On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

We study a system of nonlinear integral equations of the Hammerstein–Volterra type on a half-line in the supercritical case. We show that this system has a one-parameter family of positive integrable bounded solutions. We describe the structure of each solution in this family. The monotone dependence of the solutions on the parameter is proved.

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

Consider a system of nonlinear integral equations of Volterra-type (P)x(t) f (t) + [o g(t, s, x(s))ds.Recently R. K. Miller and G. R. Sell [1] proved some fundamental theorems of (P) under airly general assumptions on f(t) and g (t, s, x) similar o the Carathodory-ype.They showed that the cross-section F(t)=(y'y=x(t), where x is some solution of (P)} is compact in R for all t e [0, a), where a is either c or a finite number such that there is a solution x(t) of (P) or which lim sup.x(t) c.This appears to be a generalization of H. Kneser's theorem to integral equations.

Abstract. This paper presents a novel approach for solving nonlinear integral equations of Fredholm and Volterra types using a multi-layer neural network (MLNN) trained through unrestricted optimization. The learning process involves minimizing the integral equation error function using the Nelder-Mead simplex method (NMSM), enabling efficient convergence in both Fredholm and Volterra equations. Our method ensures high accuracy despite potentially time-consuming computations for complex problems, maintaining convergence and approximation precision. Furthermore, the proposed approach demonstrates robustness against high-dimensional and variant equations. Numerical experiments employing the NMSM showcase its superior performance compared to existing methods. This research offers a promising avenue for solving a system of nonlinear integral equations efficiently and accurately, with broad applicability across diverse domains.

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analogy have direct applications in various areas of physics and biology. In particular, scalar and vector equations of this nature are encountered in the dynamic theory of p-adic strings, in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of the spread of epidemic diseases. A constructive theorem of existence in the space of continuous and bounded vector functions is proved. The integral asymptotic of the constructed solution is studied. An effective iterative process for constructing an approximate solution to this system is proposed. In a certain class of bounded vector functions, the uniqueness of the solution is proved. In the class of bounded vector functions (with non-negative coordinates) with a zero limit at infinity, the absence of an identically zero solution is proved. Examples of the matrix kernel and nonlinearities are given to illustrate the importance of the obtained results. Some of the examples have applications in the above-mentioned branches of natural science.

Solving a system of nonlinear integral equations by an RBF network

One of the methods for studying the solvability of infinite systems of integral or differential equations is the application of various fixed-point theorems to operators acting in appropriate functional Banach spaces. This method is fairly well developed, frequently used, and effective in many situations. However, there are cases in which certain infinite systems of differential equations arise—linked to the modeling of significant real-world phenomena—where this method, based on situating considerations within Banach spaces, fails and cannot be applied. In this paper, we propose a slightly different approach, which involves conducting the analysis within appropriate functional Fréchet spaces. We discuss the fundamental properties of these spaces and formulate compactness criteria. The main result of this paper is a positive answer, using the proposed method, to an open problem concerning the modeling of a stochastic birth-and-death process, as formulated in one of the cited publications. The most important conclusion is that the presented computational technique, based on functional Fréchet spaces, can be regarded as a more effective alternative to methods based on Banach spaces.

The study of radiative heat transfer with scattering usually leads to the solution of singular Fredholm integral equations. The present paper presents an accurate and efficient numerical method to solve certain integral equations that govern radiative equilibrium problems in plane-parallel geometry for both grey and nongrey, anisotropically scattering media. In particular, the nongrey problem is represented by a spectral integral of a system of nonlinear integral equations in space, which has not been solved previously. The numerical technique is constructed to handle this unique nongrey governing equation as well as the difficulties caused by singular kernels. Example problems are solved and the method's accuracy and computational speed are analyzed.

The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.

In this paper, a two-dimensional problem of excitation of the perfect conducting cylindrical surface, coated with the MM, by a magnetic current filament is examined. On a cylindrical surface parallel to element of cylinder the narrow nonlinear loads (NLs) are placed. The problem is solved using nonlinear boundary conditions on the surface of NL and is reduced to the system of nonlinear integral equations relatively frequency harmonics of surface currents on NLs.

In the present work we take a system of two integral equations and prove the existence and uniqueness of their solution. We investigate four aspects of the problem, namely, error estimation and rate of convergence of the iteration leading to the solution, Ulam-Hyers stability, well-posedness and data dependence of the solution sets. We give some new de nitions pertaining to the system we analyze here. In order to establish our results we utilize the coupled contraction mapping principle due to Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65(2006), 1379-1393) and several related results which we deduce here.

For a jet incident on a porous wall at which the normal fluid speed is specified, it is found that the problem of determining the free surface of the jet is governed by a system of nonlinear integral equations relating the flow angles on the boundary, on the free surface, and on the porous wall. With a constant normal speed at the porous wall, the system reduces to an integral equation for the flow angle, which is solved numerically; and a comparison with previous results is made. Numerical results, corresponding to different nonconstant normal jet speeds along the porous wall, are also presented. The extension of this formulation to include the effect of gravity is also given.

More often than not, the non-linear problems of electrical engineering arise from discrete physical systems and are usually reducible mathematically to the solution of systems of non-linear total differential equations or to systems of non-linear integral equations. Six independent methods of solving discrete non-linear problems are given in this paper. Each method is illustrated by means of an electrical engineering problem. The illustrative examples employed pertain to a-c and d-c non-linear circuits, reluctance-induction motors, hunting of and dynamic braking of synchronous machines. References to additional methods are given in the bibliography. The 88 given references listed represent approximately ten per cent of the field, but many of the entries contain a bibliography on their respective fields. The accelerated growth of research in the field of non-linearity is due to different causes. The general advancement of science requires increasingly more precise expressions for the laws of science. Accurate non-linear equations frequently depart from the linearized or postulated linear equations which have been previously used for approximate results. The quest for perfection and generalization and the love of difficult investigations by professional mathematicians play a large part in this growth. A recent incentive is the increasingly exacting requirements of modern manufacturing. These requirements are born of the competitive necessity of producing ever improved machines and equipment in the most economical manner.

This live demonstration implements an established signal flow platform with its foundation derived from a system of nonlinear integral equations in a MATLAB simulation environment, characterizing the functional behavior of the signal that traverses from the photoreceptor to the ganglion cell in the vision processing architecture. While an increase in computational speed over the conventional method of solving a system of nonlinear ordinary differential equations (ODEs) has been confirmed for a single ganglion cell, the notion is extended to a retinal dual-pathway simulation which provides for a significantly improved adoption for organic mechanisms. There are various numerical methods in solving such a system which all have a bearing on both speed and error, and as such, systematic analyses using two common forms of integral solving methods are shown to improve the overall performance of simulating the extended pathway of the retinal model.

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.