Nonlinear Integro-differential Equations Research Articles

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.

Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM

This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than that, the existance and uniqueness of solution as well as the norm convergence with respect to ND-HAM, were proven in a Hilbert space. In addition, three numerical examples (including multi-term fractional IDEs) are presented and compared with the HAM, modified HAM and ”Generalized block pulse operational differentiation matrices method” developed in previous works by illustrating the accuracy as well as validity with respect to ND-HAM. Empirical investigations reveal that ND-HAM and the modified HAM yields the same results when control parameter ℏ is chosen as ℏ = −1 and is comparable to the standard HAM. The findings discovered that the ND-HAM is highly convenient, effective, as well as in line with theoretical results.

Chebyshev Petrov–Galerkin method for nonlinear time-fractional integro-differential equations with a mildly singular kernel

Chebyshev Petrov–Galerkin method for nonlinear time-fractional integro-differential equations with a mildly singular kernel

Controlled Filtered Poisson Processes

Filtered Poisson processes are used as models in various applications, in particular in statistical hydrology. In this paper, controlled filtered Poisson processes are considered. The aim is to minimize the expected time that the process will spend in the continuation region. The dynamic programming equation satisfied by the value function is derived. To obtain the value function, and hence the optimal control, a non-linear integro-differential equation must be solved, subject to the appropriate boundary conditions. Various cases for the size of the jumps are treated and explicit results are obtained.

High order semi-IMEX BDF schemes for nonlinear partial integro-differential equations arising in finance

In this paper, semi-implicit-explicit (Semi-IMEX) and semi-implicit multistep methods are proposed to solve nonlinear partial integro-differential equations (PIDEs), which describe the option pricing models with transaction costs or illiquid markets under Merton's jump-diffusion process. After spatial differential operators are treated by using finite difference methods and the jump integral is computed by using the composite trapezoidal rule, the consistency error and global error bounds for the semi-IMEX and semi-implicit multistep methods for abstract PIDEs are provided when the nonlinear operator satisfies the boundedness and coercivity conditions. A numerical study is carried out for different option pricing models based on the convergence properties of the schemes and the comparison of the different Greek values. Numerical experiments verify our theoretical results and illustrate the intrinsic nature of the proposed option pricing models under jump-diffusion models with transaction costs or illiquid markets.

Local existence of a classical solution to the Einstein–Maxwell–Klein–Gordon system in higher dimensions

In this paper, we study the local existence of a classical solution to the coupled Einstein and Maxwell–Klein–Gordon system in higher dimensions. This system provides a toy model to study the dynamics of the complex scalar fields under the influence of the interaction of the gravitational and electromagnetic fields. In the starting point, we introduce the metric of the space-time in the Bondi coordinate. Then, we construct the evolution equation in the form of a single first-order nonlinear integro-differential equation. Furthermore, we show that there exists a unique fixed point that is the solution of the main problem based on the contraction mapping arguments. Finally, for a given initial data, we prove the existence of a local classical solution.

Maximum principles for nonlinear integro-differential equations and symmetry of solutions

Maximum principles for nonlinear integro-differential equations and symmetry of solutions

Adaptive Control for Multi-Agent Systems Governed by Fractional-Order Space-Varying Partial Integro-Differential Equations

This paper investigates a class of multi-agent systems (MASs) governed by nonlinear fractional-order space-varying partial integro-differential equations (SVPIDEs), which incorporate both nonlinear state terms and integro terms. Firstly, a distributed adaptive control protocol is developed for leaderless fractional-order SVPIDE-based MASs, aiming to achieve consensus among all agents without a leader. Then, for leader-following fractional-order SVPIDE-based MASs, the protocol is extended to account for communication between the leader and follower agents, ensuring that the followers reach consensus with the leader. Finally, three examples are presented to illustrate the effectiveness of the proposed distributed adaptive control protocols.

On boundary value problem of the nonlinear fractional partial integro-differential equation via inverse operators

On boundary value problem of the nonlinear fractional partial integro-differential equation via inverse operators

Third-Kind Chebyshev Spectral Collocation Method for Solving Models of Two Interacting Biological Species

This paper develops numerical methods for solving a system of two nonlinear integro-differential equations that arise in biological modeling. A spectral collocation method utilizing third-kind Chebyshev polynomials forms the basis of the solution methodology, which efficiently converts the integro-differential system into a collection of nonlinear algebraic equations. To guarantee precise and effective calculation, these algebraic equations are subsequently numerically solved using Newton’s method. In comparison to current methods, the suggested approach offers notable gains in computational efficiency and precision. The spectral collocation method’s accuracy is confirmed by contrasting the outcomes with those derived from other numerical methods that are published in the literature. To further illustrate the applicability, dependability, and computational efficiency of the suggested approach in resolving complicated biological systems, a number of illustrative instances are provided. The ability of spectral collocation techniques based on third-kind Chebyshev polynomials to solve integro-differential equations in a variety of scientific and engineering applications is highlighted by this work.

