Nonlinear Fractional Integro-differential Equations Research Articles

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.

Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method

The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided.

New Existence Results for Nonlinear Fractional Integrodifferential Equations

This paper discusses a boundary value problem of nonlinear fractional integrodifferential equations of order 1 < α ≤ 2 and 1 < β ≤ 2 and boundary conditions of the form x 0 = x 1 = D c β x 1 = D c β x 0 = 0 . Some new existence and uniqueness results are proposed by using the fixed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.

Spectral collocation method for nonlinear Riemann–Liouville fractional differential system

The spectral collocation method is investigated for the system of nonlinear Riemann–Liouville fractional differential equations (FDEs). The main idea of the presented method is to solve the corresponding system of nonlinear weakly singular Volterra integral equations obtained from the system of FDEs. In order to carry out convergence analysis for the presented method, we investigate the regularity of the solution to the system of FDEs. The provided convergence analysis result shows that the presented method has spectral convergence. Theoretical results are confirmed by numerical experiments. The presented method is applied to solve multi-term nonlinear Riemann–Liouville fractional differential equations and multi-term nonlinear Riemann–Liouville fractional integro-differential equations.

Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space

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.

Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

Existence results for nonlinear fractional integro-differential equations with integral and antiperiodic boundary conditions

The objective of this paper is to investigate the existence and uniqueness for a nonlinear fractional integro-differential equations with integral and anti-periodic boundary conditions by means of fixed point theorems. The existence of solutions is obtained from the well known Krasnoselskii’s fixed point theorem, whereas the uniqueness of solutions is established via the Banach’s contraction principle. Two examples are given to validate the results.

A numerical technique for solving nonlinear fractional stochastic integro-differential equations with n-dimensional Wiener process

This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative and a suitable numerical integration rule. Error estimate of the method is investigated in detail. At the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.

New results on nonlocal fractional Volterra-Fredholm integro-differential equations

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence and uniqueness of solutions. Results of this paper are based on nonstandard assumptions and hypotheses and provide a supplementary result concerning the regularity of solutions.

Existence of positive solutions for boundary value problems of nonlinear fractional functional integro-differential equations

In this paper, we study the existence of positive solutions for a class of fractional functional integro-differential equations with two fractional derivative terms. First, we transform the boundary value problem into an equivalent integral equation, establish the operator T and prove its full continuity, then the existence theorems of positive solutions of boundary value problems is established by using the fixed point theorems of cone extension and cone compression.

Existence results for nonlinear fractional-order multi-term integro-multipoint boundary value problems

We investigate the existence of solutions for integro-multipoint boundary value problems involving nonlinear multi-term fractional integro-differential equations. The case involving three different types of nonlinearities is also briefly described. The desired results are obtained by applying the methods of modern functional analysis and are well-illustrated with examples.

On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these parameters increase the complexity, they present more acceptable solutions. With the aid of Atangana-Baleanu-Caputo (ABC) fractional differential operator, an advanced numerical-analysis approach is considered and applied in this work to deal with different classes of fuzzy integrodifferential equations of fractional order fitted with uncertain constraints conditions. The fractional derivative of ABC is adopted under the generalized H-differentiability (g-HD) framework, which uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale of the fuzzy models. Towards this end, applications of reproducing kernel algorithm are extended to solve classes of linear and nonlinear fuzzy fractional ABC Volterra-Fredholm integrodifferential equations. Based on the characterization theorem, preconditions are established under the Lipschitz condition to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integrodifferential equations. Parametric solutions of the ABC interval are provided in terms of rapidly convergent series in Sobolev spaces. Several examples of fuzzy ABC Volterra-Fredholm models are implemented in light of g-HD to demonstrate the feasibility and efficiency of the designed algorithm. Numerical and graphical representations of both classical Caputo and ABC fractional derivatives are presented to show the effect of the ABC derivative on the parametric solutions of the posed models. The achieved results reveal that the proposed method is systematic and suitable for dealing with the fuzzy fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

Existence and Attractivity Theorems for Nonlinear Hybrid Fractional Integrodifferential Equations with Anticipation and Retardation

In this paper, we establish the existence and a global attractivity results for a nonlinear mixed quadratic and linearly perturbed hybrid fractional integrodifferential equation of second type involving the Caputo fractional derivative on unbounded intervals of real line with the mixed arguments of anticipations and retardation. The hybrid fixed point theorem of Dhage is used in the analysis of our nonlinear fractional integrodifferential problem. A positivity result is also obtained under certain usual natural conditions. Our hypotheses and claims have also been explained with the help of a natural realization.

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

Multivalued Problems, Orthogonal Mappings, and Fractional Integro-Differential Equation

In this manuscript, we propose some sufficient conditions for the existence of solution for the multivalued orthogonal ℱ -contraction mappings in the framework of orthogonal metric spaces. As a consequence of results, we obtain some interesting results. Also as application of the results obtained, we investigate Ulam’s stability of fixed point problem and present a solution for the Caputo-type nonlinear fractional integro-differential equation. An example is also provided to illustrate the usability of the obtained results.

Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral type

This paper addresses the study of nonlinear fractional stochastic neutral integro-differential equations with Poisson jumps and infinite delay in Hilbert space. Initially, the existence and uniqueness of mild solution for the proposed system are verified by applying the suitable conditions of sectorial operators, fixed point technique and stochastic analysis with non-Lipschitz condition and Lipschitz condition as a special case. Also, continuous dependence of mild solutions for proposed model is determined. Finally an example is provided to illustrate the developed theory.

On a Nonlocal Multipoint and Integral Boundary Value Problem of Nonlinear Fractional Integrodifferential Equations

The aim of this paper is to give the existence as well as the uniqueness results for a multipoint nonlocal integral boundary value problem of nonlinear sequential fractional integrodifferential equations. First of all, we give some preliminaries and notations that are necessary for the understanding of the manuscript; second of all, we show the existence and uniqueness of the solution by means of the fixed point theory, namely, Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Last, but not least, we give two examples to illustrate the results.

Analysis of Fractional Integro-Differential Equations with Nonlocal Erdélyi-Kober Type Integral Boundary Conditions

Abstract In this article, we study the existence and uniqueness of solutions for nonlinear fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions. The existence results are based on Krasnoselskii’s and Schaefer’s fixed point theorems, whereas the uniqueness result is based on the contraction mapping principle. Examples illustrating the applicability of our main results are also constructed.

On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses

In this paper, we consider the initial boundary value problem for a class of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses in Banach spaces. Sufficient conditions of existence and uniqueness of PC-mild solutions for the equations are obtained via general Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and α-order solution operator.