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In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

The paper contains a brief summary of the generalization of metrical structure regarding the fixed point theorem and corresponding operator equation problems. We observed that many researcher either tried to weaken the metrical structure, the contraction condition, or both. The idea behind this paper is to look for a minimal metrical structure to establish fixed point theorems. In this connection, we present new variants of the known fixed point theorem under non-triangular metric space (namely F-contraction, (A; S)-contraction, (A; S)-contraction). We also apply the obtain result in solving various types of operator equation problems. e.g., high-order fractional differential equation with non-local boundary conditions and non-linear integral equation problems. [ABSTRACT FROM AUTHOR]

In this article, two types of contractive conditions are introduced, namely extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ , Ω - Ϝ) -contraction has been introduced, which is called (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction. In the end, the applications of an extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required. [ABSTRACT FROM AUTHOR]

Many physical problems represented as initial and boundary value problems are usually solved by transforming them into integral equations on the half-line. Therefore, this paper discusses the Galerkin and collocation methods along with the multi-Galerkin and multi-collocation methods for non-linear integral equations with compact and Wiener-Hopf kernel operators on the half-line in the space of piecewise polynomial subspaces. As Hammerstein integral equation on half-line has the unbounded domain, the finite section approximation method is applied; and afterward, we find an approximate solution for the finite section integral equation. In order to obtain improved superconvergence rates, we apply Sunil Kumar's method and then compare proposed convergence rates with the existing results of Nahid et al. [22] both theoretically and numerically. These results are further improved by applying multi-projection methods. In addition, it has shown that the proposed theory enhances the results of [22]. Finally, numerical examples are presented to demonstrate the given theoretical framework. [ABSTRACT FROM AUTHOR]

Analytic approximations are presented for the exact solutions of the Volterra type nonlinear integral equation of the second kind for classical gaseous polytropes in closed form. This equation is considered as the integral equivalent of the Lane-Emden differential equation with boundary conditions which describes the standard polytropic models in terms of a Cauchy problem. With the aid of a linear approximation of this equation and general heuristic considerations of a physical character, as well as with the aid of a graphical model and variation of the parameters of the approximating functions, approximate expressions for the Emden functions and the dimensionless density are obtained in closed form with a mean square accuracy from ~10-4 to a few percent for a series of values of the polytropic index n of practical interest (n = 0.5, 3, 4, 6, ∞). Our previous approximation for the spatial density of the isothermal model is compared with a pseudo-isothermal law describing the distribution of the density of dark matter surrounding spiral galaxies and used by various authors for studying their rotation curves. [ABSTRACT FROM AUTHOR]

Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra–Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach. [ABSTRACT FROM AUTHOR]

In this paper, we use Krasnoselskii's fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative 0 A B C D α (ℓ) = ζ (ℓ , (ℓ) , ℘ (ℓ)) , 1 < α ≤ 2 , 0 A B C D σ ℘ (ℓ) = ξ (ℓ , (ℓ) , ℘ (ℓ)) , 1 < σ ≤ 2 , for all ℓ ∈ [ 0 , 1 ] , with boundary conditions (0) = 0 , λ ′ (η) = γ ′ (1) , ℘ (0) = 0 , λ ℘ ′ (η) = γ ℘ ′ (1). We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers–Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs. [ABSTRACT FROM AUTHOR]

The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform is computed numerically, with all steps leading up to that executed by analytical manipulations or table look-up. Here, we consider fully numerical methods, where both forward and inverse transforms are computed numerically. Because the computation of the inverse transform has been studied extensively, this paper focus mainly on the forward transform. Existing methods for computing the forward transform based on exponential sums are considered along with a new method based on the formulas of Weeks. Numerical examples include a nonlinear integral equation of convolution type, a fractional ordinary differential equation, and a partial differential equation with an inhomogeneous boundary condition. [ABSTRACT FROM AUTHOR]

