a couple of approximation results for local existence and uniqueness of the solution of a IVP of nonlinear first order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on a hybrid fixed point theorem of Dhage (2022) and Dhage {et al.} (2022). An approximation result for the Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

Existence, Uniqueness and Stability of Solutions of a Variable-Order Nonlinear Integro-differential Equation in a Banach Space

Existence, Uniqueness and Stability of Solutions of a Variable-Order Nonlinear Integro-differential Equation in a Banach Space

Harnack inequalities and Hölder estimates for fully nonlinear integro-differential equations with weak scaling conditions

Hölder estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some mild assumptions. We allow the kernels of variable order and critically close to 2.