Solution Of Nonlinear Integral Equation Research Articles

A reliable treatment to solve nonlinear Fredholm integral equations with non-separable kernel

This work is devoted to solve integral equations formulated in terms of the kernel functions and Nemytskii operators. This type of equations appear in different applied problems such as electrostatics and radiative heat transfer problems. We deal with both cases separable and non-separable kernels by setting the theoretical semilocal convergence results for an adequate iterative scheme that can be useful for approximating the solution of the infinite dimensional problem. We pay special attention to non-separable kernels avoiding the solution given in previous works where the original nonlinear integral equation has been approximated by means of an equation with separable kernel. However, in this case, we introduce an approximation of the derivative operator that it is needed for applying the iterative scheme considered. Moreover, we study the localization and separation of possible solutions of nonlinear integral equation by means of a result of semilocal convergence for the iterative scheme considered. The theoretical results obtained have been tested with some applied problems showing competitive results.

Expansive mappings on bounded sets and their application to rational integral equations

The aim of this paper is to introduce a new condition of expansiveness which extends the class of the known expansive-type mappings in the domain of closed and bounded subsets of a metric space. For the sum of an expansive mapping in a proposed sense defined on a closed convex and bounded subset of a Banach space and a continuous and compact operator, there are proved some fixed point theorems. There are also provided some examples which motivate our research. Moreover, an application to the existence problem of periodic solutions of nonlinear integral equations with a rational part is provided.

On Solutions of Nonlinear Integral Equations in the Space of Functions of Shiba-Bounded Variation

On Solutions of Nonlinear Integral Equations in the Space of Functions of Shiba-Bounded Variation

The Cardinal Spline Methods for the Numerical Solution of Nonlinear Integral Equations

In this study, an effective technique is presented for solving nonlinear Volterra integral equations. The method is based on application of cardinal spline functions on small compact supports. The integral equation is reduced to a system of algebra equations. Since the matrix for the system is triangular, it is relatively straightforward to solve for the unknowns and an approximation of the original solution with high accuracy is accomplished. Several cardinal splines are employed in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined, and the convergence rate is analyzed. We compare our method with other methods proposed in recent papers and demonstrated the advantage of our method with several examples.

Existence of solution for nonlinear integral equations of Hénon type in critical case

In this paper, we will show that there exists a δ∗>0 such that when α>0,β>0 and α+β<δ∗, the following problem (0.1)fqγ−1(x)=∫Ω|x|α|y|βf(y)|x−y|n−γdy+λ∫Ωf(y)|x−y|n−γ−1dy,x∈Ω¯has at least one positive solution f∈C(Ω̄), where Ω is the unit ball in Rn(n≥3) centered at the origin, 1<γ<n, λ>0 and qγ≔2nn+γ.

Coupled coincidence point technique and its application for solving nonlinear integral equations in RPOCbML spaces

The aim of this paper is to obtain some new important consequences related to coupled coincidence points via C-class functions in the context of a regular partial ordered complete b-metric-like space (for short, RPOCbML space); this space arises from combining the results of b-metric-like space with partial metric space and adding the regularity condition. Finally, we support our theoretical results by some examples and an application about finding an analytical solution for nonlinear integral equations.

Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball

In this paper, we consider the problem \begin{document}$ \begin{equation*} f^{q-1}(x) = \int_{\Omega}\frac{|x|^{\alpha}|y|^{\beta}f(y)}{|x-y|^{n-\gamma}}dy, \; f>0, \; x\in\overline{\Omega}, \end{equation*} $\end{document} where \begin{document}$ \Omega $\end{document} is the unit ball in \begin{document}$ \mathbb{R}^n(n\geq3) $\end{document} centered at the origin, \begin{document}$ 1 and \begin{document}$ \alpha, \beta>0 $\end{document} , \begin{document}$ q_\gamma: = \frac{2n}{n+\gamma} . We will investigate the asymptotic behavior of energy maximizing positive solution as \begin{document}$ q\rightarrow (\frac{2n}{n+\gamma})^{+} = (q_\gamma)^+ $\end{document} . We also show that the energy maximizing positive solution concentrate at a point, which is located at the boundary as \begin{document}$ q\rightarrow (q_\gamma)^{+} $\end{document} . In addition, the energy maximizing positive solution is non-radial provided that \begin{document}$ q $\end{document} closes to \begin{document}$ q_\gamma $\end{document} .