The work was carried out within the context of a competitive ideology of creating the element base of digital optical computers (transphasors, optical switches, memory elements) built on a basis other than the Fabry–Perot interferometer. Mathematical models of stationary (problem I) and non-stationary (problem II) four-beam laser interaction in optically nonlinear media are considered in detail. Problem (I) is the system of ordinary differential equations with specified boundary conditions. Problem (II) is the system of integro-differential equations with boundary conditions. We introduced the original sought-for functions z(x) and u(z, t), υ(z, t) (II). As a result, the problem (I) is reduced to solving a simple transcendental equation (canonical equation of optical hysteresis). The problem (II) is reduced to solving a system of two nonlinear integral equations for the amplitudes of interference patterns (the canonical system of equations of non-stationary optical hysteresis). [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]

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]

Propagation of leaky TE-polarized electromagnetic waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear metamaterial medium is studied. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered. [ABSTRACT FROM AUTHOR]

We prove the existence of a rolling odd solution to a boundary value problem for a convolution type nonlinear singular integral equation on the whole axis with power nonlinearity. We obtain the integral asymptotic formula for the solution and describe some examples. [ABSTRACT FROM AUTHOR]

For a three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, we studied the attenuation of ultrasonic beams after the formation of shock fronts. Earlier, in the work of one of the authors for this model, submodels were obtained and studied, described by non-stationary solutions, invariant with respect to some three-dimensional subgroups of the main ten-parameter group of the differential equation that defines this model. In our paper we obtained and studied four non-stationary submodels of this model that are invariant with respect to four-parameter subgroups of the main group of this differential equation. These submodels are new submodels and have not been previously noted in the literature. They are given by invariant solutions of rank 1. Among these 4 submodels, 2 submodels describe axisymmetric ultrasonic beams, the remaining 2 describe one-dimensional ultrasonic beams. The search for invariant solutions that define these 4 submodels is reduced to solving of nonlinear integral equations, the implicit solutions of which are obtained in the form of nonlinear algebraic equations containing transcendental functions. These submodels are used to study the propagation of ultrasonic beams, for which either the acoustic pressure and its rate of change or the acoustic pressure and its gradient are given at the initial time at a fixed point. Conditions are obtained that ensure the existence and uniqueness of solutions to boundary value problems describing these processes. This makes it possible to correctly carry out numerical calculations in the study of these processes. As a result of the numerical solution of these boundary value problems for some values of the parameters characterizing these processes, pressure distribution graphs were obtained. In all cases, ultrasonic beams are weaken monotonically with time and completely fade away in a finite time. At each point, the time of complete attenuation of ultrasonic beams is found. The obtained and studied new submodels are another step towards the creation of a database of physically significant submodels of the three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, which describes the attenuation of ultrasonic beams after the formation of shock fronts. • We study 3-dl generalized Khokhlov–Zabolotskaya–Kuznetsov​ model of nonlinear hydroacoustics in a cubic nonlinear medium. • For this model four invariant submodels are obtained. • The solutions describing these submodels are reduces to the solving of the nonlinear integral equation. • Implicit solutions of these integral equations were obtained in the form of nonlinear algebraic equations containing transcendental functions. • These 4 submodels were used to study the boundary value problems. • The existence and uniqueness of the solutions of boundary value problems are established. • For some values of the parameters the solutions of these boundary value problems are numerically solved. [ABSTRACT FROM AUTHOR]

In this paper, we derive the equivalent fractional integral equation to the nonlinear implicit fractional differential equations involving Ψ‐Hilfer fractional derivative subject to nonlocal fractional integral boundary conditions. The existence of a solution, Ulam–Hyers, and Ulam–Hyers–Rassias stability have been acquired by means of an equivalent fractional integral equation. Our investigations depend on the fixed‐point theorem due to Krasnoselskii and the Gronwall inequality involving Ψ‐Riemann–Liouville fractional integral. Finally, examples are provided to show the utilization of primary outcomes. [ABSTRACT FROM AUTHOR]

We consider propagation of leaky TE waves in a rod filled with nonlinear metamaterial medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0,T). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L2- and L ∞ - norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse scattering problem of determining the shape of a cavity with a penetrable inhomogeneous medium of compact support from one source and a knowledge of measurements placed on a curve inside the cavity. First, the boundary value problem of the partial differential equations can be transformed into an equivalent system of nonlinear and ill‐posed integral equations for the unknown boundary. Then, we apply the regularized Newton iterative method to reconstruct the boundary and prove the injectivity for the linearized system. Finally, we present some numerical examples to show the feasibility of our method. [ABSTRACT FROM AUTHOR]