Analyzing inverse backward problem in nonlinear integro-differential equation with memory kernel

This paper focuses on the backward problem related to an integro-differential equation with a general convolutional derivative in time and nonlinear source terms. The existence, uniqueness, and regularity of the mild solution to the proposed problem are established under certain assumptions in a suitable space. The proposed problem is ill-posed in the sense of Hadamard. Moreover, the Fourier truncation method is used to construct a regularized solution. Finally, the convergence rate between the regularized solution and the exact solution is determined.

Qualitative analysis of a coupled system with nonlinear mixed fractional integro-differential equation involving Caputo fractional derivative

Qualitative analysis of a coupled system with nonlinear mixed fractional integro-differential equation involving Caputo fractional derivative

Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method

Abstract We investigate a class of third-order nonlinear integro-differential equations (IDEs) with parallel computing of intelligent Internet of Things and wireless networks for numerical solutions. A numerical scheme based on the Haar wavelet has been established to compute the approximate solution for the problem under our consideration. By utilizing the mentioned tool, we discretize the involved derivatives and integrals. In this way, a sophisticated scheme is derived. Formulations for maximum root mean square and absolute errors have been given. Also, the convergent method has been discussed. In engineering, such as structural dynamics and control systems, third-order IDEs can improve modelling precision and solution effectiveness. Various examples have been testified by the aforementioned method. Additionally, by using different Gauss and collocation points (CPs), the aforementioned error terms were recorded. The convergence rate using distinct numbers of CPs has also been calculated, which is nearly equal to 2.

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.

Error estimates for the robust α-stable central limit theorem under sublinear expectation by a discrete approximation method

In this work, we develop a numerical method to study the error estimates of the α-stable central limit theorem under sublinear expectation with α∈(0,2), whose limit distribution can be characterized by a fully nonlinear integro-differential equation (PIDE). Based on the sequence of independent random variables, we propose a discrete approximation scheme for the fully nonlinear PIDE. With the help of the nonlinear stochastic analysis techniques and numerical analysis tools, we establish the error bounds for the discrete approximation scheme, which in turn provides a general error bound for the robust α-stable central limit theorem, including the integrable case α∈(1,2) as well as the non-integrable case α∈(0,1]. Finally, we provide some concrete examples to illustrate our main results and derive the precise convergence rates.

Solving nonlinear neutral delay integro-differential equations via general linear methods

General linear methods are adapted for solving nonlinear neutral delay integro-differential equations. The sufficient conditions for the stability and asymptotic stability of (k,l)-algebraically stable general linear methods are derived. At last, a numerical test is given to validate the theoretical results.

Optimizing linear/non-linear Volterra-type integro-differential equations with Runge–Kutta 2 and 4 for time efficiency

A novel non-adaptative time-efficient (TE) numerical scheme based on the Runge–Kutta (RK) algorithm is devised for solving linear and non-linear Volterra-type integro-differential equations (VTIDE) characterized by a specific class of convolution memory, K(t,s)=aκ(t−s), where a,κ∈R. This kernel is commonly encountered in viscoelasticity and various applied sciences. The integration governing the convolution term is elegantly reformulated, allowing for the implementation of a backstage integration within the RK scheme’s main body. This approach circumvents the need for full integration of the convolution during each iteration, thereby significantly reducing computational time to O(Nt) for Nt iterations. Furthermore, this formulation facilitates the adoption of an implicit scheme by selecting appropriate methods and data stencils. We demonstrate this concept using an implicit trapezoidal method applied to a linear VTIDE, accompanied by stability analyses. Additionally, a complex VTIDE is constructed featuring nonlinearities both within and outside the convolutions, as well as a derivative-of-dependent-variable integrant. This setup enables the synergy of differentiation, integration, and RK schemes to generate data for intricate VTIDE types. Consequently, both the scheme and the equation exhibit uniqueness denoted as TE-RK and TE-RK-VTIDE. We illustrate how this development simplifies the implementation of implicit schemes and the calculation of stability regions concerning the convolution. The TE-RK scheme is tested on VTIDE using RK2 and RK4, yielding plots that exhibit strong agreement, thus validating our approach. Furthermore, we successfully apply the TE-RK scheme to a well-known nonlinear logistic equation. Plots for various step sizes are generated alongside corresponding error graphs, demonstrating consistent trends in error behavior.

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 $H<1/2$.


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.