Solutions Of Fractional Integral Equations Research Articles

The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

Solution of fractional integral equations via fixed point results

In this paper, we introduce two new concepts of F-contraction, called dual F^{*}-weak contraction and triple F^{*}-weak contraction, which generalize the existing contractions in the sense of Wardowski, Jleli and Samet as well as Skof. These new generalizations embed their roots in the aim devoted to extending the generalized Banach contraction conjuncture to the class of F-contraction type mappings with the use of multiple F-type functions. Furthermore, we establish the existence of a unique fixed point for such contractions under certain conditions. Fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates that can be expressed as functions of one independent variable. We apply our main result to weaken certain conditions on the fractional integral equations. Finally, we discuss the significance of our obtained results in comparison with certain renowned ones in the literature.

Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations

Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations

Integral Equations of Non-Integer Orders and Discrete Maps with Memory

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

Fractional Order Integrals for the Sustainable Development Model

Sustainable development is now engaged in many leading countries. The notion is connected with the activity of countries, their economies, social and political institutions. One of the criteria of sustainability is the economic sphere, the study of which will allow you to control and predict its activity. It is known that the description of continuous processes that arise in economic problems, using definite integrals. In the case when the studied continuous development is carried out in small jumps, when it is the description of the use of fractional integral and differential equations. However, their exact solution is possible only in special cases. Therefore, the construction of approximate methods for solving fractional integral and differential equations is of great importance. In this paper, we give an approximate solution of fractional integral equations of the first kind, where the fractional integral is constructed using the Weyl fractional integral. It turns out that the constructed operator of fractional integration is symmetrical and positively definite. These properties are built by the operator enables to introduce the scalar product on the basis of the specified operator and the corresponding energy space. Next, using the same methodology that is used in the finite element method, approximate solution is constructed, constructed a rough scheme of the method and rationale of the proposed method to construct the energy space. The integral equation of the standard form is given as a specific example, as well as an example solution to illustrate the effectiveness of the proposed method.

Best Approximations of Solutions of Fractional-integral Equations with the Riemann-Liouville Operator

The article is devoted to the best approximations of solutions of integral equations that are defined on the line segment and have a fractional Riemann-Liouville integral in the main part. These approximations are constructed with the “generalized” projection method using the apparatus of algebraic polynomials. At the same time, the Fredholm property of an integral equation operator in a special pair of Holder spaces of the desired elements and right-hand sides plays an important role.

An iterative numerical method for fractional integral equations of the second kind

In this paper we propose an iterative numerical method for approximating solutions of fractional integral equations of the second kind, based on Picard iteration and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates of the approximations. Numerical examples are given, which illustrate the good approximations.

Positive solutions of fractional integral equations by the technique of measure of noncompactness

In the present study, we work on the problem of the existence of positive solutions of fractional integral equations by means of measures of noncompactness in association with Darbo’s fixed point theorem. To achieve the goal, we first establish new fixed point theorems using a new contractive condition of the measure of noncompactness in Banach spaces. By doing this we generalize Darbo’s fixed point theorem along with some recent results of (Aghajani et al. (J. Comput. Appl. Math. 260:67-77, 2014)), (Aghajani et al. (Bull. Belg. Math. Soc. Simon Stevin 20(2):345-358, 2013)), (Arab (Mediterr. J. Math. 13(2):759-773, 2016)), (Banaś et al. (Dyn. Syst. Appl. 18:251-264, 2009)), and (Samadi et al. (Abstr. Appl. Anal. 2014:852324, 2014)). We also derive corresponding coupled fixed point results. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.

Estimation of unknown function of a class of integral inequalities and applications in fractional integral equations

In this paper, we establish a class of iterated integral inequalities, which includes a nonconstant term outside the integrals. The upper bound of the embedded unknown function is estimated explicitly by adopting novel analysis techniques, such as: change of variable, amplification method, differential and integration. The derived result can be applied in the study of qualitative properties of solutions of fractional integral equations.

Solving fractional integral equations by the Haar wavelet method

Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.