Order Volterra Integro-differential Equations Research Articles

A new application of the reproducing kernel method for solving linear systems of fractional order Volterra integro-differential equations

In this article, a new implementation of the reproducing kernel method is presented for solving systems of fractional-order Volterra integro-differential equations. Unlike previous implementations, this method does not rely on the Gram-Schmidt process. The reproducing kernel method utilizes various components, including space, inner product, bases, and points. Furthermore, the system of fractional-order Volterra integro-differential equations involves Caputo’s fractional derivative and Volterra integral. However, when using the reproducing kernel method to solve these systems, challenges such as longer execution time and lower accuracy may arise compared to other methods. The present method has overcome these challenges with features such as easy implementation, high accuracy, and lower execution time.

A developed technique of homotopy analysis method for solving nonlinear systems of Volterra integro-differential equations of fractional order

In this paper, a developed technique of homotopy analysis method (HAM) is presented to solve nonlinear system of fractional order Volterra integro-differential equations. The fractional derivative is described in the Caputo form. The main thing we have done in this paper is that we have simplified the process of evaluating the fractional order integrations in the deformation equations than the standard HAM. The proposed technique is tested by some illustrative examples in order to demonstrate its accuracy, applicability and efficiency.

On the convergence analysis of efficient numerical schemes for singularly perturbed second order Volterra integro-differential equations

On the convergence analysis of efficient numerical schemes for singularly perturbed second order Volterra integro-differential equations

On Stability and Asymptotic Stability of Solutions for a Linear Fourth Order Volterra Integro-Differential Equation on the Half-Axis

On Stability and Asymptotic Stability of Solutions for a Linear Fourth Order Volterra Integro-Differential Equation on the Half-Axis

Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay

Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay

Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

<abstract> <p>In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.</p> </abstract>

Walsh functions and their applications to solving nonlinear fractional Volterra integro-differential equation

In this article‎, ‎we extended an efficient computational‎ ‎method based on Walsh operational matrix to find an approximate‎ ‎solution of nonlinear fractional order Volterra integro-differential‎ ‎equation‎, ‎First‎, ‎we present the fractional Walsh operational‎ ‎matrix of integration and differentiation‎. ‎Then by applying this‎ ‎method‎, ‎the nonlinear fractional Volterra integro-differential‎ ‎equation is reduced into a system of algebraic equation‎. ‎The‎ ‎benefits of this method are the low-cost of setting up the equations‎ ‎without applying any projection method such as collocation‎, ‎Galerkin‎, ‎etc‎. ‎The results show that the method is very accuracy and ‎efficiency.‎

Numerical solution of two-dimensional fractional order Volterra integro-differential equations

The present paper is concerned with the implementation of the optimal homotopy asymptotic method to find the approximate solutions of two-dimensional fractional order Volterra integro-differential equations. The technique’s applicability and validity are tested through some numerical examples. The fractional order derivatives are calculated using Caputo’s sense. Results obtained by the proposed technique are compared with the Legendre wavelet method. The proposed method provides us with efficient and more accurate solutions than the other existing methods in the literature. Error analysis and convergence of the proposed method are also provided in the paper.

New approach to approximate the solution for the system of fractional order Volterra integro-differential equations

The main aim of this article is the extension of Optimal Homotopy Asymptotic Method to the system of fractional order integro-differential equations. The systems of fractional order Volterra integro-differential equations (SFIDEs) are taken as test examples. The fractional order derivatives are defined in the Caputo fractional form and the optimal values of auxiliary constants are calculated using the well-known method of least squares. The results obtained by proposed scheme are very encouraging and show close resemblance with exact values. Hence it will be more appealing for the researchers to apply the proposed scheme to different fractional order systems arising in different fields of sciences especially in fluid dynamics and bio-engineering.

On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis

In this work we consider a class of fractional order Volterra integro-differential equations of first kind where the fractional derivative is considered in the Caputo sense. Here, we consider the initial value problem and the boundary value problem separately. For simplicity of the analysis, we reduce each of these problems to the fractional order Volterra integro-differential equation of second kind by using the Leibniz’s rule. We have obtained sufficient conditions for the existence and uniqueness of the solutions of initial and the boundary value problems. An operator based method has been considered to approximate their solutions. In addition, we provide a convergence analysis of the adopted approach. Several numerical experiments are presented to support the theoretical results.

Numerical solution to Volterra Integro-Differential Equations of the second kind by hybrid one-step block method

The hybrid block one-step method of order four is presented and implemented to solve first order Volterra Integro-Differential Equations (VIDEs). The technique is developed using Lagrange interpolation method. The numerical solutions of VIDEs will be solved at two-point concurrently using the proposed numerical method. Properties of the method such as consistency, order and region of absolute stability are explored. After that, the linear and nonlinear of VIDEs has been solved and the numerical results show that the accuracy and efficiency of the hybrid block method is better than the other methods which is RK4 and VIM.

