Type Integral Equations Research Articles

Coincidence Point Results on Relation Theoretic F w , R g -Contractions and Applications

Motivated by the ideas of F -weak contractions and F , R g -contractions, the notion of F w , R g -contractions is introduced and studied in the present paper. The idea is to establish some interesting results for the existence and uniqueness of a coincidence point for these contractions. Further, using an additional condition of weakly compatible mappings, a common fixed-point theorem and a fixed-point result are proved for F w , R g -contractions in metric spaces equipped with a transitive binary relation. The results are elaborated by illustrative examples. Some consequences of these results are also deduced in ordered metric spaces and metric spaces endowed with graph. Finally, as an application, the existence of the solution of certain Voltera type integral equations is investigated.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model. Mathematical Subject Classification (2010): 45P05, 45G10, 47H99

On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain

In this paper, we consider a singular Volterra type integral equation of the second kind, to which some boundary value problems of heat conduction in domains with a boundary varying with time are reduced by the method of thermal potentials. The peculiarity of such problems is that the domain degenerates into a point at the initial moment of time. Accordingly, a distinctive feature of the integral equation under study is that the integral of the kernel, as the upper limit of integration tends to the lower one, is not equal to zero. This circumstance does not allow solving this equation by the method of successive approximations. We constructed the general solution of the corresponding characteristic equation and found the solution of the complete integral equation by the Carleman–Vekua method of equivalent regularization. It is shown that the corresponding homogeneous integral equation has a nonzero solution.

Solutions of Fredholm type integral equations via the classical Schauder fixed point theorem

We examine the existence of solutions for more general integral equations with respect to other work in Caballero Mena, Nalepa and Sadarangani (2014). Our techniques are based on the classical Schauder fixed point theorem and a new result about the relative compactness in Hölder spaces. Moreover, an example is presented to illustrate the obtained results.

Numerical solution of integral equations with Monte Carlo-collocation (MC-Coll) method and its application in neutron transport

In this study, we introduce a new procedure for the numerical solution of linear and nonlinear integral equations by combining the Monte Carlo simulation and collocation method. One of the major advantages of this method is reducing calculations by converting the integral equation to a system of algebraic equations. The numerical examples of various types of integral equations indicate the efficiency and accuracy of the proposed method. The results show this method is able to solve linear and nonlinear integral equations and also nonlinear weakly singular equations, with remarkable accuracy. Finally, the method is generalized for solving integral-differential equation of the neutron transport for the stationary case.

Rational Type Compatible Single-Valued Mappings via Unique Common Fixed Point Findings in Complex-Valued b-Metric Spaces with an Application

In this paper, we establish some new generalized rational type common fixed point results for compatible three self-mappings in complex-valued b-metric space, in which a one self-map is continuous. In support of our results, we present some illustrative examples to verify the validity of our main work. Moreover, we present the application of two Urysohn integral type equations (UITEs) for the existence of a common solution to support our work. The UITEs are v 1 p = ∫ k 1 k 2 Q 1 p , r , v 1 r d r + ℏ 1 p and v 2 p = ∫ k 1 k 2 Q 2 p , r , v 2 r d r + ℏ 2 p , where p ∈ k 1 , k 2 , v 1 , v 2 , ℏ 1 , ℏ 2 ∈ V , where V = C k 1 , k 2 , ℝ n is the set of all real-valued continuous functions defined on k 1 , k 2 and Q 1 , Q 2 : k 1 , k 2 × k 1 , k 2 × ℝ n ⟶ ℝ n .

An analytical approach to rotor blade modulation

Scattering of electromagnetic plane waves normally incident on a flat perfect electric conductor strip in arbitrary rotational motion in its own plane is investigated in the context of Hertzian Electrodynamics. This canonical problem is an analytical-asymptotic approach that reveals the parametric dependence of interference of electromagnetic waves with rotating blades, a subject which is known as rotor blade modulation. Scattering by a stationary perfect electric conductor strip under TE and TM incidences stand as two sub-problems that are incorporated in the main formulation. These sub-problems are formulated by Wiener–Hopf Technique and solved using a spectral iteration method that converges rapidly in the solution of corresponding coupled dual Fredholm type integral equations in case of electrically wide strips. Numerical implementations that simulate electromagnetic scattering by helicopters blades and wind turbines are provided and discussed.

Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel

Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

SOLUTION OF INTEGRAL EQUATIONS BY BESSEL WAVELET TRANSFORM

The integral equations of the first kind arise in many areas of science and engineering fields such as image processing and electromagnetic theory. The wavelet transform technique to solve integral equation allows the creation of very fast algorithms when compared with known algorithms. Various wavelet methods are used to solve certain type of integral equations. To find the most accurate and stable solution of the integral equation Bessel wavelet is the appropriate method. To study the properties of solution of integral equations on distribution spaces Bessel wavelet transform is also used. In this paper, we accomplished the concept of Hankel convolution and continuous Bessel wavelet transform to solve certain types of integral equations (Volterra integral equation of first kind, Volterra integral equation of second kind and Abel integral equation). Also distributional wavelet transform and generalized convolution will be applied to find the solution of certain Integral equations.