On the numerical solution of nonlinear integral equations on non-rectangular domains utilizing thin plate spline collocation method

This article investigates an approximate scheme to solve nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The integral equations considered in the current paper are considered together with either smooth or weakly singular kernels. The offered method utilizes thin plate splines as a basis in the discrete collocation method. We can regard thin plate splines as a type of the free shape parameter radial basis functions. These basis functions establish an accurate and stable technique to estimate an unknown function by using a set of scattered points on the solution domains. Since the thin plate splines have limited smoothness, the integrals appeared in the scheme cannot be estimated by classical integration rules. Therefore, we introduce a special precise quadrature formula on non-rectangular domains to compute these integrals. The proposed scheme does not require any mesh generations, so it is meshless and does not depend on the domain form. Error analysis is also provided for the method. The performance and convergence of the new approach are tested on four two-dimensional integral equations given on the wing, mushroom, pentagon and fish-like domains.

Solution of Nonlinear Integral Equation via Fixed Point of Cyclic $$\alpha _{L}^{ \psi }$$αLψ-Rational Contraction Mappings in Metric-Like Spaces

In this paper, we introduce the notions of $$\alpha _{L}^{\psi }$$-rational contractive and cyclic $$\alpha _{L}^{\psi }$$- rational contractive mappings and establish the existence and uniqueness of fixed points for such mappings in complete metric-like spaces (dislocated metric spaces). The results presented here substantially generalize and extend several comparable results in the existing literature. As an application, we prove new fixed point results for $$\psi L$$-graphic and cyclic $$\psi L$$-graphic rational contractive mappings. Moreover, some examples and an application to integral equation are presented here to illustrate the usability of the obtained results.

A New Approach to the Solution of Non-Linear Integral Equations via Various FBe-Contractions

In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended F B e -contraction, the extended F B e -expanding contraction, and the extended generalized F B e -contraction. Thereafter, we propose a simple and efficient solution for non-linear integral equations using the fixed point technique in the setting of a B e -metric space. Moreover, to address conceptual depth within this approach, we supply illustrative examples where necessary.

Fixed Point Results for φ − (γ, η, n, m)−Contractions with Applications to Nonlinear Integral Equations

The aim of this paper is to introduce a new class of pair of contraction mappings, called φ − (γ, η, n, m)-contraction pairs, and obtain common fixed point theorems for a pair of mappings in this class, satisfying a weakly compatible condition. As an application, we use mappings of this class to find the existence of solutions for nonlinear integral equations on the space of continuous functions and in some of its subspaces. Moreover, some examples are given here to illustrate the applicability of these results.

Invariant submodels of model of nonlinear diffusion in inhomogeneous medium with nonstationary absorption or source

We carried out a complete group classification of the nonlinear differential equation of the model, describing the nonlinear diffusion process (or the process of heat propagation) in an inhomogeneous medium with nonstationary absorption or source. We found all models whose equations admit a nontrivial continuous Lie group of the transformations. We researched invariant submodels of the obtained models. Some invariant solutions, describing these submodels were found either explicitly, or their search was reduced to solving of the nonlinear integral equations. In particular, we obtained an invariant solution that describes the nonlinear diffusion process (or heat distribution process) with two fixed “black holes”, and invariant solution that describes the nonlinear diffusion process (or heat distribution process) with a fixed “black hole”, and with a moving “black hole”. The solutions with “black holes” can be used to describe the process of destruction of glaciers under the influence of the external heat source, for example a solar energy. Using other invariant solution, we studied a diffusion process (or heat distribution process) for which at the initial moment of the time at a fixed point a concentration (a temperature) and its gradient are specified. The solution of the boundary value problem describing this process reduces to the solution of nonlinear integral equations. We have established the existence and uniqueness of solutions of this boundary value problem under certain conditions. The mechanical significance of the obtained solutions is as follows: 1) they describe specific nonlinear diffusion processes (or heat distribution processes), 2) they can be used as tests in numerical calculations in studies of nonlinear diffusion (or heat distribution) in an inhomogeneous medium with nonstationary absorption or source, 3) they allow to assess the degree of adequacy of a given mathematical model to real physical processes, after carrying out experiments corresponding to these decisions, and estimating the arising deviations.

