Fractional Integral Equations Research Articles

Applications of measure of noncompactness for solvability of Hadamard fractional integral equations

Applications of measure of noncompactness for solvability of Hadamard fractional integral equations

A Computational Method for Solving Nonlinear Fractional Integral Equations

A Computational Method for Solving Nonlinear Fractional Integral Equations

Existence of solutions for fractional functional integral equations of Hadamard type via measure of noncompactness

Existence of solutions for fractional functional integral equations of Hadamard type via measure of noncompactness

Numerical Treatment for the Caputo-Fabrizio Fractional Smoking Model using an Efficient Numerical Technique

Smoking is a prevalent social behavior widely practiced worldwide, especially in settings such as schools and some significant gatherings. Based on the World Health Organization (WHO), smoking is the third leading source of human mortality and the primary preventable cause of disease. This study introduces a highly efficient simulation technique to analyze and solve the Caputo-Fabrizio (CF) fractional smoking model. We numerically solve the fractional integral equations (FIEs) with Simpson’s 1/3 rule, an effective numerical integration technique. We concentrate on clarifying the stability/convergence of the proposed strategy. We juxtapose the outcomes derived using the Runge-Kutta method (RK4) with those obtained through the implemented methodology. The findings indicate that the used technique provides a simple and efficient instrument for simulating the solution of these models. The principal advantage of the proposed method is its dependence on a limited number of straightforward steps, devoid of long-term consequences or reliance on a perturbation parameter.

A study on the solvability of fractional integral equation in a Banach algebra via Petryshyn's fixed point theorem

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra 0 $ ]]> C [ 0 , a ] , a > 0 . Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

Bernstein polynomials iterative method for weakly singular and fractional Fredholm integral equations

Bernstein polynomials iterative method for weakly singular and fractional Fredholm integral equations

Existence of solution for some nonlinear g-Caputo fractional-order differential equations based on Wardowski–Mizoguchi–Takahashi contractions

In this study, we prove the existence and uniqueness of a solution to a g-Caputo fractional differential equation with new boundary value conditions utilizing the combined Wardowski–Mizoguchi–Takahashi contractions via reduction of this equation to a fractional integral equation. We provide an example to demonstrate our findings.

β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of β-Banach spaces. Moreover, we examine the β–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained.

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

Picard and Adomian decomposition methods for a fractional quadratic integral equation via generalized fractional integral

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadraticintegral equation via generalized fractional integral. To achieve this, we introduce an operator that possessesfixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming thegiven equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove theuniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition methodis used, to solve the resulting fractional quadratic integral equation. This technique rapidly provides convergentsuccessive approximations of the exact solution to the given fractional quadratic integral equation, therefore, weinvestigate the convergence of approximate solutions, using the Adomian decomposition method. Finally, we providesome examples, to demonstrate our results. Our findings contribute to the current understanding of fractionalquadratic integral equation and their solutions and have the potential to inform future research in this area.

Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators

In the context of complete strong b-metric-like spaces, we prove new multi-fixed point solutions for the pair of multivalued, dominated operators that fulfill the generalized nonlinear contractions on a closed ball. We employ a mix of two different types of mappings in our approach: one is a class of multi-dominated mappings, while the other is a weaker class of strictly increasing mappings. Additionally, some new fixed-point results concerning the multi-graph-dominated structure in graph contraction are presented. To validate the hypothesis of acquired results, a few sample cases are presented. A numerical experiment has been carried out to approximate the fixed point. In addition, to demonstrate the originality of our findings, we proposed simple and efficient solutions to the system of fractional differential equations and nonlinear Volterra-type integral equations.

On new common fixed point theorems via bipolar fuzzy [Formula: see text]-metric space with their applications.

This research work is devoted to investigating new common fixed point theorems on bipolar fuzzy [Formula: see text]-metric space. Our main findings generalize some of the existence outcomes in the literature. Furthermore, we illustrate our findings by providing some applications for fractional differential and integral equations.

Sparse wavelet Galerkin method: Application for fractional Pantograph problem

The multiwavelets Galerkin method is utilized to solve the fractional Pantograph equation. The method is based on rendering the desired problem into the equivalent integral equation with a weakly singular kernel (fractional integral equation). In this paper, for the first time, the wavelet properties are considered to solve and reduce the computational load for these equations. Compared to the use of other bases, even wavelets generated by a single generator, particularly Alpert’s multiwavelets, have been found to exhibit superior efficiency due to their unique properties. Our investigation supports the theoretical finding. The illustrative examples are also reported, demonstrating the accuracy and flexibility of the method.

Study of Hybrid Problems under Exponential Type Fractional-Order Derivatives

In this investigation, we develop a theory for the hybrid boundary value problem for fractional differential equations subject to three-point boundary conditions, including the antiperiodic hybrid boundary condition. On suggested problems, the third-order Caputo–Fabrizio derivative is the fractional operator applied. In this regard, the corresponding hybrid fractional integral equation is obtained by the Caputo–Fabrizio operator’s properties with the Green function’s aid. Then, we apply Dhage’s nonlinear alternative to the Schaefer type to prove the existence results. Finally, two examples are provided to confirm the validity of our main results.

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α>1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.

Generalizations of Riemann–Liouville fractional integrals and applications

The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs.

Infinite system of nonlinear fractional integral equations in two variables via measure of noncompactness in Banach algebra $$C(I\times I,E)$$

Infinite system of nonlinear fractional integral equations in two variables via measure of noncompactness in Banach algebra $$C(I\times I,E)$$

Solvability of infinite system of implicit fractional integral equations of two variables in tempered sequence spaces via generalized Darbo’s fixed point theorem

Solvability of infinite system of implicit fractional integral equations of two variables in tempered sequence spaces via generalized Darbo’s fixed point theorem

Applications of a new measure of noncompactness to the solvability of systems of nonlinear and fractional integral equations in the generalized Morrey spaces

Applications of a new measure of noncompactness to the solvability of systems of nonlinear and fractional integral equations in the generalized Morrey spaces