Stability analysis through the Bielecki metric to nonlinear fractional integral equations of $ n $-product operators

The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two‐dimensional nonlinear fractional Fredholm and Volterra integral equations. For this aim, the two‐variable shifted fractional‐order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are derived. These operational matrices and shifted fractional‐order Jacobi collocation method are utilized to reduce the understudy equations to systems of nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the convergence analysis and error bound of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing numerical methods in the literature recently.

Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order

Spaces of continuous functions over a Ψ-space

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.

The Numerical Solution of Integral Equations on the Half-Line

The authors prove two local attractivity and asymptotic stability results for a hybrid functional nonlinear fractional integral equation under weak Lipschitz and compactness type conditions. It is shown that comparable solutions of the equation are uniformly locally ultimately attractive and asymptotically stable on unbounded intervals of the real line. Their proofs rely on a recent measure theoretic fixed point theorem of Dhage.

This manuscript is devoted to ensure the existence of a solution to nonlinear fractional integral equations with three variables under a measure of noncompactness. In order to accomplish our main goal, we develop a new fixed point theorem that generalizes Darbo’s fixed point theorem by utilizing a measure of noncompactness and a new contraction operator. A related tripled FP theorem is also obtained. Finally, we use this generalized Darbo’s fixed point theorem to solve a nonlinear fractional integral equation involving three variables, and an example to demonstrate our results is presented.

We set up the existence of a symmetric outcome of a system of simultaneous nonlinear fractional integral equations, that arises in motion of water wave on smooth surface, with the help of a common fixed point theorem satisfying a generalized FG-contractive condition. To accomplish this, we introduce first the concept of generalized FG-contractive condition for two pairs of self-mappings in a complete metric space and then we establish requisites for common fixed point results for weakly compatible mappings followed by a suitable example.

Numerial Methods for Volterra Integral Equations with Singular Kernels

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg‐de Vries equation in the form of . The authors first convert this equation into a forced generalized Kadomtsev‐Petviashvili equation, , and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green′s function method. The integral representations generate compact operators in a Banach space of real‐valued continuous functions. The Schauder′s fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

The fractional differential transform method is employed here for solving first kind Abel integral equation. Abel integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. An analytic technique for solving Abel integral equation of first kind by the proposed method is introduced here. Also illustrative examples with exact solutions are considered to show the validity and applicability of the proposed method. Numerical results reveal that the proposed method works well and has good accuracy. The method introduces a promising tool for solving many linear and nonlinear fractional integral equation.

In this chapter we develop multiscale methods for solving the Hammerstein equation, and the nonlinear boundary integral equation resulting from a reformulation of a boundary value problem of the Laplace equation with nonlinear boundary conditions. Fast algorithms are proposed using the MAM, in conjunction with matrix truncation strategies and techniques of numerical integration for integrals appearing in the process of solving equations. We prove that the proposed methods require only linear (up to a logarithmic factor) computational complexity and have the optimal convergence order. In the section that follows we discuss the critical issues in solving nonlinear integral equations. This will shine a light on the ideas developed later in this chapter. In Section 10.2, we introduce the MAM for solving Hammerstein equations and provide a complete convergence analysis for the proposed method. In Section 10.3, we develop the MAM for solving the nonlinear boundary integral equation as a result of a reformulation of a boundary value problem of the Laplace equation with nonlinear boundary conditions. We present numerical experiments in Section 10.4. Critical issues in solving nonlinear equations Nonlinear integral equations portray many mathematical physics problems. The Hammerstein equation is a typical kind of nonlinear integral equation. Moreover, boundary value problems of the Laplace equation serve as mathematical models for many important applications. Making use of the fundamental solutions of the equation, we can formulate the boundary value problems as integral equations defined on the boundary (see, Section 2.2.3). For linear boundary conditions, the resulting boundary integral equations are linear, the numerical methods of which have been studied extensively. Nonlinear boundary conditions are also involved in various applications. In these cases, the reformulation of the corresponding boundary value problems leads to nonlinear integral equations. The nonlinearity introduces difficulties in the numerical solution of the equation, which normally requires an iteration scheme to solve it locally as a linearized integral equation.

Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise.

In this paper, we present some coupled hybrid fixed point theorems for partially condensing mixed monotone mappings in a partially ordered metric space which include among others the coupled hybrid fixed point theorems of Bhaskar and Lakshmikantham (Nonlinear Anal TMA 65:1379–1393, 2006), Berinde (Nonlinear Anal 74:7347–7355, 2011), Dhage (Differ Equ Appl 8:77–97, 2016) and Dhage and Dhage (Nonlinear Stud 21(4):675–656, 2014) as special cases with a different method. An application of the main hybrid fixed point principle is given to coupled nonlinear hybrid fractional functional integral equations for proving an algorithm for the existence and attractivity of the approximate solutions on a unbounded interval under certain mixed algebraic and analytical conditions.

The technique of Volterra–Stieltjes integral equations in the application to infinite systems of nonlinear integral equations of fractional orders