Fractional Volterra Integro-differential Equations Research Articles

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.

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.

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

Analytical and numerical techniques for solving a fractional integro-differential equation in complex space

&lt;p&gt;In this article, we describe the existence and uniqueness of a solution to the nonlinear fractional Volterra integro differential equation in complex space using the fixed-point theory. We also examine the remarkably effective Euler wavelet method, which converts the model to a matrix structure that lines up with a system of algebraic linear equations; this method then provides approximate solutions for the given problem. The proposed technique demonstrates superior accuracy in numerical solutions when compared to the Euler wavelet method. Although we provide two cases of computational methods using MATLAB R2022b, which could be the final step in confirming the theoretical investigation.&lt;/p&gt;

Convergence analysis and numerical implementation of projection methods for solving classical and fractional Volterra integro-differential equations

In this article, we discuss the convergence analysis of the classical first-order and fractional-order Volterra integro-differential equations of the second kind with a smooth kernel by reducing them into a system of fractional Fredholm integro-differential equations (FFIDEs). For that, we first reformulate the given equation into a system of fractional Volterra integro-differential equations and then transform it into the system of FFIDEs using a simple transformation. We develop a general framework of the newly defined iterated Galerkin method for the reduced system of equations and investigate the existence and uniqueness of the approximate solutions in the given Banach space. We provide the error estimates and convergence analysis for the iterated Galerkin approximate solutions in the supremum norm without any limiting conditions. Further, we provide the superconvergence results for classical first-order and fractional-order Volterra integro-differential equations by proposing a general framework of multi-Galerkin and iterated multi-Galerkin methods for the reduced system of equations. Moreover, we prove that the order of convergence of the proposed methods increases theoretically and numerically with the increasing order of the fractional derivatives. Finally, numerical implementations and illustrative examples are provided to demonstrate our theoretical aspects.

Enhanced moving least squares method for solving the stochastic fractional Volterra integro-differential equations of Hammerstein type

Enhanced moving least squares method for solving the stochastic fractional Volterra integro-differential equations of Hammerstein type

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

On stability analysis of a fractional volterra integro delay differential equation in the context of Mittag–Leffler kernel

This paper deals with a nonlinear multi-derivative fractional Volterra integro-differential equation with finite delay involving a non-singular Mittag–Leffler kernel. Krasnoselskii's fixed-point theorem is used to obtain the current result. Also, we establish the generalization of Pachapate's inequality and provide its applications for the existence, uniqueness, stability, and data dependence of the suggested problem. All the results are justified by an example.

A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

SHIFTED LEGENDRE FRACTIONAL PSEUDO-SPECTRAL INTEGRATION MATRICES FOR SOLVING FRACTIONAL VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS AND ABEL’S INTEGRAL EQUATIONS

Shifted Legendre polynomials (SLPs) with the Riemann–Liouville fractional integral operator have been used to create a novel fractional integration tool. This tool will be called the fractional shifted Legendre integration matrix (FSL B-matrix). Two algorithms depending on this matrix are designed to solve two different types of integral equations. The first algorithm is to solve fractional Volterra integro-differential equations (VIDEs) with a non-singular kernel. The second algorithm is for Abel’s integral equations. In addition, error analysis for the spectral expansion has been proven to ensure the expansion’s convergence. Finally, several examples have been illustrated, including an application for the population model.

An $ hp $-version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels

&lt;abstract&gt;&lt;p&gt;We present a multi-step spectral collocation method to solve Caputo-type fractional integro-differential equations (FIDEs) involving weakly singular kernels. We reformulate the problem as the second type Volterra integral equation (VIE) with two different weakly singular kernels. Based on these integral equations, we construct a multi-step Legendre-Gauss spectral collocation scheme for the problem. The $ hp $-version convergence is established rigorously. To demonstrate the effectiveness of the suggested method and the validity of the theoretical results, the results of some numerical experiments are presented.&lt;/p&gt;&lt;/abstract&gt;

Fractional Linear Volterra Integro-Differential Equations in Banach Spaces

The paper presents the foundations of the theory of linear fractional Volterra integro-differential equations of convolution type in Banach spaces. It is established that the existence of a fractional resolvent operator for such equations is equivalent to the well-posedness of the formulation of the initial problem for them. Within the framework of this approach, a theorem of the Hille–Yosida type is proved.

Collocation Method for Solving Two-Dimensional Fractional Volterra Integro-Differential Equations

Collocation Method for Solving Two-Dimensional Fractional Volterra Integro-Differential Equations

On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels

On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels

Solution of optimal control problems governed by volterra integral and fractional integro-differential equations

In this manuscript, we investigate two categories of optimal control problems (OCPs), OPCs via fractional Volterra integro-differential equations and Volterra integral equations. Touchard wavelets as an appropriate class of bases are defined to develop a new hybrid scheme for the considered problems. To this approach, Riemann–Liouville fractional integral operator (RLFIO) of Touchard wavelets is achieved exactly using the Hypergeometric functions. Next, by approximating the fractional derivative of the state variables and control variables using the mentioned wavelet functions, applying RLFIO, collocation method, and Gauss–Legendre quadrature formula, the considered problems are inserted into systems of algebraic equations, which can be solved using “FindRoot” package in Mathematica software. Numerical results are presented that validate the theory and show the effectiveness of the established technique.

Laplace Transform for Solving System of Integro-Fractional Differential Equations of Volterra Type with Variable Coefficients and Multi-Time Delay

This study is the first to use Laplace transform methods to solve a system of Caputo fractional Volterra integro-differential equations with variable coefficients and a constant multi-time delay. This technique is based on different types of kernels, which we will explain in this paper. Symmetry kernels, which have properties of difference kernels or simple degenerate kernels, are able to compute analytical work. These are demonstrated by solving certain examples and analyzing the effectiveness and precision of cause techniques.

Lipschitz continuity in the Hurst index of the solutions of fractional stochastic volterra integro-differential equations

The problem of investigating the continuity in the Hurst index arises naturally in statistical inferences related to fractional Brownian motion. In this paper, based on the techniques of the Malliavin calculus, we introduce a method to deal with this problem. We first provide an explicit bound on the difference between two non-smooth functionals of Malliavin differentiable random variables. Then, we apply the obtained bound to show Lipchitz continuity of fractional stochastic Volterra integro-differential equations and its additive functionals.

Modification of Sumudu Decomposition Method for Nonlinear Fractional Volterra Integro-Differential Equations

In this paper, modification of Sumudu decomposition method is successfully applied to find the approximate solution of nonlinear Volterra integro-differential equation of fractional order. The proposed method is based on the combining of two powerful techniques, Sumudu decomposition method and Daftardar-Gejji and Jafari (DGJ) method. Several illustrative examples are given to demonstrate the validity, reliability, and efficiency of the proposed technique.

A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations

&lt;abstract&gt;&lt;p&gt;A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.&lt;/p&gt;&lt;/abstract&gt;

Existence and uniqueness of mild and strong solutions of nonlinear fractional integrodifferential equation

In this paper, we will discuss some results on the existence and uniqueness of mild and strong solution of initial value problem of fractional order subjected to non-local conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family under Caputo sense. Furthermore, we also prove that solution of Nonlinear Fractional Volterra Integrodifferential Equations and Nonlinear Fractional Mixed Integrodifferential Equations With Nonlocal Conditions is unique. Moreover, examples demonstrate the validity of the obtained main result and we obtain the solution for an equation, and proved that this solution is unique.