A reliable strategy for a category of third-kind nonlinear fractional integro-differential equations

Purpose This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Design/methodology/approach The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations. Findings Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods. Originality/value The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

The aim of this paper is to obtain approximate solution of a class of nonlinear fractional Fredholm integro-differential equations by means of sinc-collocation method which is not used for solving them in the literature before. The fractional derivatives are defined in the Caputo sense often used in fractional calculus. The important feature of the present study is that obtained results are stated as two new theorems. The introduced method is tested on some nonlinear problems and it seems that the method is a very efficient and powerful tool to obtain numerical solutions of nonlinear fractional integro-differential equations.

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

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.‎

In this article, we introduce a class of stochastic matrix control functions to stabilize a nonlinear fractional Volterra integro-differential equation with Ψ-Hilfer fractional derivative. Next, using the fixed-point method, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability of the nonlinear fractional Volterra integro-differential equation in matrix MB-space.

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

In this article, we consider a boundary value problem for a nonlinear partial integro-differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with positive spectral parameter in a negative rectangular domain. The partial integro-differential equation of mixed type depends on another real spectral parameter in integral part of the mixed equation. With respect to first variable this equation is a fractional integro-differential equation in the positive part of the considering segment, and is a second-order integro-differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method of separation variables and Fredholm method of degenerate kernels, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of solution of the problem are proved for regular values of the spectral parameters.

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.

This paper is concerned with the controllability of nonlinear higher order fractional delay integrodifferential equations with time varying delay in control, which involved Caputo derivatives of any different orders. A formula for the solution expression of the system is derived by using Laplace transform. A necessary and sufficient condition for the relative controllability of linear fractional delay dynamical systems with time varying delays in control is proved, and a sufficient condition for the corresponding nonlinear integrodifferential equation has obtained. Examples has given to verify the results.

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

Numerical simulation for coupled systems of nonlinear fractional order integro-differential equations via wavelets method

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.

This study explored the challenges of solving nonlinear stochastic fractional integro-differential equations on time scales, which are critical in various applied mathematics and engineering problems. The complexity of these equations and the lack of efficient analytical methods highlighted the need for an improved computational approach.The aim of the research was to develop a hybrid spectral-Adomian decomposition method to solve these types of equations with enhanced accuracy and efficiency. The focus was on bridging existing methodological gaps and providing a robust tool for researchers dealing with stochastic fractional systems. The method involved integrating the spectral method with the Adomian decomposition technique to create a hybrid algorithm. Numerical experiments were carried out on benchmark problems to assess the performance of the proposed method, and computational tests were implemented using MATLAB.The results indicated that the hybrid spectral-Adomian method achieved superior accuracy and faster convergence compared to conventional methods. The numerical solutions closely matched exact results where available, confirming the method’s validity and reliability.The study concluded that the hybrid method is a highly effective approach for solving nonlinear stochastic fractional integro-differential equations on time scales. It demonstrated significant improvements over traditional numerical schemes, making it a valuable addition to computational mathematics.This research contributed a new hybrid numerical method that combines the strengths of spectral accuracy with the flexibility of the Adomian decomposition method. The findings are expected to benefit the field of applied mathematics, particularly in advancing solution techniques for complex stochastic systems, aligning with the journal’s scope.