Nonlinear Fractional Integro-differential Equations Research Articles

On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditions

On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditions

Existence and uniqueness of mild solutions for nonlinear hybrid Caputo fractional integro-differential equations via fixed point theorems

We prove the existence and uniqueness of mild solutions for initial valueproblems of nonlinear hybrid first order Caputo fractionalintegro-differential equations. The main tool employed here is theKrasnoselskii and Banach fixed point theorems. An example is also given toillustrate the main results. In addition, the case of the Higher orderCaputo fractional integro-differential equations is studied.

Some qualitative properties of nonlinear fractional integro-differential equations of variable order

The existence-uniqueness criteria of nonlinear fractional integro-differential equations of variable order with multiterm boundary value conditions are considered in this work. By utilizing the concepts of generalized intervals combined with the piecewise constant functions, we transform our problem into usual Caputo’s fractional differential equations of constant order. We develop the necessary criteria for assuring the solution's existence and uniqueness by applying Schauder and Banach fixed point theorem. We also examine the stability of the derived solution in the Ulam-Hyers-Rassias (UHR) sense and provide an example to demonstrate the credibility of the results.

Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations

Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations

Fast‐time complete controllability of nonlinear fractional delay integrodifferential evolution equations with nonlocal conditions and a parameter

In this work, new controllability results for a class of nonlinear fractional delay integrodifferential evolution equations with nonlocal conditions and a parameter have been derived under a new concept that we define as fast‐time complete controllability. Nonlinearity here is only supposed to be continuous rather than Lipschitz continuous by contrast. The major tools we adopt are resolvent operator theory and the theory of nonlinear functional analysis. Theoretical and practical applications are presented to illustrate the effectiveness of the new abstract results.

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

Fractional Chebyshev cardinal wavelets: application for fractional quadratic integro-differential equations

This paper introduces a new set of the basis functions called the fractional Chebyshev cardinal wavelets and details their properties. These wavelets have a greater degree of freedom than the classical Chebyshev cardinal wavelets. Moreover, they retain the cardinality and the spectral accuracy of these wavelets. The fractional derivative and integral matrices of these fractional basis functions are obtained exactly in the explicit forms. In this study, we aim to devise an efficient and powerful approximation method using these new basis functions. Then, we employ them for a new category of nonlinear fractional quadratic integro-differential equations. By employing their fractional integral matrix and their cardinality, the problem under study is transformed into solving a nonlinear system of algebraic equations. The error analysis of the presented technique is first investigated theoretically and then computational efficiency is examined for two numerical examples.

Hyers–Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives

• In this paper, we investigate the boundary value problem with Riemann-Liouville fractional differential equations for a class of impulsive implicit nonlinear fractional order integro-differential equations. • We are using some sufficient conditions to achieve the existence and uniqueness results of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem and Kranoselskii’s fixed point theorem. • Moreover, we present different types of stability such as Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability using the classical technique of functional analysis. • To illustrate our theoretical results, at the end we give an example. In this article, we investigate the existence, uniqueness, and stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives. We analyze the existence and uniqueness of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem. Moreover, we present different types of stability using the classical technique of functional analysis. To illustrate our theoretical results, at the end we give an example.

A qualitative study on generalized Caputo fractional integro-differential equations

The aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form (I_{a}^{rho }f)(t)=int _{a}^{t}f(s)s^{rho -1},ds. Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.

A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

Theoretical and computational analysis of nonlinear fractional integro-differential equations via collocation method

In this article, a class of nonlinear Volterra-Fredholm fractional integro-differential equations is considered, both theoretical and computational aspects. The respective theoretical results are devoted to the existence of a solution via fixed point approach. Further, for the computational aspect, the Proposed Methodology of Haar wavelet collocation. This method minimizes a system of nonlinear algebraic equations, which is developed by Broyden’s method. In literature, the proposed method is taken for checking the convergence with help of some numerical examples. Calculate mean square root and maximum absolute errors for various collocation point numbers. The final outcomes show that the applied Haar method is effective, and the convergence rate for different collocation point is roughly equal to 2.

Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

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.

The Modified Sadik Decomposition Method to Solve a System of Nonlinear Fractional Volterra Integro-Differential Equations of Convolution type

In this work, an incorporated form of Sadik transform and Adomian decomposition method which is called the Sadik decomposition method is presented. The method is applied to solve a system of nonlinear fractional Volterra integro-differential equations in the convolution form. To avoid collecting the noise terms that lead the method to fail for seeking the solution, the proposed method is modified by selecting a suitable initial solution. The obtained results are expressed in the explicit form of a power series with easily computable terms. In addition, illustrative examples are shown to demonstrate the effectiveness of the method.

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.‎

New Existence of Solutions for Fractional Integro-Differential Equations with Nonseparated Boundary Conditions

Results reported in this article prove the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations supplemented by nonseparated boundary value conditions. We consider a new norm to establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle. Some examples are provided to illustrate the results.

Existence and stability of solution for nonlinear differential equations with [formula omitted]-Hilfer fractional derivative

In this paper, we investigate a nonlinear ψ-Hilfer fractional integro-differential equation on an unbounded domain [a,+∞). Firstly, the existence and uniqueness of solution for the equation are proved by means of Banach contraction mapping principle based on suitable growth conditions in an appropriate Banach space. Moreover the stability of Ulam–Hyers–Rassias, Ulam–Hyers and Semi-Ulam–Hyers–Rassias to the initial value problem is obtained.

Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay

The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results.

To study existence of at least three weak solutions to a system of over-determined Fredholm fractional integro-differential equations

In this article, the weak solutions of a class of nonlinear system of fractional boundary value problems including integral equations is investigated. Each equation in the system is either a nonlinear fractional partial integro-differential equation containing a gradient of a nonlinear source term or a Fredholm integral equation of the second kind. It is proved that there exist at least three distinct weak solutions by applying the critical point theory and the variational structure. In order to achieve this purpose, we use the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Furthermore, our main result is demonstrated by an illustrative example to show its legitimacy and applicability.

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

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.