An integro-differential Dirac system with a convolution kernel consisting of four independent functions is considered. We prove that the kernel is uniquely determined by specifying the spectra of two boundary value problems with one common boundary condition. The proof is based on the reduction of this nonlinear inverse problem to solving some nonlinear integral equation, which we solve globally. On this basis we also obtain a constructive procedure for solving the inverse problem along with necessary and sufficient conditions for its solvability in an appropriate class of kernels. [ABSTRACT FROM AUTHOR]

Abstract In this work the performance of four new and different approaches are presented for numerical solution of some nonlinear boundary value problems which arise in modelling a tabular adiabatic chemical reactor. Quasi-linearization and converting differential equations to integral equation techniques and derivative and integration operational matrices of Cubic B-spline wavelets via some projection methods are applied to reducing the nonlinear problem to some algebraic system. For study of effect of involved parameters in main problem and for showing the accuracy and efficiency of the introduced methods some cases of main problem are given and findings are compared with the results of alternative methods for numerical solving of this class of equations. [ABSTRACT FROM AUTHOR]

This article studies the existence and multiplicity of componentwise positive solutions for systems of nonlinear Hammerstein integral equations. In this system one nonlinear term is uniformly superlinear or uniformly sublinear, and the other is locally uniformly superlinear or locally uniformly sublinear. Discussions are undertaken by means of the fixed point index theory in cones. As applications, we show the existence and multiplicity of component-wise positive solutions for systems of second-order ordinary differential equations with the Dirichlet boundary value conditions and mixed boundary value conditions, respectively. [ABSTRACT FROM AUTHOR]

The solvability of the boundary value problem for a system of second-order nonlinear partial differential equations with given boundary conditions which describes the equilibrium of elastic inhomogeneous shallow shells with free edges in the framework of the Timoshenko shear model is considered. The boundary value problem is reduced to a single nonlinear equation whose solvability is established by using the contraction mapping principle. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse rough surface scattering problem in near-field optical imaging. This problem is actually to reconstruct the scattering surface as well as its impedance coefficient from multifrequency near-field data, and can be reduced into an integral scheme by employing an integral representation. We solve this integral scheme by a non-linear integral equation method, and further develop a fast inversion algorithm for reconstructing both the rough surface and the impedance coefficient. Numerical experiments are presented to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Abstract The main purpose of our research is to find as many as possible the solutions of the equation of the general model of nonlinear diffusion in an inhomogeneous medium and to establish their physical meaning and to apply the obtained solutions to the description nonlinear diffusion processes in an inhomogeneous medium. To achieve of this purpose the basic submodels (possessing nontrivial symmetry properties) of the general model are obtained and researched. The formulas of the production of the new solutions for the equations of these submodels are obtained. For these submodels all invariant submodels are found. The essentially distinct invariant solutions (not connected with a help of the point transformations) describing these invariant submodels are found either explicitly, or their search is reduced to the solving of the nonlinear integral equations. The physical meanings of these solutions are established. Some of the 25 explicitly found solutions describe a diffusion process only for a finite period of time, others — for an infinite period of time. Some solutions describe a nonlinear diffusion process either with fixed or evolving "black holes", in the vicinity of which the concentration infinitely increases. The presence of the arbitrary constants in the integral equations, that determine other 27 solutions opens up the new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process. For such invariant submodels, we are studied the diffusion processes, for which at the initial instant of the time at a fixed point either a concentration and rate of its change, or concentration and its gradient are given. The solving of the boundary value problems describing these processes reduces to the solving of nonlinear integral equations. The existence and uniqueness of the solutions of these boundary value problems under certain conditions are established. A mechanical relevance of the obtained solutions is as follows: 1) these solutions describe specific nonlinear diffusion processes in an inhomogeneous medium, 2) these solutions can be used as test solutions in the numerical calculations, which perform in the studies of the real diffusion processes, 3) these solutions make it possible to assess the degree of adequacy of a given mathematical models to the real physical processes, after carrying out experiments corresponding to these solutions, and estimating the resulting deviations. The obtained results can be used to study the diffusion of substances, the diffusion of conduction electrons and other particles, the diffusion of physical fields, the propagation of heat in an inhomogeneous medium. Highlights • We researched all invariant solutions of the models of nonlinear diffusion in an inhomogeneous medium. • 25 of these solutions are found explicitly. • The search for the remaining 27 solutions reduces to the solving of the nonlinear integral equations. • We researched some diffusion processes. • We established the existence and the uniqueness of the boundary value problems, describing some diffusion processes. • We pointed the mechanical relevance of the solutions proposed. [ABSTRACT FROM AUTHOR]

