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In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton's methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques. [ABSTRACT FROM AUTHOR]

This article introduces a measure of noncompactness in the generalized Morrey space. We study the applications of our new definition in investigating the conditions for the existence of solutions for systems of nonlinear integral equations. We can extend many useful theorems in L p (R N) for functions belonging to the generalized Morrey spaces. Compared to the L p (R N) spaces, the advantage of studying in the Morrey spaces is that we can research no compact support functions in our problems. Finally, significant examples are presented to show the efficiency of the main results. [ABSTRACT FROM AUTHOR]

In this paper, a combination approach based on Bernoulli polynomials and Gauss-Jacobi quadrature formula is developed to solve the system of nonlinear variable-order fractional Volterra integral equations (V-O-FVIEs). For this, we extend the constant coefficient in the Gauss-Jacobi formula to the variable coefficient and used it in our method. The method converts the system of V-O-FVIEs into the corresponding nonlinear system of algebraic equations. In addition, we use Gronwall inequality and the collectively compact theory to prove the existence and uniqueness of the solution of the original equation and the approximate equation, respectively. The convergence analysis and the error estimation of proposed method are discussed. Finally, some numerical examples illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In many physical problems, it is important to capture exponentially small effects that lie beyond‐all‐orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans‐series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans‐series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans‐series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial‐over‐power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial‐over‐power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences. [ABSTRACT FROM AUTHOR]

A nonlinear integral equation arising from a parametric closure of the third spatial moment in the single-species model of logistic dynamics of U. Dieckmann and R. Law is analyzed. The case of piecewise constant kernels is studied, which is important for further computer modeling. Sufficient conditions are found that guarantee the existence of a nontrivial solution to the equilibrium equation. The use of constant kernels makes it possible to obtain more accurate results compared to earlier works, in particular, more accurate estimates for the L1 norm of the solution and for the closure parameter are obtained. [ABSTRACT FROM AUTHOR]

Two weakly nonlinear integral equations of motion commonly used in the literature to study the nonlinear dynamics of straight and initially not rectilinear Euler‐Bernoulli beams, respectively, are further investigated. Attention is focused on the process known as "static condensation", which consists of neglecting the axial inertia in the exact, fully nonlinear system of equations of motion to determine the axial displacement as a function of the transversal one. The novelty of the paper relies on showing that, contrarily to expectation and somehow surprisingly, the integral equation for beams with a not rectilinear initial configuration cannot be obtained by the static condensation process starting from the exact, fully nonlinear, equations of motion, apart from a very particular and specific case. On the contrary, it is confirmed the well‐known result that for rectilinear beams the integral equation can be obtained by the static condensation. This highlights a major difference between the two integral equations in terms of reliability and allows us a better understanding of the integral equation of rectilinear beams, underlying its stronger mathematical background than the classical counterpart for not straight beams (i.e., its being obtainable from the exact, fully nonlinear, equations of motion via static condensation), which provides it with a "special" behavior and makes it more trustworthy. [ABSTRACT FROM AUTHOR]

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [ABSTRACT FROM AUTHOR]

In this article, we consider Lucas series for N = 2, 5, 6 in numerical solutions of linear and nonlinear Volterra integral equations of the second kind. The series is used to transform the equation into a system of nonlinear algebraic equations, and the unknown parameters are determined. The application of this method has shown that the Lucas series is a powerful and active candidate that can be used to approximate the solution of linear and nonlinear Volterra integral equations of the second kind. The method gave a good performance, as can be seen from the four sample problems considered. The numerical results reveal computational efficiency of the method and it is also seen to be highly accurate and converge to the exact solution in some cases. Approximate and exact solutions are plotted to further confirm the accuracy of the method. [ABSTRACT FROM AUTHOR]

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of 2m × 2m equations, we have simplified the problem to finding the solution for a set of 2 × 2 nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam-Hyers stability and generalized Ulam-Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method. [ABSTRACT FROM AUTHOR]

In the present paper, a new efficient technique is described for solving nonlinear mixed partial integrodifferential equations with continuous kernels. Using the separation of variables, the nonlinear mixed partial integrodifferential equation is converted to a nonlinear Fredholm integral equation. Then, using different numerical methods, the Bernoulli polynomial method and the Chebyshev polynomials of the sixth kind, the nonlinear Fredholm integral equation has been reduced into a system of nonlinear algebraic equations. The Banach fixed-point theory is utilized in order to have a conversation about the nonlinear mixed integral equation's solution, namely, its existence and uniqueness. In addition, we talk about the convergence and stability of the solution. Finally, a comparison between the two different methods and some other famous methods is presented through various examples. All the numerical results are calculated and obtained using the Maple software. [ABSTRACT FROM AUTHOR]