On the numerical solution of nonlinear integral equation arising in conductor like screening model for realistic solvents

The aim of this work is to introduce an efficient algorithm for the numerical solution of nonlinear integral equation arising from chemical phenomenon which is a famous equation in chemistry engineering. A procedure is described for transforming the nonlinear integral equation by using Chebyshev polynomials to nonlinear system of algebraic equations. Also, we present a convergence analysis and error bound for presented method. In addition, some numerical results are reported to evaluate the validity and applicability of the method and also comparison has been done with existing results.

Spline quasi-interpolating projectors for the solution of nonlinear integral equations

In this paper we use spline quasi-interpolating projectors on a bounded interval for the numerical solution of nonlinear integral equations. In particular, we propose a spline quasi-interpolating projection method with high order of convergence and a spline quasi-interpolating collocation method, both in case of smooth kernels and in case of Green’s function type ones. We explicitly construct the approximate solutions and we get results related to the convergence orders. Finally, we provide numerical tests, that confirm the theoretical results.

On surjective second order non-linear Markov operators and associated nonlinear integral equations

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.

On new evolution of Ri\u2019s result via w-distances and the study on the solution for nonlinear integral equations and fractional differential equations

The aim of this work is to establish a new fixed point theorem for generalized contraction mappings with respect to w-distances in complete metric spaces. An illustrative example is provided to advocate the usability of our results. Also, we give a numerical experiment for approximating a fixed point in these examples. As an application, the received results are used to summarize the existence and uniqueness of the solution for nonlinear integral equations and nonlinear fractional differential equations of Caputo type.

The Approximate Solution of Nonlinear Integral Equations with the RH Wavelet Bases in a Complex Plane

This study has been conducted to calculate the one-dimensional nonlinear Volterra–Fredholm and mixed Volterra–Fredholm integral equation of second kind in a complex plane, regarding the use of the wavelet. As far as we are aware, there have been no studies conducted so far to solve this kind of integral equation in the complex plane. The main feature of this method is that, it approximates the integral equations without solving any linear or algebra systems. In “Approximation of the Solutions with RH Functions” section, the integral operator for the RH wavelet and its use will be presented in our numerical methods. “Error Analysis” section, will discuss about the unique solution of the mentioned problems. Furthermore, an upper bound for the error analysis will be given. Finally, some problem examples and their solution will be proposed in “Numerical Examples” section.

A Bernstein polynomial approach for solution of nonlinear integral equations

A Bernstein polynomial approach for solution of nonlinear integral equations

Approximate Solution of Nonlinear Integral Equations of the Second Kind by Using Homotopy Perturbation Method

In this paper, we apply Homotopy perturbation method (HPM) for obtaining approximate solution of nonlinear Fredholm integral equation of the second kind. Finally, some numerical examples are provided, and the obtained numerical approximations are compared with the corresponding exact solution.&#x0D; Dhaka Univ. J. Sci. 65(2): 151-155, 2017 (July)

Extension of some theorems to find solution of nonlinear integral equation and homotopy perturbation method to solve it

In this paper, the concept of contraction via the measure of non-compactness on a Banach space is investigated by generalizing some results which have been previously discussed in literatures. Furthermore, to validity of the theorems and homotopy perturbation method (HPM), as a technical solution, they are applied on some nonlinear singular integral equations.