Integral Equations Of Fractional Order Research Articles

Integral representation for hypergeometric function of the Mittag-Leffler type

The Mittag-Leffler function has gained importance and popularity through its applications. When solving differential equations of fractional order and integral equations of fractional order. Also, the Mittag-Leffler function plays an important role in various fields of applied mathematics and engineering sciences, such as chemistry, biology, statistics, thermodynamics, mechanics, quantum physics, computer science, signal processing [1-6].

On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces

We provide and prove some new fundamental properties of the Erdélyi–Kober (EK) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces (Lp) and Orlicz spaces (Lφ). We employ these properties with the concept of the measure of noncompactness (MNC) associated with the fixed-point hypothesis (FPT) in solving a quadratic integral equation of fractional order in Lp,p≥1 and Lφ. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems.

Locally Attractivity Solution of Coupled Fractional Integral Equation in Banach Algebras

In this paper we prove the Locally Attractivity solution of coupled hybrid functional integral equation of fractional order (CQHFIE) in Banach algebras. Our main result is based on the hybrid fixed point theory used theorem due to Dhage B. C.

Discontinuous solutions of delay fractional integral equation via measures of noncompactness

<abstract><p>This article considers the existence and the uniqueness of monotonic solutions of a delay functional integral equation of fractional order in the weighted Lebesgue space $ L_1^N({\mathbb{R}}^+) $. Our analysis uses a suitable measure of noncompactness, a modified version of Darbo's fixed point theorem, and fractional calculus in the mentioned space. An illustrated example to show the applicability and significance of our outcomes is included.</p></abstract>

К вопросу о решении смешанной краевой задачи для уравнения с производными дробного порядка с различными началами

An approach to solving boundary value problems for an ordinary differential equation of fractional order containing a composition of left- and right-handed Riemann-Liouville and Caputo fractional differentiation operators is proposed. The approach is based on the reduction of the equation under consideration to the study of integral equations of fractional order with involution. As an example, a mixed problem has been solved. In particular, the proposed approach allows us to improve the previously obtained conditions for the solvability of this problem.

Nonexistence Results of Global Solutions for Fractional Order Integral Equations on the Heisenberg Group

Nonexistence Results of Global Solutions for Fractional Order Integral Equations on the Heisenberg Group

Solvability of Implicit Fractional Order Integral Equation in ℓ p 1 ≤ p < ∞ Space via Generalized Darbo’s Fixed Point Theorem

We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in ℓ p 1 ≤ p < ∞ space. The fundamental tool in the presentation of our proofs is the measure of noncompactness mnc approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.

Analyzing the existence of solution of a fractional order integral equation: A fixed point approach

Abstract This paper signifies the beauty of fixed point theory in the context of attaining the sufficient condition for the existence of the solution of a non-linear functional-integral equation in an unbounded interval. Here a non-linear integral equation (NLIE) involving a fractional operator is taken in the form of an operator equation. Utilizing the perception of measure of noncompactness (MNC) along with certain relevant assumptions, it is proved that the operator equation satisfies the Darbo condition for the product of operators in a Banach algebra. Finally, the result obtained is verified by the assistance of the numerical example.

Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation

Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ− fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results.

On some properties of Riemann-Liouville fractional operator in Orlicz spaces and applications to quadratic integral equations

This article demonstrates some properties of the Riemann-Liouville (R-L) fractional integral operator like acting, continuity, and boundedness in Orlicz spaces L?. We apply these results to examine the solvability of the quadratic integral equation of fractional order in L?. Because of the distinctive continuity and boundedness conditions of the operators in Orlicz spaces, we look for our concern in three situations when the generating N-functions fulfill ??, ?2, or ?3-conditions. We utilize the analysis of the measure of noncompactness with the fixed point hypothesis. Our hypothesis can be effectively applied to various fractional problems.

Application of some special operators on the analysis of a new generalized fractional Navier problem in the context of q-calculus

The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t. q-RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital α-admissible and α-ψ-contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.

Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equation

Abstract In this work, we study the existence of one and exactly one solution x ∈ C [ 0 , 1 ] x\in C\left[0,1] , for a delay quadratic integral equation of Volterra-Stieltjes type. As special cases we study a delay quadratic integral equation of fractional order and a Chandrasekhar cubic integral equation.

Two-dimensional Euler polynomials solutions of two-dimensional Volterra integral equations of fractional order

This paper proposes a method based on two-dimensional Euler polynomials combined with Gauss-Jacobi quadrature formula. The method is used to solve two-dimensional Volterra integral equations with fractional order weakly singular kernels. Firstly, we prove the existence and uniqueness of the original equation by Gronwall inequality and mathematical induction method. Secondly, we use two-dimensional Euler polynomials to approximate the unknown function of the original equation, and the Gauss-Jacobi quadrature formula is used to approximate the integrals in the original equation. Thirdly, we prove the existence and uniqueness of the solution of approximate equation, and the error analysis of the proposed method is given. Finally, some numerical examples illustrate the efficiency of the method.

Solvability of generalized fractional order integral equations via measures of noncompactness

In this article, we work on the existence of solution of generalized fractional integral equations of two variables. To achieve our main objective, we establish a new fixed point theorem using measure of noncompactness and a new contraction operator which generalized the Darbo’s fixed point theorem (DFPT). Also we obtain the corresponding coupled fixed point theorem. Finally we apply this generalized DFPT on the generalized fractional integral equations of two variables and illustrate our findings with the help of an example.

Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra

Here, we investigate the existence result for a nonlinear quadratic functional integral equation of fractional order using a fixed point theorem of Dhage. The continuous dependence of solution on the delay functions will be studied. As an application, an existence theorem for the fractional hybrid differential equations is proved. Also, we study a general quadratic integral equation of fractional order.

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.

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.

Existence results for a quadratic integral equation of fractional order by a certain function

Existence results for a quadratic integral equation of fractional order by a certain function

Solving two-dimensional non-linear quadratic integral equations of fractional order via operational matrix method

Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

Algebraic Fractional Order Differentiator Based on the Pseudo-State Space Representation

Abstract The aim of this paper is to design an algebraic fractional order differentiator for a class of commensurate fractional order linear systems modeled by the pseudo-state space representation. For this purpose, a new algebraic method is introduced by designing an operator which can transform the considered system into a fractional order integral equation by eliminating unknown initial conditions. Based on the obtained equation, the desired fractional derivative is exactly given by a new algebraic formula using a recursive way. Then, a digital fractional order differentiator is introduced in discrete noisy cases. Finally, numerical results are given to illustrate the accuracy and the robustness of the proposed method.