We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given. [ABSTRACT FROM AUTHOR]

A numerical method to solve the nonlinear Volterra integral equations of the first kind with discontinuous kernels is proposed. Usage of operational matrices for this kind of equation is a cost-efficient scheme. Shifted Legendre polynomials are applied for solving Volterra integral equations with discontinuous kernels by converting the equation to a system of nonlinear algebraic equations. The convergence analysis is given for the approximated solution and numerical examples are demonstrated to denote the precision of the proposed method. [ABSTRACT FROM AUTHOR]

The objective of the present work is to develop a numerical method for solving a nonlinear integral equation given by y (t) = f (t) + ∫ 0 1 k (t , s) 1 [ y (s) ] α d s , where y(t)∈C2[0,1] with y(t)>0 , f(t)∈C[0,1] with f(t)≥0 , the kernel function k(t,s) is non-negative and continuous on [0,1]×[0,1] and α>0. The existence of a continuous positive solution of this equation is well-established in the literature. However, there is no reported method to solve it numerically for any α>0. To attain the desired objective, the renowned Chebyshev collocation method is used. Having unknown Chebyshev coefficients, this method converts the integral equation into a matrix equation that produces a set of nonlinear algebraic equations. To solve these equations, computationally, the well-established Newton's method is employed. To validate the effectiveness and precision of the method, various numerical examples with well-defined exact solutions are examined. Obtained numerical solutions confirm the accuracy and validity of the numerical method. [ABSTRACT FROM AUTHOR]

The nonlinear Volterra–Fredholm integral Equation (NVFIE) with a singular kernel is discussed such that the kernel of position can take the Hilbert kernel form, Carleman function, logarithmic form, or Cauchy kernel. Using the quadrature method, the NVFIE with a singular kernel leads to a system of nonlinear integral equations. The existence and unique numerical solution of this system is discussed, as is the truncation error of the numerical solution. The solution of the nonlinear integral equation system is obtained using the spectral relations and techniques of the Chebyshev polynomial method. Finally, we will discuss examples of when the kernel takes various forms to demonstrate this technique's high accuracy and simplicity. Some numerical results and estimating errors are calculated and plotted using the program Wolfram Mathematica 10. [ABSTRACT FROM AUTHOR]

This manuscript contains several new notions including intuitionistic fuzzy  N b  metric space, intuitionistic fuzzy quasi- S b -metric space, intuitionistic fuzzy pseudo- S b -metric space, intuitionistic fuzzy quasi- N -metric space and intuitionistic fuzzy pseudo  N b  fuzzy metric space. We prove decomposition theorem and fixed-point results in the setting of intuitionistic fuzzy pseudo  N b  fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications. [ABSTRACT FROM AUTHOR]

We consider a generic type of nonlinear Hammerstein-type integral equations with the particularity of having non-differentiable kernel of Nemystkii type. So, in order to solve it we consider a uniparametric family of iterative processes derivative free, with the main advantage that for a special value of the involved parameter the iterative method obtained coincides with Newton's method, that is due to the fact of evaluating the divided difference operator when the two values are the same. We perform a qualitative convergence study by choosing an auxiliary point, that allow us to obtain the existence and separation of solutions of the given equation, that is, local and semilocal convergence balls can be obtained. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend Boyd and Wong's fixed point theorem in complete generalized metric spaces and to apply this result under the so-called G-weak topology in generalized Banach algebras. Some tools will be introduced and involved in our work as the generalized measures of weak non-compactness and the sequential condition (P G). Our results will extend many known theorems in the literature. Also, we will give an application for an infinite system of nonlinear integral equations defined on the generalized Banach algebra (C ([ 0 , 1 ] , c 0) × C ([ 0 , 1 ] , c 0)) , where c0 is the space of all real sequences converging to zero. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the uniqueness and certain qualitative properties of the solution of a class of integral equations with sum-difference kernel on the positive half-line and with a convex nonlinearity. This class of equations arises in a particular case in the dynamical theory of -adic closed-open strings for the scalar field of tachyons. Such equations also play a very important role in the study of the existence and uniqueness of solutions of nonlinear integral equations in the mathematical theory of the geographical distribution of an epidemic within the framework of the Diekmann–Kaper model. We prove the uniqueness theorem for the solution of the equation under consideration for a class of nonnegative (nonzero) and bounded functions on , thereby obtaining a definitive solution of Vladimirov's open problem on the uniqueness of rolling solutions of nonlinear -adic equations. Under an additional constraint on the kernel of the equation, we also prove that the solution is a concave function on whose derivative belongs to the space . At the end of the paper, we give specific model equations from the above-mentioned applications, to which our results are applied. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a system of nonlinear integral equations resulting from the three-parameter closure of the third spatial moments in the logistic dynamics model of U. Dieckmann and R. Law in the multi-species case. Specifically, the conditions under which the solution of this system is stable with respect to the closure parameters are investigated. To do this, the initial system of equations is represented as a single operator equation in a special Banach space, after which the generalized fixed point principle is applied. [ABSTRACT FROM AUTHOR]

