Nonlinear Integro-differential Equations Research Articles

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

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

Numerical simulation, existence and uniqueness for solving nonlinear mixed partial integro-differential equations with discontinuous kernels

Numerical simulation, existence and uniqueness for solving nonlinear mixed partial integro-differential equations with discontinuous kernels

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.

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 straight forward 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.

Correction: Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Correction: Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

About a new nonlinear integro-differential equation of Volterra-Strum-Liouville type with a weakly singular kernel

This present paper proposes an analytical and numerical study of a new integro-differential Volterra-Strum-Liouville equation of the second type, having a weak singularity. We provide conditions that guarantee the existence and uniqueness of the solution of the nonlinear problem. Then, we develop a numerical technique using the production integration method. The numerical application shows the efficiency of the proposed procedure.

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

The chemostat reactor: A stability analysis and model predictive control

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system’s model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches.

Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

Development of integrated radial basis function Kriging interpolation for linear and nonlinear parabolic integro-differential equations

In this study, we explore linear and nonlinear parabolic integro-differential equations in one and two dimensions. We employ a semi-implicit scheme to discretize the temporal variable and discretize the spatial variable using an integrated radial basis function based on the moving Kriging interpolation (MKI) method. Unlike the global integrated radial basis function (IRBF) method, our proposed approach is particularly well-suited for addressing large-scale problems. Moreover, issues such as the selection of shape parameters leading to matrices with high condition numbers can be mitigated by devising an algorithm to determine optimal shape parameters, resulting in lower condition numbers. To assess the accuracy of the proposed method, we utilize two criteria, namely, L2 and H1. We compare the computed results obtained from the IRBF-MKI method with those from two grid and finite element methods. To demonstrate the accuracy and efficiency of our approach, we consider irregular areas and scattered Halton points in the two-dimensional case.

New algorithm on linearization-discretization solving systems of nonlinear integro-differential Fredholm equations

This article deals with a new strategy for solving a certain type of nonlinear integro-differential Fredholm equations with a weakly singular kernel. We build our new algorithm starting with the linearization phase using Newton's iterative process, then with the discretization phase we apply the Kantorovich's projection method. The discretized linear scheme will be approximated by the product integration method in the weak singular terms, and the other regular integrals will be approximated by the Nyström method. The process of convergence of our new algorithm is carried out under certain predefined and necessary conditions. Finally, we give practical examples where, the results show the efficiency of our new algorithm for solving systems of weakly singular nonlinear integro-differential equations.

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

Abstract Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Numerical solutions for nonlinear mixed integro-differential equation and the neutron transport equation

In this work, I will discuss the solution of the new in a nonlinear integro-differential equation (NI-DE) form. The NI-DE is established from the nuclear transport equation, which is a linearized derivative of the equation developed by Boltzmann for the kinetic theory of gases. In this study, I will introduce the basic equations and boundary conditions for the formulation of the problem. Then, the NI-DE is established from the planar geometry problem of the neutron transport equation. In addition, I will prove the existence of a unique solution to the problem. In addition, I will use the numerical method to obtain a system of NI-DEs. Therefore, I will prove, under certain conditions, the existence of a unique solution to this system. Finally, I will present two different effective methods to solve the problem numerically, and I will discuss the results.