Solution Of Nonlinear Integral Equation Research Articles

Solvability of mixed type operator equations

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.

On a fixed point theorem of Krasnosel'skii type and application to integral equations

AbstractThis paper presents a remark on a fixed point theorem of Krasnosel'skii type. This result is applied to prove the existence of asymptotically stable solutions of nonlinear integral equations.

An approximation method for the solution of nonlinear integral equations

A Chebyshev collocation method has been presented to solve nonlinear integral equations in terms of Chebyshev polynomials. This method transforms the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients. Finally, some examples are presented to illustrate the method and results discussed.

Wavelet Galerkin method for numerical solution of nonlinear integral equation

In this paper, the continues Legendre wavelets constructed on the interval [0, 1] are used to solve the nonlinear Volterra and Fredholm integral equation of the second kind.The nonlinear part of the integral equation is approximated by Legendre wavelets, and the nonlinear integral equation is reduced to a system of nonlinear equations. Numerical examples illustrates the pertinent features of the method.

On the existence of continuous solutions of nonlinear integral equations

In this paper, we study the existence of continuous solutions over compact intervals of some nonlinear integral equations. A special interest is devoted to the study of the existence as well as the uniqueness of nonlinear Volterra equations.

Existence and approximate solutions of nonlinear integral equations

We investigate the existence of continuous solutions on compact intervals of some nonlinear integral equations. The existence of such solutions is based on some well-known fixed point theorems in Banach spaces such as Schaefer fixed point theorem, Schauder fixed point theorem, and Leray-Schauder principle. A special interest is devoted to the study of nonlinear Volterra equations and to the numerical treatment of these equations.

A solution of a nonlinear integral equation

In this work, we will prove the existence of integrable solution of nonlinear integral equation, of type Hammerstein–Volterra of the second kind, by using the technique of measure of weak noncompactness and Schauder fixed point theorem. Also, the solution of the integral equation is obtained and studied.

On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock–Kurzweil integrals

We prove some existence theorems for nonlinear integral equations of the Urysohn type $x(t)=\varphi (t)+\lambda \int _0^a f(t,s,x(s))\, ds$ and Volterra type $x(t)=\varphi (t)+\int _0^tf(t,s,x(s))\, ds$, $t\in I_a=[0,a]$, where $f$ and $\varphi $ are func

Approximate Solution of Nonlinear Integral Equations of the Theory of Developing Systems

Approximate Solution of Nonlinear Integral Equations of the Theory of Developing Systems

On periodic solutions of nonlinear integral equations modelling infectious disease on measure chain

The modelling of the spread of infectious disease is carried out for time t on a measure chain T. Our approach unifies the continuous case (t∈ R) and the discrete case (t∈ Z) . The model is described by the integral equation x(t)= ∫ t−τ t f(s,x(s)) Δs, t∈ T ∩[τ,∞) where x( t) represents the proportion of the population infected at time t, f( t, x( t)) denotes the proportion of the population newly infected per unit time, and τ is the length of time an individual remains infectious. Using the measure chain calculus, we shall develop criteria for the existence of a nontrivial and nonnegative periodic solution for the modelling equation. The criteria can be implemented numerically, for this we shall give an algorithm as well as illustrative examples.

On Sufficient Conditions for the Convergence of a Modified Collocation–Iterative Method for the Solution of Nonlinear Equations

We consider the problem of application of the collocation–iterative method to the solution of nonlinear integral equations. We construct an algorithm of the method and establish sufficient conditions for the convergence of the method.

Positons: Slowly Decreasing Analogues of Solitons

We present an introduction to positon theory, almost never covered in the Russian scientific literature. Positons are long-range analogues of solitons and are slowly decreasing, oscillating solutions of nonlinear integrable equations of the KdV type. Positon and soliton–positon solutions of the KdV equation, first constructed and analyzed about a decade ago, were then constructed for several other models: for the mKdV equation, the Toda chain, the NS equation, as well as the sinh-Gordon equation and its lattice analogue. Under a proper choice of the scattering data, the one-positon and multipositon potentials have a remarkable property: the corresponding reflection coefficient is zero, but the transmission coefficient is unity (as is known, the latter does not hold for the standard short-range reflectionless potentials).

Elliptic solitons with free constants and their isospectral deformations

New exact solvable elliptic potentials with free constants for the spectral problems of the third order are found. A time dependence of such potentials gives their isospectral deformations and solutions of nonlinear integrable equations.

Rekonstruktion der Leitfähigkeitsverteilung mit einem Impedanztomographen

Abstract We have built an electrical impedance tomograph which makes it possible to reconstruct impedance distributions in the interior of a cylinder by means of measurements of currents and potentials on its boundary. The reconstuction algorithm is based on Newton's algorithm for the solution of non-linear integral equations, where the electric fields are calculated by the methods of finite elements and finite integrals. The results can be represented graphically on a PC. The resolution is fine enough to envisage applications in medical diagnostcs.

Multiple nonnegative solutions of nonlinear integral equations on compact and semi-infinite intervals

Existence results are presented which guarantee the existence of multiple nonnegative continuous solutions of a nonlinear integral equation on both a compact interval and semi-infinite interval

Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations

The spectral determinant $D(E)$ of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the $A_3$-related $Y$-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to give an alternative integral expression for $D(E)$. Generalising this result, we conjecture a relationship between the $x^{2M}$ anharmonic oscillators and the $A_{2M-1}$ TBA systems. Finally, spectral determinants for general $|x|^{\alpha}$ potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models.

Analytic solution of nonlinear integral equations of Hammerstein type

The decomposition method is applied to Hammerstein integral equations.

Error estimation and optimization in C-space of Monte Carlo iterative solution of nonlinear integral equations

Article Error estimation and optimization in C-space of Monte Carlo iterative solution of nonlinear integral equations was published on January 1, 1998 in the journal Monte Carlo Methods and Applications (volume 4, issue 1).

On special function solutions to nonlinear integrable equations

A method for constructing special function solutions of nonlinear integrable equations is given. It is based on Darboux transformations and the integral representation of special functions. Some explicit examples including the q-hypergeometric function are presented.

Almost periodic solutions of nonlinear integral equations

Almost periodic solutions of nonlinear integral equations