Order Integro-differential Equations Research Articles

Perturbation iteration transform method for solving fractional order integro-differential equation

In this study, the Perturbation Iteration transform method, namely PITM, is in short presented and implemented for solving a class of fractional integro-differential equations. The fractional derivative will be in the Atangana–Baleanu Caputo fractional derivative sense (ABC). The (PITM) is consists of merging Laplace transform method and the perturbation iteration algorithm (PIM). The proposed method furnish the solution in the form of a fastly convergent series. Some illustrative examples are presented to illustrate that the PITM is a powerful, efficient and accurate method and it can be enforced to other nonlinear problems.

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar–Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar–Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilizing ATDM.

Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

Controllability of Hilfer Fractional Semilinear Integro-Differential Equation of Order α ∊ (0, 1), β ∊ [0, 1

In this paper, Hilfer fractional differential equation is studied. Firstly, we used Laplace transform and semigroup theory to find the mild solution of the system. Then exact controllability of proposed system is established using the Arzela-Ascoli theorem and Schauder fixed point theorem. To illustrate the developed theory we provide an example at the end.

Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

The aim of this paper is to investigate some types of stability such as generalized Hyers-Ulam- Rassias stability(G-H-U-R-stabile) and the relation with Hyers-Ulam(H-U-stabile) stable and Hyers-Ulam-Rassias stable (H-U-R-stabile) and generalized Hyers-Ulam stable (G-H-U- stable) to obtain which one guarantee to satisfy stability of equations included a nonlinear function some of them contains a delay time of solution and the other contain a vector of different order of derivatives for the solution to n-time and vector of fractional order of integrals with different fractional orders and that was the for using a claculse of fractional calculus to satisfies the issue of this techniques. Moreover, the nonlocal initial values for the proposal equation of nonlinear impulsive higher order differential – fractional integral delay time equations which are adding more interesting for nonlinear analytic object of nonlinear higher order integro – fractional order impulsive classes, and the impulsive difference of the equation has some necessary conditions to prove the results of solution to be stable with certain type has related with other types. The necessary and sufficient conditions which assumed on this nonlinear higher order integro-differential impulsive equation have been achieved the stability with interesting certain estimates obtain through the proving technique. Also the uniqueness of solution has been studied with same conditions was presented for stability and used for that issue a contraction fixed point theorem.

Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions

In this paper, we derive a spectral collocation method for solving fractional-order integro-differential equations by using a kind of Müntz orthogonal functions that are defined on 0,1 and have simple and real roots in this interval. To this end, we first construct the operator of Riemann–Liouville fractional integral corresponding to this kind of Müntz functions. Then, using the Gauss–Legendre quadrature rule and by employing the roots of Müntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. We also construct an upper bound for the truncation error of Müntz orthogonal functions, and we analyze the error of the proposed collocation method. Numerical examples are included to demonstrate the validity and accuracy of the method.

Superconvergence of Legendre spectral projection methods for [formula omitted]th order integro-differential equations with weakly singular kernels

In this article, we apply Legendre spectral Galerkin, Legendre spectral multi-projection methods and their iterated versions to find the approximate solution of mth order Fredholm integro-differential equations with weakly singular kernel. Motivated by Mandal et al. (2023), we use Cauchy repeated integral theorem to transform the integro-differential equation to an single integral equation and obtain superconvergence results by iterated Legendre spectral Galerkin method, in spite of the singularity in the kernel function and unbounded differential operator. We have further improved the convergence rate of the approximate solution by using iterated Legendre spectral multi-projection method. Global Legendre polynomials are used as a basis for the approximating space, to reduce the computational cost of our proposed methods. In this article, we have derived theoretical error bounds and obtained the global convergence rates for all the discussed methods in both L2 and infinity norms. Numerical methods are implemented on examples to justify the theoretical results.