In this paper, we establish new fixed point results for the sum of two operators A and B, where the operator A is assumed to be weakly compact and (ws)-compact, while B is a weakly condensing and expansive operator defined on unbounded domains under different boundary conditions as well as other additional assumptions. In addition, we get new generalized forms of the Krasnosel'skii fixed point theorem in a Banach space by using the concept of measure of weak noncompactness of De Blasi. Later on, we give an application to solve a nonlinear Hammerstein integral equation in L1-space. [ABSTRACT FROM AUTHOR]

In this paper we present a numerical solution method for the Bernoulli free boundary value problem for the Laplace equation in three dimensions. We extend a nonlinear integral equation approach for the free boundary reconstruction (Kress, 2016) from the two-dimensional to the three-dimensional case. The idea of the method consists in reformulating Bernoulli’s problem as a system of boundary integral equations which are nonlinear with respect to the unknown shape of the free boundary and linear with respect to the boundary values. The system is linearized simultaneously with respect to both unknowns, i.e., it is solved by Newton iterations. In each iteration step the linearized system is solved numerically by a spectrally accurate method. After expressing the Fréchet derivatives as a linear combination of single- and double-layer potentials we obtain a local convergence result on the Newton iterations and illustrate the feasibility of the method by numerical examples. [ABSTRACT FROM AUTHOR]

We consider propagation of surface TE waves in the Goubau line (GL) assuming that the dielectric cover is non-linear and inhomogeneous. The problem at hand is reduced to a non-linear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating TE waves for the chosen nonlinearity (Kerr law) is proved by the method of contraction. Conditions under which several higher-order waves can propagate are obtained, and the intervals of the corresponding propagation constants are determined. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. In numerical experiment two types of nonlinearities are considered and compared: Kerr nonlinearity and nonlinearity with saturation. New propagation modes are found. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the existence, nonexistence, uniqueness, and multiplicity of positive solutions for a class of eigenvalue problems of nonlinear fractional differential equations with a nonlinear integral term and a disturbance parameter in the boundary conditions. By using fixed point index theory we give the critical curve of eigenvalue λ and disturbance parameter μ that divides the range of λ and μ for the existence of at least two, one, and no positive solutions for the eigenvalue problem. Furthermore, by using fixed point theorem for a sum operator with a parameter we establish the maximum eigenvalue interval for the existence of the unique positive solution for the eigenvalue problem and show that such a positive solution depends continuously on the parameter λ for given μ. In particular, we give estimates for the critical value of parameters. Two examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper studies the existence of extremal solutions for nonlinear fractional differential equations with nonlinear integral boundary conditions and explores an explicit algorithm which converges to the extremal solutions of the problem at hand. An example is discussed for the illustration of the main work. [ABSTRACT FROM AUTHOR]

An approximating functions method for solving a nonlinear Volterra integral equation is presented. The equation form is the same for various media and it does not depend on the initial signal. Other key features of this approach are the natural description of non-stationary and nonlinear phenomena, the unified definition of inner and outer problems and the inclusion of initial and boundary conditions in the same equation. It permits universal modelling algorithms for a wide range of nonlinear electrodynamic problems. A computer model based on the algorithm is developed for 1D + 1T case and the transformation of an electromagnetic pulse in a nonlinear layer is investigated. [ABSTRACT FROM AUTHOR]

