Fractional Integral Equations Research Articles

Bivariate barycentric rational interpolation method for two dimensional fractional Volterra integral equations

The advantages of the barycentric rational interpolation (BRI) introduced by Floater and Hormann include the stability of interpolation, no poles, and high accuracy for any sufficiently smooth function. In this paper we design a transformed BRI scheme to solve two dimensional fractional Volterra integral equation (2D-FVIE), whose solution may be non-smooth since its derivatives may be unbounded near the integral domain boundary. The transformed BRI method is constructed based on bivariate BRI and some smoothing transformations, hence inherits the advantages of the BRI even for a singular function. First, the smoothing transformations are employed to change the original 2D-FVIE into a new form, so that the solution of the new transformed 2D-FVIE has better regularity. Then the transformed equation can be solved efficiently by using the bivariate BRI together with composite Gauss–Jacobi quadrature formula. Last, some inverse transformations are used to obtain the solution of the original equation. The whole algorithm is easy to be implemented and does not require any integral computation. Besides, we analyze the convergence behavior via the transformed equation. Several numerical experiments are provided to illustrate the features of the proposed method.

On Nonlinear Hybrid Fractional Differential Equations with Atangana-Baleanu-Caputo Derivative

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.

Further results on the existence of solutions for generalized fractional quadratic functional integral equations

This paper discusses some existence results for at least one continuous solution for generalized fractional quadratic functional integral equations. Some results on nonlinear functional analysis including Schauder fixed point theorem are applied to establish the existence result for proposed equations. We improve and extend the literature by incorporated some well-known and commonly cited results as special cases in this topic. Further, we prove the existence of maximal and minimal solutions for these equations.

Existence of local fractional integral equation via a measure of non-compactness with monotone property on Banach spaces

In this paper, we discuss fixed point theorems for a new χ-set contraction condition in partially ordered Banach spaces, whose positive cone mathbb{K} is normal, and then proceed to prove some coupled fixed point theorems in partially ordered Banach spaces. We relax the conditions of a proper domain of an underlying operator for partially ordered Banach spaces. Furthermore, we discuss an application to the existence of a local fractional integral equation.

On the fuzzy stability results for fractional stochastic Volterra integral equation

<p style='text-indent:20px;'>By a fuzzy controller function, we stable a random operator associated with a type of fractional stochastic Volterra integral equations. Using the fixed point technique, we get an approximation for the mentioned random operator by a solution of the fractional stochastic Volterra integral equation.</p>

Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions

In this paper, we find fractional Riemann–Liouville derivatives for the Takagi–Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi–Landsberg functions, which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of weighted Takagi–Landsberg functions of order H > 0 on [0; 1] coincides with the class of H-Hölder continuous functions on [0; 1]. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a Hölder continuous function. This result allows us to get a new formula of a Riemann–Stieltjes integral. The application of such series representation is a new method of numerical solution of the Volterra and linear integral equations driven by a Hölder continuous function.

Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

Existence of positive solutions for weighted fractional order differential equations

Abstract In this paper, we deliberate two classes of initial value problems for nonlinear fractional differential equations under a version weighted generalized of Caputo fractional derivative given by Jarad et al. (2020a) [25]. We get a formula for the solution through the equivalent fractional integral equations to the proposed problems. The existence and uniqueness of positive solutions have been obtained by using lower and upper solutions. The acquired results are demonstrated by building the upper and lower control functions of the nonlinear term with the aid of Banach and Schauder fixed point theorems. The acquired results are demonstrated by pertinent numerical examples along with the Bashforth Moulton prediction correction scheme and Matlab.

State Dependent Versions of The Space-Time Fractional Poisson Process

Abstract In this paper, we introduce and study two counting processes by considering state dependency on the order of fractional derivative as well as on the exponent of backward shift operator involved in the governing difference-differential equations of the state probabilities of space-time fractional Poisson process. The Adomian decomposition method is employed to obtain their state probabilities and then their Laplace transforms are evaluated. Also, the compound versions of these state dependent models are studied and the corresponding governing fractional integral equations of their state probabilities are obtained.