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl's type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations. [ABSTRACT FROM AUTHOR]

In this paper, we present some results of coupled fixed points for the system of non-linear integral equations in Banach space. Our results enlarge the results of newer papers. Additionally, we prove the applicability of those results to the solvability of the system of non-linear integral equations. Finally, we give an example to validate the applicability of our results. [ABSTRACT FROM AUTHOR]

This paper discusses two efficient collocation methods for solving the Hammerstein integral equations on the semi-infinite domain, where the underlying solutions decay to zero at infinity. These methods are based upon modified Legendre rational and exponential functions, and reduce the Hammerstein integral equation to a nonlinear algebraic system. The error between the approximate and exact solutions in the usual L 2 -norm is estimated. Finally, some numerical experiments are presented to examine and demonstrate the effectiveness and accuracy of the proposed methods in comparison to other approaches. [ABSTRACT FROM AUTHOR]

We present new numerical algorithms for solving the structural inverse gravimetry problem for the case of multiple surfaces. The inverse problem of finding the multiple surfaces that divide the constant density layers is an ill‐posed one described by a nonlinear integral equation of the first kind. To solve it, it is necessary to apply the regularization ideas. The new regularized variants of the gradient type methods with the weighting factors are constructed, namely, the steepest descent and conjugate gradient method. We suggest the empirical rule for choosing the regularization parameters. On the basis of the constructed methods, we elaborate the parallel algorithms and implement them in the multicore CPU using the OpenMP technology. A set of experiments with the disturbed data is performed to test the gradient algorithms and study performance of the developed code. For the test problems with quasi‐real data, these new regularized algorithms increase the accuracy and speed up computation in comparison with the unregularized ones. By using the 8‐core CPU, we achieve the speedup of 8 times. [ABSTRACT FROM AUTHOR]

The stellar dynamic models considered here deal with triples () of three functions: the distribution function , the local density , and the Newtonian potential , where , ( are the space-velocity coordinates), and is a function of the local energy . Our first result is an answer to the following question: Given a (positive) function on a bounded interval , how can one recognize as the local density of a stellar dynamic model of the given type ("inverse problem")? If this is the case, we say that is "extendable" (to a complete stellar dynamic model). Assuming that is strictly decreasing we reveal the connection between and , which appears in the nonlinear integral equation and the solvability of Eddington's equation between and (Theorem 4.1). Second, we investigate the following question ("direct problem"): Which induce distribution functions of the form of a stellar dynamic model? This leads to the investigation of the nonlinear equation in an approximative and constructive way by mainly numerical methods. [ABSTRACT FROM AUTHOR]

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space B C (R +) using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to B C (R +) is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space B C (R +) are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to B C (R +) for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to B C (R +). This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with N dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity. [ABSTRACT FROM AUTHOR]

The Sumudu decomposition method was used and developed in this paper to find approximate solutions for a general form of fractional integro-differential equation of Volterra and Fredholm types. The Caputo definition was used to deal with fractional derivatives. As the method under consideration depends mainly on writing non-linear terms, which are often found inside the kernel of the integral equation, writing it in the form of Adomian's polynomials in the well-known way. After applying the Sumudu transformation to both sides of the integral equation, the solution was written in the form of a convergent infinite series whose terms can be alternately calculated. The method was applied to three examples of non-linear integral equations with fractional derivatives. The results that were presented in the form of tables and graphs showed that the method is accurate, effective and highly efficient. [ABSTRACT FROM AUTHOR]

Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and uses some finite dimensional linearization process to solve numerically the discrete problem. Its convergence is proved under mild hypotheses on the nonlinearity and the quadrature rule of the singularity subtraction scheme. The New Approach is based on linearization of the problem in its infinite dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. It is more efficient than the former, as three numerical experiments confirm. [ABSTRACT FROM AUTHOR]

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods. [ABSTRACT FROM AUTHOR]

In this paper, a direct operator method is presented for the exact closed-form solution of certain classes of linear and nonlinear integral Volterra–Fredholm equations of the second kind. The method is based on the existence of the inverse of the relevant linear Volterra operator. In the case of convolution kernels, the inverse is constructed using the Laplace transform method. For linear integral equations, results for the existence and uniqueness are given. The solution of nonlinear integral equations depends on the existence and type of solutions ofthe corresponding nonlinear algebraic system. A complete algorithm for symbolic computations in a computer algebra system is also provided. The method finds many applications in science and engineering. [ABSTRACT FROM AUTHOR]

Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of solutions of system of nonlinear integral equations in the setting of tempered sequence space. Moreover, a very interesting and illustrative example is designed to visualize our findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator A , the solutions of the integral system are the fixed points of A. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator A has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space (X , { | ⋅ | p } p ∈ Λ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 휏-Krein–Šmulian property, where 휏 is a weaker Hausdorff locally convex topology of 푋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 휏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ (R +) × C ⁢ (R +) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result. [ABSTRACT FROM AUTHOR]

This paper proposes a discrete collocation method based on the fuzzy Lagrange interpolation and the fuzzy Gauss-Legendre quadrature formula for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations. Firstly, the existence and uniqueness of the solution of the original equation are proved by using a two-dimensional Gronwall inequality and an iterative method. Secondly, the discrete collocation method is utilized to convert the linear and nonlinear fuzzy Volterra integral equations into the corresponding linear and nonlinear systems of algebraic equations. Finally, the error estimation and convergence analysis of the proposed numerical method are discussed in terms of the uniform modulus of continuity. Some numerical experiments illustrate that the scheme possesses a higher accuracy and yields a better approximation when compared with other known methods. [ABSTRACT FROM AUTHOR]

This paper introduces a collocation method for numerically solving Hammerstein generalized integral equations on the whole line. The approach utilizes mapped Gegenbauer functions to transform the Hammerstein integral equation into a system of nonlinear algebraic equations. Additionally, convergence results are obtained using the standard L 2 -norm. Finally, the effectiveness of the proposed method is demonstrated through several numerical experiments, which also serve to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

We investigate the existence and multiplicity of positive solutions for a system of Riemann‐Liouville fractional differential equations with sequential derivatives, positive parameters and sign‐changing singular nonlinearities, subject to nonlocal uncoupled boundary conditions which contain various fractional derivatives and Riemann‐Stieltjes integrals. We apply the nonlinear alternative of Leray‐Schauder type and the Guo‐Krasnosel'skii fixed point theorem in the proof of our main existence results. [ABSTRACT FROM AUTHOR]

We propose two numerical techniques for solving Bratu-like equations arising in the electrospinning and vibration-electrospinning process. Due to the presence of parameter δ as well as strong nonlinearity, these problems pose difficulties in obtaining their solutions. The first numerical method is based on the Bernstein polynomials, and the second method is based on the Gegenbauer-wavelets method. To establish numerical algorithms, we consider the equivalent integral form of the Bratu-like equation with suitable boundary conditions. Using the approximation theory based on the Bernstein polynomials and the Gegenbauer wavelets combined with the collocation technique, we convert the integral equation into a nonlinear system of equations. The Newton–Raphson method is implemented to analyze the resulting nonlinear system of equations. Three examples of Bratu-like equations are provided to demonstrate the accuracy, applicability, and efficiency of the respective techniques. The obtained results of numerical solutions and residual errors are compared. We observe that the collocation technique based on the Bernstein polynomial provides a better result than the Gegenbauer wavelets and other known methods. [ABSTRACT FROM AUTHOR]

In this paper we study some nonlinear integral equations and give conditions for the functions and kernel involved in such equations under which we assure the existence, uniqueness, and continuity of solutions in the space of functions of bounded K-variation in the sense of Korenblum, (KBV ([a,b])). [ABSTRACT FROM AUTHOR]

This paper proposes a numerical procedure for solving the nonlinear partial mixed Volterra-Fredholm integro-differential equations by two-dimensional triangular function (2D-TFs). The integration and differentiation in two-dimensional spaces have been presented for an operational matrix on triangular functions, whereas by converting the nonlinear partial mixedVolterra-Fredholm integro-differential equation to a system of algebraic by using these matrices. Some numerical examples, have been proposed to obtain the accuracy and effectiveness of the method. [ABSTRACT FROM AUTHOR]

The convergence of normal S-iterative method to solution of a nonlinear Fredholm integral equation with modified argument is established. The corresponding data dependence result has also been proved. An example in support of the established results is included in our analysis. [ABSTRACT FROM AUTHOR]

This research aims to investigate a novel coincidence point (cp) of generalized multivalued contraction (gmc) mapping involved a directed graph in b-metric spaces (b-ms). An example and some corollaries are derived to strengthen our main theoretical results. We end the manuscript with two important applications, one of them is interested in finding a solution to the system of nonlinear integral equations (nie) and the other one relies on the existence of a solution to fractional integral equations (fie). [ABSTRACT FROM AUTHOR]

Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods. [ABSTRACT FROM AUTHOR]

We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the ω condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study. [ABSTRACT FROM AUTHOR]

A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds. [ABSTRACT FROM AUTHOR]