A new numerical method for fractional order Volterra integro-differential equations

Abstract In this paper, a new numerical method based on the Bernoulli wavelet approximation is presented for the solution of fractional order Volterra integro-differential equations. The operational matrix of fractional derivative of order λ in the Caputo sense derived together with the Gaussian quadrature rule implemented to reduce the Volterra integro-differential equations to system of algebraic equations. Convergence analysis and error estimation of the proposed method have been discussed. The efficiency and accuracy of the proposed method have been depicted through some illustrative examples.

Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations

Abstract This paper focuses on providing a novel high-order algorithm for the numerical solutions of fractional order Volterra integro-differential equations using Atangana–Baleanu approach by employing the reproducing kernel approximation. For this purpose, we investigate couples of Hilbert spaces and kernel functions, as well as, the regularity properties of Atangana–Baleanu derivative, and utilize that the representation theorem of its solution. To remove the singularity in the kernel function, using new Atangana–Baleanu approach the main operator posses smoothing solution with a better regularity properties and the reproducing kernel algorithm is designed for the required equation. The convergence properties of the proposed algorithm are also studied which proves that the new strategy exhibits a high-order of convergence with decreasing error bound. Some numerical examples of single and system formulation illustrate the performance of the approach. Summary and some notes are also provided in the case of conclusion and highlight.

Convergence Analysis of Legendre-Collocation Spectral Methods for Second Order Volterra Integro-Differential Equation with Delay

Convergence Analysis of Legendre-Collocation Spectral Methods for Second Order Volterra Integro-Differential Equation with Delay

Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz–Jacobi Tau method

This paper presents a spectral Tau method based on Müntz–Jacobi basis functions for approximating the solutions of fractional order Volterra integro-differential algebraic equations. We examine the solvability of the considered equation, and we show that the exact solutions suffer from a singularity at the origin. Numerical solvability and convergence analysis of the proposed approximate approach are also studied, and the error bounds are obtained in a weighted L 2 -norm. Some illustrative examples are presented to validate the theoretical results.

A Computational Method for Volterra Integro-Differential Equation

In this paper, we examine the initial value problem for a linear first order Volterra integro-differential equation. In order to solve the problem computationally, we present a novel finite difference method, which is based on the method of integral identities with the use of the basis functions and interpolating quadrature rules with remainder term in integral form. Furthermore, as a consequence of error analysis the method is proved to be first-order convergent in the discrete maximum norm. Finally, an example is provided to support our theoretical results.

An efficient algorithm for solving Volterra integro-differential equations based on Alpert’s multi-wavelets Galerkin method

This paper is concerned with the application of the multi-wavelets Galerkin method to linear and nonlinear second order Volterra integro-differential equations. Using the operational matrices of integration and wavelet transform matrix, we reduce the problem to a set of algebraic equations. For linear type, we use thresholding to reduce the computational effort. We demonstrate that our solutions may be efficiently represented in a multi-wavelets basis, due to flexible vanishing moments property of this type of multi-wavelets. The L 2 convergence of the scheme for the proposed equation has been investigated. A series of numerical tests is provided to demonstrate the validity and applicability of the technique.

A Fitted Approximate Method for a Volterra Delay-Integro-Differential Equation with Initial Layer

This study is concerned with the finite-difference solution of singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay. The method is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. The emphasis is on the convergence of numerical method. It is shown that the method displays uniform convergence in respect to the perturbation parameter. Numerical results are also given.

Numerical Solutions of Non-Linear System of Higher Order Volterra Integro-Differential Equations using Generalized STWS Technique

In this article, we deal with non-linear system of higher order Volterra integro-differential equations and their numerical solutions using the Single Term Walsh Series (STWS) method. The connections between STWS coefficients of the unknown functions and its derivatives are derived. The non-linear system of Volterra integro-differential equations are converted into a system of non-linear algebraic equations using the Single Term Walsh Series coefficients. Solving these system of algebraic equations, we obtain the discrete numerical solutions of the non-linear Volterra integro-differential equations. Numerical examples are presented to show the efficiency and applicability of this STWS method for solving the non-linear system of higher order Volterra integro-differential equations.

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Indeed, interesting properties of artificial neural networks approach made this non-parametric model a powerful tool in solving various complicated mathematical problems. The current research attempts to produce an approximate polynomial solution for special type of fractional order Volterra integro-differential equations. The present technique combines the neural networks approach with the power series method to introduce an efficient iterative technique. To do this, a multi-layer feed-forward neural architecture is depicted for constructing a power series of arbitrary degree. Combining the initial conditions with the resulted series gives us a suitable trial solution. Substituting this solution instead of the unknown function and employing the least mean square rule, converts the origin problem to an approximated unconstrained optimization problem. Subsequently, the resulting nonlinear minimization problem is solved iteratively using the neural networks approach. For this aim, a suitable three-layer feed-forward neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. In other words, discretizing the differential domain with a classical rule produces some training rules. By importing these to designed architecture as input signals, the indicated learning algorithm can minimize the defined criterion function to achieve the solution series coefficients. Ultimately, the analysis is accompanied by two numerical examples to illustrate the ability of the method. Also, some comparisons are made between the present iterative approach and another traditional technique. The obtained results reveal that our method is very effective, and in these examples leads to the better approximations.