Integro-differential Equations Research Articles

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Representations of the solutions for volterra integro-differential equations in hilbert spaces

Representations of the solutions for volterra integro-differential equations in hilbert spaces

RESULTS ON IMPULSIVE ψ-CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

In this research article, we investigate the existence and uniqueness of solutions for a type of boundary value problems involving fractional integro-differential equations with the ψ-Caputo fractional derivative. Our method utilizes several classical fixed point theorems, such as the Banach contraction principle and Krasnoselskii's fixed point theorem. We establish our main results through theoretical analysis and present a specific case study to illustrate the practical significance of our findings.

Existence of mild solutions for non-instantaneous impulsive ξ-Caputo fractional integro-differential equations

The aim of this paper is to investigate the existence of mild solutions for a nonlocal ξ-Caputo fractional non-instantaneous impulses semilinear integro-differential equation in a Banach space. The proofs are based on some fixed point theorems for condensing maps. As an application, an example is given to illustrate our theoretical results.

Numerical computational technique for solving Volterra integro-differential equations of the third kind using meshless collocation method

The primary goal of this study is to give an approximate algorithm for solving Volterra integro-differential equations (VIDEs) of the third kind using meshless collocation techniques. The basic framework of the novel approach is based on a collocation scheme and radial basis functions (RBFs) created on scattered points. This technique requires no background approximation cells, and the algorithm is powerful, has greater stability, and does not require much computer memory. This approach represents the solution of VIDEs of the third kind by interpolating the RBFs based on the Gauss–Legendre quadrature formula. The problem is reduced to a system of algebraic equations that can be easily solved. A description of the technique for the proposed equations is provided. Furthermore, the error analysis of this scheme is examined. A few numerical experiments are presented to prove the reliability and precision of the suggested approach for solving VIDEs of the third kind. Certain problems were compared with analytical solutions, the moving least squares method, and other methods to prove the effectiveness and applicability of the approach described.

Synthesis of uniformly distributed optimal control with nonlinear optimization of oscillatory processes

In the article the problem of synthesizing uniformly distributed optimal control for nonlinear optimization of oscillatory processes described by integro-differential partial differential equations with the Volterra integral operator was explored. The study was conducted according to the Bellman-Egorov scheme and an algorithm for constructing a uniformly distributed optimal control in the form of a functional from the state of the controlled process was developed. Sufficient conditions for the solvability of the synthesis problem in nonlinear optimization were established.

Existence of extremal solutions for a class of fractional integro-differential equations

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

Numerical implementation of solving a boundary value problem with parameter for Fredholm integro-differential equation

The boundary value problem with parameter for Fredholm integro-differential equation with degenerate kernel is investigated in this paper. The aim of the paper is to establish the solvability conditions, to construct analytical and numerical solutions of the considered problem. The basis for achieving the goal is the ideas of Dzhumabayev parameterization method, classical numerical methods of solving Cauchy problems and numerical integration techniques. A problem with parameters is obtained by introducing an additional parameter and a new unknown function. A system of equations with respect to parameters is compiled according to the initial data of the considered equation and boundary conditions. The unknown function is found as a solution of the Cauchy problem for the ordinary differential equation. The equivalence of the original problem and the problem with parameters, the conditions of unique solvability are established and the formula for finding an analytical solution is obtained. Test examples of finding analytical and approximate solutions of the original problem are given.

An optimization-based equilibrium measure describes fixed points of non-equilibrium dynamics: application to edge of chaos

Abstract Understanding neural dynamics is a central topic in machine learning, non-linear physics and neuroscience. However, the dynamics is non-linear, stochastic and particularly non-gradient, i.e., the driving force can not be written as gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory, but the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed form solutions in general. From the aspect of associated Fokker-Planck equation, the steady state solution is generally unknown. Here, we treat searching for the steady states as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running an approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, the distribution of the original steady states can be achieved. The resultant stationary state of the dynamics follows exactly the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

On Fuzzy Fractional Volterra-Fredholm Model Under the Uncertainty θ-Operator of the AD Technique: Theorems and Applications

This article investigates the proper existence conditions and uniqueness results for a class of fuzzy fractional Caputo Volterra-Fredholm integro-differential equations (FFCV-FIDE) with initial conditions. The findings are based on Banach's contraction principle and Schaefer's fixed point theorem. Furthermore, the solution to the posed problem is found using the Adomian decomposition technique (ADT). We support the concept with several examples. The relationship between the upper and lower reduced approximation of the fuzzy solutions was demonstrated numerically and graphically using MATLAB.

Theoretical and Numerical Studies of Fractional Volterra-Fredholm Integro-Differential Equations in Banach Space

This paper examines the theoretical, analytical, and approximate solutions of the Caputo fractional Volterra-Fredholm integro-differential equations (FVFIDEs). Utilizing Schaefer's fixed-point theorem, the Banach contraction theorem and the Arzel\`{a}-Ascoli theorem, we establish some conditions that guarantee the existence and uniqueness of the solution. Furthermore, the stability of the solution is proved using the Hyers-Ulam stability and Gronwall-Bellman's inequality. Additionally, the Laplace Adomian decomposition method (LADM) is employed to obtain the approximate solutions for both linear and non-linear FVFIDEs. The method's efficiency is demonstrated through some numerical examples.

The existence of a unique solution and stability results with numerical solutions for the fractional hybrid integro-differential equations with Dirichlet boundary conditions

In this paper, we investigate the fractional hybrid integro-differential equations with Dirichlet boundary conditions. We first prove the existence of a unique solution for the equation using a fixed point technique. Our main goal is to obtain the best approximation using optimal controllers. After studying the stability, we present the reproducing kernel Hilbert space numerical method to obtain approximate solutions to the equation. We finally conclude with numerical results.

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

Computational Approach for Two-Dimensional Fractional Integro-Differential Equations

Computational Approach for Two-Dimensional Fractional Integro-Differential Equations

Existence solutions for nondensely integro-differential equations with state-dependent delay

Existence solutions for nondensely integro-differential equations with state-dependent delay

Multiple soliton solutions and other travelling wave solutions to new structured (2+1)-dimensional integro-partial differential equation using efficient technique

The Ito equation belongs to the Korteweg–de Vries (KdV) family and is commonly employed to predict how ships roll in regular seas. Additionally, it characterizes the interaction between two internal long waves. In the 1980s, Ito extended the bilinear KdV equation, resulting in the well-known (1+1)-dimensional and (2+1)-dimensional Ito equations. In this study finds numerous classes of exact solutions for a new structured (2 + 1)-dimensional Ito integro-differential equation using the help of the Mathematica software. The Improved Modified Extended Tanh Function Scheme (IMETFS) is utilised to address the aforementioned equation analytically. Bright, dark, and singular soliton solutions are produced. Additionally, periodic, exponential, rational, singular periodic, and Weierstrass elliptic doubly periodic results are achieved. The method employed includes the nonlinear evolution equations that arise in a variety of real-world situations, and it is efficient, applicable, and simple to handle. For certain obtained solutions, specific options of free constants are presented in 3D, 2D, and contour graphical depictions.

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.