The nonlinear buckling analysis of drill string in a rigid well is presented in this paper. Considering effects of friction and boundary constraints, this problem could be taken as a model of a rod laterally constrained in a rigid cylinder (horizontal, oblique and vertical rigid cylinder could be included). After introducing a new variable, the resulting coupled nonlinear integral–differential equations are successfully solved by employing the extended system shooting method. Examples with various friction coefficients and combinations of boundary conditions are proposed. It is found that the axial frictional force plays a more significant part on buckling load for horizontal well than vertical one. Compared to experimental data or results obtained by using the discrete singular convolution algorithm (DSC) and the finite element method (FEM), the accuracy of the formulations and solution procedures, is verified. What’s more, the nonlinear buckling behaviors of two instances of vertical scientific wells are analyzed. The present results are useful for practical design applications related to calculation of buckling loads and selection of bottom hole assembly (BHA) elements. [ABSTRACT FROM AUTHOR]

(0.1) {Δu+λup-1eup=0,u>0inΩ;u=0on∂Ω, where Ω is a bounded domain in =R² with smooth boundary, λ>0 is a small parameter and 0

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a novel technique is being formulated for the numerical solution of integral equations (IEs) as well as integro-differential equations (IDEs) of first and higher orders. The present approach is an improved form of the Haar wavelet methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). The proposed modifications resulted in computational efficiency and simple applicability of the earlier methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). In addition to this, the new approach is being extended from IDEs of first order to IDEs of higher orders with initial- and boundary-conditions. Unlike the methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013) (where the kernel function is being approximated by two-dimensional Haar wavelet), the kernel function in the present case is being approximated by one-dimensional Haar wavelet. The modified approach is easily extendable to higher order IDEs. Numerical examples are being included to show the accuracy and efficiency of the new method. [Copyright &y& Elsevier]

Abstract: In this paper we derive new far-field boundary conditions for the time-dependent Navier–Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedness of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier–Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations. We perform computations with a high-order finite difference scheme on summation-by-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent. [Copyright &y& Elsevier]

We consider initial-boundary value problems for the KdV equation ut + ux + 6uux+uxxx = 0 on the half-line x ⩾ 0. For a well-posed problem, the initial data u(x, 0) as well as one of the three boundary values {u(0, t ), ux(0, t ), uxx(0, t )} can be prescribed; the other two boundary values remain unknown.We provide a characterization of the unknown boundary values for the Dirichlet as well as the two Neumann problems in terms of a system of nonlinear integral equations. The characterizations are effective in the sense that the integral equations can be solved perturbatively to all orders in a well-defined recursive scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this study, the shape reconstruction of conducting cylinders by making use of electromagnetic scattered fields is presented. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated in to an optimization problem. A hybrid evolutionary algorithm suitable for the optimization of large-domain electromagnetic problems is proposed for the solution of the inverse scattering problem arising in microwave-imaging applications. The hybrid technique, called genetical swarm optimization (GSO), combines genetic algorithms (GA) and particle swarm optimization (PSO). In such a framework, this paper is aimed at assessing the effectiveness of the proposed approach in locating and shaping of unknown two-dimensional perfect conducting cylinders. Reconstructions of conducting scatterers from experimental inverse scattering data are finally presented to demonstrate the accuracy and efficiency of the hybrid technique. [Copyright &y& Elsevier]

We introduce the notion of ɛ-unique bounded solution to the nonlinear differential equation x′ = f( x) − h( t), where f: ℝ → ℝ is a continuous function and h( t) is an arbitrary continuous function bounded on ℝ. We derive necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions to this equation. [ABSTRACT FROM AUTHOR]

Abstract: We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small , where is the interest rate and is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry. [Copyright &y& Elsevier]

Abstract: By establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique, we investigate the systems of nonlinear mixed integro-differential equations with deviating arguments and mixed boundary value conditions. Sufficient conditions are established for the existence of extremal system of solutions of the given problem. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and multiplicity of solutions for two-point boundary value problems for even order ordinary differential equation. By using the critical point theory and the property of projective operator, we establish some conditions on the nonlinearity which guarantee that the nonlinear Hammerstein integral equation with quasi-positive definite kernel has at least one solution, two nontrivial solutions and finite pairs of solutions, respectively. As an application, we get the corresponding results for two-point boundary value problems for even order ordinary differential equation. Moreover, an example is presented to illustrate our main results. [Copyright &y& Elsevier]

In this article, the authors establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential inclusions involving the Caputo fractional derivative and nonlinear integral conditions. Both cases of convex and nonconvex valued right hand sides are considered. The topological structure of the set of solutions also examined. [ABSTRACT FROM AUTHOR]

