Integro-differential Equations Research Articles

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

On the qualitative behaviors of Volterra-Fredholm integro differential equations with multiple time-varying delays

This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays.

Properties of a Partial Fredholm Integro-differential Equations with Nonlocal Condition and Algorithms

Properties of a Partial Fredholm Integro-differential Equations with Nonlocal Condition and Algorithms

Study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments

In this paper, we give new results on the study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments, considered in a Banach space. Using the method suggested by Hernández et al. (2010), which is based on the theory of resolvent operators for integral equations and fixed point theorems, we prove the main results on the existence and uniqueness of the mild solution of this type of problems which have not been studied before. We also give an example to which our main results are applicable.

Numerical Solution of Stochastic Fractional Integro-Differential Equations: The Poly-sinc Collocation Approach

Numerical Solution of Stochastic Fractional Integro-Differential Equations: The Poly-sinc Collocation Approach

Capsize criteria in beam seas: Melnikov analysis vs. safe basin erosion

We study the problem of capsizing of a rolling ship in harmonic and random beam seas, by means of the Melnikov method and safe basin calculations. In the random excitation case, we consider an integro-differential Cummins-type model equation that takes into account the hydrodynamic memory. The non-linear restoring moment is modeled by a high-order polynomial and the Melnikov criteria are evaluated numerically. We also examine a variant of the classical Melnikov method in which the damping terms are incorporated into the unperturbed system and the perturbation takes a modified form. We compare the theoretical predictions with direct (Monte-Carlo) simulations of the safe basins for two existing ships. In the random excitation case, we quantify the erosion by means of a mean integrity index. For weak damping, the classical and modified Melnikov curves coincide and are in good agreement with the onset of the safe basin erosion. As damping increases the Melnikov curves become less and less conservative with respect to the onset of erosion. Finally, the potential of the Melnikov method to provide a ship classification tool is discussed.

A Numerical Method for Investigating Fractionali Volterra-Fredholm Integro-Differential Model

In this article, we investigate the fractional Volterra-Fredholm integro-differential equations. These equations appear in several applications such as control theory, biology, and particle dynamics in physics. We derive a numerical method based on the operational matrix method to solve this class of integro-differential equations. We prove the existence and uniqueness of the ex-act solution. Additionally, we demonstrate the uniform convergence of the numerical solutions to the exact solution. We present several numerical examples to show the numerical efficiency of the proposed method. In the first example, we choose a linear problem and find that the approximate solution converges to the exact solution when the number of block pulse functions is very large. In the next two examples, we consider the nonlinear case and compute the L2-local truncation error since exact solutions are not available. The error was of order 10−12. Furthermore, we sketch the graph of the approximate solutions for different values of the fractional derivative to observe the influence of the fractional derivative on the profile of the solutions. Theoretical and numerical results show that the proposed method is accurate and can be applied to other nonlinear problems in science.

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Representations of Solutions for Volterra Integro-Differential Equations in Hilbert Spaces

Representations of Solutions for Volterra Integro-Differential Equations in Hilbert Spaces

A Novel Operational Matrix Method for Solving the Fractional Delay Integro-Differential Equations with a Weakly Singular Kernel

A Novel Operational Matrix Method for Solving the Fractional Delay Integro-Differential Equations with a Weakly Singular Kernel

A Fitted Approximate Method for Solving Singularly Perturbed Volterra–Fredholm Integrodifferential Equations with Integral Boundary Condition

A Fitted Approximate Method for Solving Singularly Perturbed Volterra–Fredholm Integrodifferential Equations with Integral Boundary Condition

Euler wavelets method for optimal control problems of fractional integro-differential equations

This research introduces an efficient direct method for finding the numerical solution to optimal control problems involving linear and nonlinear fractional integro-differential equations using orthonormal Euler wavelets. Using orthonormal Euler wavelets, fractional integral, and product operational matrices have been obtained. By using these fractional operational matrices and collocation points, optimal control problems of fractional Volterra integro-differential equations have been reduced to a system of linear or non-linear equations. We establish the convergence analysis and error bound of the proposed method. Orthonormal Euler wavelets have been utilized to solve numerical examples and determine the approximate value of the cost function by approximating state and control functions. This has been done in order to verify the effectiveness of the proposed numerical technique.

Existence of Solutions for a Class of Integro-Differential Equations With the Logarithmic Laplacian and Transport

Existence of Solutions for a Class of Integro-Differential Equations With the Logarithmic Laplacian and Transport

Higher-order Haar wavelet method for solution of fourth-order integro-differential equations

This paper presents a numerical approach to solve third and fourth order intego-differential equations (IDEs). In order to ascertain the numerical solution for third and fourth order IDEs of second kind, the newly introduced Higher order Haar wavelet method (HOHWM) has been employed to improve the numerical result and rate of convergence compared to classical Haar wavelet approach. Some examples available in the literature have been solved to verify the HOHWM’s effectiveness. To ensure that the approach presented is legitimate, applicable and achieves its objective, the maximum absolute error of each test problem is calculated at a test point.

Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative

Abstract In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

Tempered nonlocal integrodifferential equations with nonsmooth solution data

This work on the time discretization for the solution of the tempered nonlocal integro-differential equation with smooth and nonsmooth initial data are extended. We find the difficulty arising from theoretical analysis can be overcome by Laplace transform technique. A mount of proof skills have been provided based on Laplace transform technique for this kind of equations. The Laplace transform method is employed effectively to show that the proposed interpolating quadrature scheme is convergence for smooth and non-smooth initial data in both homogeneous and inhomogeneous cases.

Boundary value problems for integro-differential and singular higher-order differential equations

Abstract We investigate third-order strongly nonlinear differential equations of the type ( Φ ( k ( t ) u ″ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}^{^{\prime\prime} }\left(t)))^{\prime} =f\left(t,u\left(t),u^{\prime} \left(t),{u}^{^{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Φ \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Φ ( k ( t ) v ′ ( t ) ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v^{\prime} \left(t)))^{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v^{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.

Cordes-Nirenberg type results for nonlocal equations with deforming kernels

We derive Cordes-Nirenberg type results for nonlocal elliptic integro-differential equations with deforming kernels comparable to sections of a convex solution of a Monge-Ampère equation. Under a natural integrability assumption on the Monge-Ampère solution, we prove a stability lemma allowing the ellipticity class to vary. Using a compactness method, we then derive Hölder regularity estimates for the gradient of the solutions.

Numerical simulation of Volterra PIDE with singular kernel via modified cubic exponential and uniform algebraic trigonometric tension B-spline DQM

In this paper, two different numerical techniques are employed to solve the Volterra partial integro-differential equation (PIDE) with a weakly singular kernel: Uniform Algebraic Trigonometric tension (UAT) B-spline and Exponential B-spline. These techniques are further modified under certain conditions. The presented techniques transform the discretized Volterra PIDE into a linear algebraic system of equations. In this process, the forward difference formula is utilized to address the time derivative, while the differential quadrature method (DQM) is used for the spatial order derivative. The fusion of modified cubic exponential and modified cubic UAT tension B-spline with DQM is considered. The effectiveness of the methods is assessed via various types of errors considered through three distinct examples. Additionally, the validity of these results is shown by comparison with previous findings in the same environment. The comparison demonstrates that the modified cubic UAT tension B-spline produces less inaccuracy than the other. This work provides robust results that advance research toward developing more advanced and computationally efficient numerical techniques.

Integro-Differential Equations in Banach Spaces and Analytic Resolving Families of Operators

Integro-Differential Equations in Banach Spaces and Analytic Resolving Families of Operators