Integral Equations Of Mixed Type Research Articles

Legendre spectral projection methods for weakly singular Hammerstein integral equations of mixed type

This work provides the Legendre spectral projection (Galerkin and collocation), iterated Legendre spectral projection, Legendre spectral multi-projection and iterated Legendre spectral multi-projection methods to approximate the solution of weakly singular Hammerstein integral equations of mixed type. The convergence rates of approximate solutions to the exact solutions are obtained for all the above four methods in both $$L^{2}$$ and infinity norm. The comparison of convergence rates for all these methods have been discussed. We also have shown that iterated Galerkin improves over Galerkin, multi-Galerkin improves over iterated Galerkin and iterated multi-Galerkin improves over multi-Galerkin in $$L^2$$ norm using Legendre polynomial bases.

" Existence of asymptotically stable solutions to a nonlinear integral equation of mixed type"

" Existence of asymptotically stable solutions to a nonlinear integral equation of mixed type"

Erratum to: Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type

Erratum to: Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type

Fixed point theorems in locally convex spaces and a nonlinear integral equation of mixed type

AbstractIn this paper, we provide a new approach for discussing the solvability of a class of operator equations by establishing fixed point theorems in locally convex spaces. Our results are obtained extend some Krasnosel’skii type fixed point theorems. As an application, we investigate the existence and global attractivity of solutions for a general nonlinear integral equation of mixed type of Urysohn and Volterra.

On a Class of Quadratic Urysohn–Hammerstein Integral Equations of Mixed Type and Initial Value Problem of Fractional Order

In this paper, we study the existence of monotonic solution of a general nonlinear functional quadratic Urysohn–Hammerstein integral equations of mixed type in the class of Lebesgue integrable functions on the half-axis (\({{\mathbb{R}}^+ = [0, \infty))}\). As an application of our results, we prove the existence of solution for the initial value problem of fractional order. The main tools used are Darbo fixed point theorem associated with the measure of noncompactness and fractional calculus.

Convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type

In this paper, we consider the discrete Legendre spectral Galerkin and discrete Legendre spectral collocation methods to approximate the solution of mixed type Hammerstein integral equation with smooth kernels. The convergence of the discrete approximate solutions to the exact solution is discussed and the rates of convergence are obtained. We have shown that, when the quadrature rule is of certain degree of precision, the rates of convergence for the Legendre spectral Galerkin and Legendre spectral collocation methods are preserved. We obtain superconvergence rates for the iterated discrete Legendre Galerkin solution. By choosing the collocation nodes and quadrature points to be same, we also obtain superconvergence rates for the iterated discrete Legendre collocation solution.

Existence and uniqueness of the solution of fuzzy-valued integral equations of mixed type

In this paper, existence theorems for the fuzzy Volterra-Fredholm integral equations of mixed type (FVFIEMT) involving fuzzy number valued mappings have been investigated. Then, by using Banach's contraction principle, sufficient conditions for the existence of a unique solution of FVFIEMT are given. Finally, illustrative examples are presented to validate the obtained results.

Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equations of mixed type. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, $$\mathcal {O}(n^{-r})$$ in $$L^{2}$$ -norm and $$\mathcal {O}(n^{\frac{1}{2}-r})$$ in infinity norm, and the iterated Legendre Galerkin solution converges with the order $$\mathcal {O}(n^{-2r})$$ in both $$L^{2}$$ -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $$\mathcal {O}(n^{-r})$$ in both $$L^{2}$$ -norm and infinity norm, n being the highest degree of Legendre polynomial employed in the approximation and r being the smoothness of the kernels.

Numerical solution of Hammerstein integral equations of mixed type using the Sinc-collocation method

In this paper, numerical solution of Hammerstein integral equations of mixed type by means of Sinc collocation method is presented. This proposed approximation reduces these kind of nonlinear Hammerstein integral equations of mixed type to a nonlinear system of algebraic equations. We prove that the Sinc procedure for Hammerstein integral equations of mixed type converges to the real solution at an exponential rate of O(e−kN) where k>0 is a constant. Some numerical examples are provided to illustrate the accuracy and the exponential rate of convergence in the proposed method.

Global Asymptotic Stability for Nonlinear Functional Integral Equation of Mixed Type

The existence results of global asymptotic stability of the solution are proved for functional integral equation of mixed type. The measure of noncompactness and the fixed-point theorem of Darbo are the main tools in carrying out our proof. Furthermore, some examples are given to show the efficiency and usefulness of the main findings.

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225–234, 2005; Appl. Math. E-Notes 7:214–221, 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian’s accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.

Numerical solution of the dynamic model of a chemical reactor by Hybrid functions

Abstract The dynamic model of chemical reactors can be transformed to a mixed Hammerstein integral equation. This paper present an efficient numerical procedure for solving the nonlinear Hammerstein integral equations of mixed type. Our method by use of a Hybrid function and some matrix properties of these functions convert these equations to an algebraic equation. These matrices are sparse so all calculation can be easily implemented. The reliability of the proposed scheme are demonstrated by numerical experiments.

Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

A Taylor polynomial approach for solving Volterra-Hammerstein integral equations of mixed type

Integral equations of mixed Voltera-Fredholm type arise in various physical and biological problems. In the present paper, we obtain the approximate solution of the nonlinear Volterra-Hammerstein integral equations of mixed type in terms of Taylor polynomial. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

On global attractivity of solutions of a functional-integral equation

We prove an existence theorem for a quadratic functional-integral equation of mixed type. The functional-integral equation studied below contains as special cases numerous integral equations encountered in nonlinear analysis. With help of a suitable measure of noncompactness, we show that the functional integral equation of mixed type has solutions being continuous and bounded on the interval [0,1) and those solutions are globally attractive.

Existence of extremal solutions for discontinuous functional integral equations

In this work the existence of extremal solutions of nonlinear discontinuous functional integral equations of mixed type x ( t ) = q ( t ) + ∑ i = 1 3 ∫ 0 σ i ( t ) k i ( t , s ) f i ( s , x ( η i ( s ) ) ) d s is proved under mixed Lipschitz, Carathéodory and monotonicity conditions.

An existence result of asymptotically stable solutions for an integral equation of mixed type

In the present Note an existence result of asymptotically stable solutions for the integral equation x (t) = q (t) + Z t 0 K (t; s; x (s)) ds + Z 1 0 G (t; s; x (s)) ds is presented.

Some Algebraic and topological random fixed point theorems with applications to nonlinear random integral equations

In this paper some algebraic and topological random fixed point theorems are proved involving the three random operators on a Banach algebra and they are further applied to a certain nonlinear functional random integral equation of mixed type for proving the existence as well as existence of extremal random solutions under the generalized Lipschizicity, Carath´eodory and monotonicity conditions.

Local fixed point theory involving three operators in Banach algebras

The present paper studies the local versions of a fixed point theorem of Dhage (1987) in Banach algebras. An application of the newly developed fixed point theorem is also discussed for proving the existence results to a nonlinear functional integral equation of mixed type.

On existence of extremal solutions of nonlinear functional integral equations in Banach algebras

An algebraic fixed point theorem involving the three operators in a Banach algebra is proved using the properties of cones and they are further applied to a certain nonlinear integral equations of mixed type x(t)=k(t,x(μ(t)))+[f(t,x(θ(t)))](q(t)+∫0σ(t)v(t,s)g(s,x(η(s)))ds) for proving the existence of maximal and minimal solutions. Our results include the earlier fixed point theorems of Dhage (1992 and 1999) as special cases with a different but simple method.