Hyers‐Ulam‐Rassias stability results for some nonlinear fractional integral equations using the Bielecki metric

In this article, the Bielecki metric on the space is used to analyze the different types of stability results of nonlinear fractional integral equation with delay in its corresponding fractional boundary value problems. Sufficient conditions are obtained to prove stability results for fractional nonlinear Volterra and Fredholm integral equations with delay, given by Ulam, Hyer, and Rassias. Further, those stability results are extended to the fractional integral equations where the domain of integration is an unbounded interval. Two numerical examples are provided to assert the obtained stability results.

Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations

Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations

Hyers-Ulam stability of fuzzy fractional Volterra integral equations with the kernel ψ−function via successive approximation method

In this paper, we investigative the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of fuzzy fractional Volterra integral equations (FFVIEs) involving the kernel ψ−function. We propose conditions for the existence, uniqueness, and stability of FFVIE by employing the method of successive approximation. To visualize the theoretical results, some examples and numerical simulations are given.

Numerical analysis of fractional Volterra integral equations via Bernstein approximation method

In this study, Bernstein approximation method has been applied along with Riemann–Liouville fractional integral operator to solve both the second and the first kind of fractional Volterra integral equations (2nd FVIEs and 1st FVIEs respectively). In order to show the applicability and efficiency of the proposed technique, some convergence analysis has been provided. Illustrative numerical experiments with comparison are included to indicate the validity and practicability of the method. All of the numerical calculation in this research has been done on a personal computer implementing some programs written in MATLAB.

Research on the Application of Fractional differential integral equation in the calculation of shooting rate and the Construction of influencing factors

Research on the Application of Fractional differential integral equation in the calculation of shooting rate and the Construction of influencing factors

The fractional Green's function by Babenko's approach

The goal of this paper is to derive the fractional Green's function for the first time in the distributional space for the fractional-order integro-differential equation with constant coefficients. Our new technique is based on Babenko's approach, without using any integral transforms such as the Laplace transform along with Mittag-Leffler function. The results obtained are not only much simpler, but also more generalized than the classical ones as they deal with distributions which are undefined in the ordinary sense in general. Furthermore, several interesting applications to solving the fractional differential and integral equations, as well as in the wave reaction-diffusion equation are provided, some of which cannot be achieved by integral transforms or numerical analysis.

A new general integral transform for solving integral equations

IntroductionIntegral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. ObjectivesIn this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\\_transforms, Pourreza, Aboodh and etc. MethodsA new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. ResultsIt is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. ConclusionIt has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

An Efficient Method Based on Framelets for Solving Fractional Volterra Integral Equations

This paper is devoted to shedding some light on the advantages of using tight frame systems for solving some types of fractional Volterra integral equations (FVIEs) involved by the Caputo fractional order derivative. A tight frame or simply framelet, is a generalization of an orthonormal basis. A lot of applications are modeled by non-negative functions; taking this into account in this paper, we consider framelet systems generated using some refinable non-negative functions, namely, B-splines. The FVIEs we considered were reduced to a set of linear system of equations and were solved numerically based on a collocation discretization technique. We present many important examples of FVIEs for which accurate and efficient numerical solutions have been accomplished and the numerical results converge very rapidly to the exact ones.

On the existence of continuous solutions of a nonlinear quadratic fractional integral equation

We prove an existence theorem for a nonlinear quadratic integral equation of fractional order, in the Banach space of real functions defined and continuous on a closed interval. This equation contains as a special case numerous integral equation studied by other authors. Finally, we give an example for indicating the natural realizations of our abstract result presented in this paper.

New stochastic operational matrix method for solving stochastic Itô–Volterra integral equations characterized by fractional Brownian motion

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the proposed scheme.

A discussion on a generalized Geraghty multi-valued mappings and applications

This research intends to investigate the existence results for both coincidence points and common fixed point of generalized Geraghty multi-valued mappings endowed with a directed graph. The proven results are supported by an example. We also consider fractional integral equations as an application.