In this paper, we derive some results concerning the continuous dependence on parameters of optimal solutions to an optimal control problem that involves a quasilinear hyperbolic differential equation with a boundary condition and a nonlinear integral functional of action; continuous dependence of such kind is sometimes referred to as stability or sensitivity. To present a sufficient condition for the continuous dependence, we use the Kuratowski–Painlevé upper limit of a sequence of sets. Also offered is a technical interpretation of the results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence of solutions of the operator equations in the Banach space . It is assumed the vector-valued function is nonlinear Pettis-integrable. Some additional assumptions imposed on are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type under the Pettis integrability assumption imposed on . [Copyright &y& Elsevier]

Abstract: We derive a characterization of the spectrum of the sinh-Gordon model in terms of certain nonlinear integral equations. There exists a large class of solutions to these equations which allows a continuation between the infrared and the ultraviolet limits, respectively. We present nontrivial evidence for the claim that the class of solutions in question describes the spectrum of the sinh-Gordon model completely in both of these limits. The evidence includes some nontrivial relations to Liouville theory. [Copyright &y& Elsevier]

Pointwise bounds are obtained for the solution of an initial boundary value problem for the resonant nonlinear Schrödinger equations. The context is that of a straight-line region with prescribed moving boundaries, expanding or noncontracting, upon which zero (Dirichlet) conditions are imposed. [ABSTRACT FROM AUTHOR]

Recently, for high-power ultrasonic beams, it was experimentally found that as a result of self-action, their self-focusing occurs. With self-focusing, a powerful ultrasonic beam is noticeably narrowed, has a nonlinear narrowing, and it is significantly amplified at the focus. A generalization of the three-dimensional Khokhlov–Zabolotskaya–Kuznetsov model in a cubic nonlinear medium in the presence of dissipation with a special nonlinearity coefficient describes the propagation of ultrasonic beams after self-focusing. This work is devoted to the study of submodels of this model without dissipation, which are invariant with respect to four-parameter subgroups of the main group of the equation of this model. These submodels are defined by invariant solutions of rank 0 or 1, which describe one-dimensional, plane and axisymmetric ultrasonic beams. Solutions of rank 0 and some solutions of rank 1 are found explicitly. Some of them, at each fixed moment of time, contain a destroying element in the form of an ultrasonic needle or an ultrasonic knife, which at each fixed moment of time are localized in a limited domain, on the surface of which the acoustic pressure is zero. These submodels describe one-dimensional, plane, or axisymmetric ultrasonic beams. Finding other invariant solutions of rank 1 is reduced to solving nonlinear integral equations. These invariant solutions are used to study the propagation of ultrasonic beams after self-focusing, for which either the acoustic pressure and its gradient, or the acoustic pressure and the rate of its change are given at the initial moment of time at a fixed point. Under certain additional conditions, the existence and uniqueness of the solutions to boundary value problems describing these processes are established. This makes it possible to correctly carry out numerical calculations when studying these processes. The graphs of the pressure distribution obtained as a result of the numerical solution of these boundary value problems for some values of the parameters characterizing the indicated processes are presented. These graphs show that for these submodels in the ultrasonic beams a monotonic increase in acoustic pressure occurs over time. • We researched invariant submodels of the generalization of Khokhlov–Zabolotskaya–Kuznetsov model in a cubic-nonlinear medium without dissipation. • We researched a propagation of the intensive acoustic waves after self-focusing. • We established the existence and the uniqueness of the boundary value problems, describing some wave processes. • The graphs of the pressure distribution are given. • We pointed the mechanical relevance of the solutions proposed. [ABSTRACT FROM AUTHOR]

We consider a second order nonlinear differential equation with nonlocal (integral) condition. The spectrum of it differs essentially from the known ones [ABSTRACT FROM AUTHOR]

The article examines the stabilization of solutions of an initial-boundary value problem for a nonlinear integro-differential equation. Dealing on mathematical modeling, analysis and numerical solution of diffusion problems for magnetic fields in media are a notable problem in applied mathematics. Nonlinear differential and integro-differential equations and systems of such equations arise in the modeling of such processes.