Periodic Solutions for an Impulsive System of Fractional Order Integro-Differential Equations with Maxima

Periodic Solutions for an Impulsive System of Fractional Order Integro-Differential Equations with Maxima

A hybrid technique for approximating the solution of fractional order integro differential equations

In this article, we present an effective approach for solving nonlinear fractional order integro-differential equations. The fractional order derivative will be in the Caputo sense. For this, we propose an approach combined the least squares method together with the Laplace transform and the shifted Legendre polynomials. Using the proposed approach we will converts the nonlinear fractional order integro-differential equation into a se of (N+1) algebraic equations, where the solution to the resultant equation provides us with the unknown coefficients of the infinite series that have been used to approximate the solution to the considered fractional order integro-differential equations. From the results of the given examples, one can conclude that the recommended approach in this article is simple to implement and accurate.

On Existence and Uniqueness of Some Fractional Order Integro-Differential Equation

In this study, a sufficient condition for the existence and uniqueness of some fractional order Integral-Differential equations is obtained. Therefore, the fixed point method is used to solve the differential equation problem involving nonlinear degree integrals. In addition, the results found is supported by examples.

Solving nth-order Integro-differential Equations by Novel Generalized Hybrid Transform

Recently, Shehu has introduced an integral transform called Shehu transform, which generalizes the two well-known integrals transforms, i.e. Laplace and Sumudu transform. In the literature, many integral transforms were used to compute the solution of integro-differential equations (IDEs). In this article, for the first time, we use Shehu transform for the computation of solution of $n^{\text{th}}$-order IDEs. We present a general scheme of solution for $n^{\text{th}}$-order IDEs. We give some examples with detailed solutions to show the appropriateness of the method. We present the accuracy, simplicity, and convergence of the proposed method through tables and graphs.

Stability Analysis of First Order Integro-Differential Equations With the Successive Approximation Method

Stability Analysis of First Order Integro-Differential Equations With the Successive Approximation Method

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations.

On the null controllability of integer order integro-differential equations

Abstract This paper is addressed to the study of the null controllability for integer order integro-differential equations. Unlike the known results for partial differential equations, we need to consider the equation involving a $\beta -$power of the Laplace operator $(-\varDelta )^\beta $ and an integral term. The key point is to construct a suitable state space of the controlled system at the final time. We first discuss a class of hyperbolic integro-differential equation. We prove that the controlled system is null controllable by an Ingham-type estimate. Also, the controllability time is given. On the other hand, by reduction to absurdity, we deduce that the null controllability property fails for a class of parabolic integro-differential equation with $\beta \in \mathbb{N}^+$.

Extensions of some differential inequalities for fractional integro-differential equations via upper and lower solutions

This paper deals with some differential inequalities for generalized fractional integro-differential equations by using the technique of upper and lower solutions. The fractional differential operator is taken in Caputo’s sense and the nonlinear term divided into two parts depends on the fractional integrals of an unknown function with two different fractional orders. The results are studied by employing a variety of coupled upper and lower solutions. These theorems have some potential for extending the iterative techniques to fractional order integro-differential equations and to coupled systems of integro-differential fractional equations to obtain the existence of solutions as well as approximate solutions for the considered problem.

Impulsive fractional order integrodifferential equation via fractional operators.

In this paper, we establish the existence and uniqueness of the solution to fractional equations abstract integrodifferential equation with impulsive as [Formula: see text] where [Formula: see text] and we used the fixed point theorems due to Banach space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. Finally, in support, an example is presented to validate the obtained results.

Modification of Homotopy Perturbation Algorithm Through Least Square Optimizer for Higher Order Integro-Differential Equations

In this manuscript, modification of homotopy perturbation method (HPM) is proposed for integro-differential equations by coupling the least square method (LSM) with HPM. Improved accuracy in a very few iterations is the general advantage of this technique. The proposed method is applied to different higher order integro-differential equations of linear and nonlinear nature, and results are compared with exact as well as available solutions from the literature. Numerical and graphical analysis reveal that the proposed algorithm is reliable for integro-differential equations and hence can be utilized for more complex problems.

Study on Urban Proximity Prediction Based on Doppler Radar Gust Front Characteristics and Urban Microclimate Characteristics

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations

We consider systems of higher order integro-differential equations in which the matrix multiplying the highest derivative of the unknown vector function is identically singular in the domain where the system is considered. We give criteria for the solvability of such systems and initial value problems for these systems as well as examples illustrating the theoretical results.

Controllability and Observability Criteria for Systems Described by Fractional Differential Equations

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.