Rational Type Fuzzy-Contraction Results in Fuzzy Metric Spaces with an Application

This paper aims to introduce the new concept of rational type fuzzy-contraction mappings in fuzzy metric spaces. We prove some fixed point results under the rational type fuzzy-contraction conditions in fuzzy metric spaces with illustrative examples to support our results. This new concept will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an application of a nonlinear integral type equation to get the existing result for a unique solution to support our work.

On a New Class of Singular Integro-differential Equations

In this paper for a new class of model and non-model partial integro-differential equations with singularity in the kernel, we obtained integral representation of family of solutions by aid of arbitrary functions. Such type of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of the theory of analytic functions. In the process of our research the new types of singular integro-differential operators are introduced and main property of entered operators are learned. It is shown that the solution of studied equation is equivalent to the solution of system of two equations with respect to x and y, one of which is integral equation and the other is integro-differential equation. Further, non-model integro-differential equations are studied by regularization method. This regularization method for non-model equation is based on selecting and analysis of a model part of the equation and reduced to the solution of two second kind Volterra type integral equations with weak singularity in the kernel. It is shown that the presence of a non-model part in the equation does not affect to the general structure of the solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important. In the cases, when the solution of given integro-differential equation depends on any arbitrary functions, a Cauchy type problems are investigated.

Fixed points of $(\psi, \phi)-$contractions and Fredholm type integral equation

In this paper, we establish a fixed point theorem for controlled rectangular $b-$metric spaces for mappings that satisfy $(\psi, \phi)-$contractive mappings. Also, we give an application of our results as an integral equation.

A new convolution operator for the linear canonical transform with applications

The linear canonical transform plays an important role in engineering and many applied fields, as it is the case of optics and signal processing. In this paper, a new convolution for the linear canonical transform is proposed and a corresponding product theorem is deduced. It is also proved a generalized Young’s inequality for the introduced convolution operator. Moreover, necessary and sufficient conditions are obtained for the solvability of a class of convolution type integral equations associated with the linear canonical transform. Finally, the obtained results are implemented in multiplicative filters design, through the product in both the linear canonical transform domain and the time domain, where specific computations and comparisons are exposed.

Coupled fixed point theorems in partially ordered complex valued metric spaces with application

In this paper, we prove some coupled fixed point theorems for nonlinear contraction type mappings in complete complex valued metric spaces endowed with partial order. We support our results by establishing an illustrative example. Also we give an application of this results to the solution of the Urysohn type integral equations.

On Fuzzy b-Metric-Like Spaces

The aim of this manuscript is to introduce the concept of fuzzy b-metric-like spaces and discuss some related fixed point results. Some examples are imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the obtained results, an application is provided to solve a first kind of Fredholm type integral equations.

Common Fixed Points Technique for Existence of a Solution of Urysohn Type Integral Equations System in Complex Valued b-Metric Spaces

In this paper we give some common fixed point theorems for Ćirić type operators in complex valued b-metric spaces. Also, some corollaries under this contraction condition are obtained. Our results extend and generalize the results of Hammad et al. In the second part of the paper, in order to strengthen our main results, an illustrative example and some applications are given.

Riemann-Hilbert problems and soliton solutions of nonlocal real reverse-spacetime mKdV equations

We would like to analyze a kind of nonlocal reverse-spacetime integrable PT-symmetric multicomponent modified Korteweg-de Vires (mKdV) equations by making a group of nonlocal reductions, and establish their associated Riemann-Hilbert problems which determine generalized Jost solutions of higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the associated Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. The Riemann-Hilbert problems in the reflectionless case are solved explicitly, and the resulting formulation of solutions enables us to present solitons to the nonlocal reverse-spacetime integrable PT-symmetric mKdV equations.

Utilization of Chebyshev collocation approach for differential, differential-difference and integro-differential equations

Numerical solutions to differential and integral equations have been a very active field of research. The necessary tools to solving differential equations can add much fuel for acceleration of scientific development. Therefore, it is quite challenging for scientists, especially mathematicians, to work on the solution methodologies for differential equations. In this study, we have discussed the use of Chebyshev collocation method for solving a large variety of problems including various types of integral equations, integro-differential equations, and differential-difference equations. The method under consideration relies on a trial solution which is a linear combination of basic functions derived from Chebyshev polynomials. The trial solution is then used to obtain a residue function whose vanishing weighted residual can be used to obtain a system of algebraic equations which gives the values of constants used in the trial solution. Plugging the constants back into the trial solution, we obtain an approximate solution to the problem at hand. We have provided the comparison between exact and approximate solutions for different problems and found an exceptionally good agreement in results. The comparison has been provided through plots and tables to portray the accurateness of Chebyshev collocation method. The behaviour of the solutions is also displayed with the help of the graphs.

Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions

This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.