Volterra-Hammerstein Integral Equations Research Articles

Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations

In this work, we propose an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra-Hammerstein integral equations. This extension is based on the operational Tau method with arbitrary polynomial basis functions for constructing the algebraic equivalent representation of the problem. This representation is an special semi lower triangular system whose solution gives the components of the vector solution. We will show that the operational Tau matrix representation for the integration of the nonlinear function can be represented by an upper triangular Toeplitz matrix. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results. Our numerical experiments show that the accuracy of the Tau approximate solution is independent of the selection of the basis functions.

A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type

A numerical method for solving Fredholm–Volterra Hammerstein integral equations is presented. This method is based on replacement of the unknown function by truncated series of well known Chebyshev expansion of functions.The quadrature formula which we use to calculate integral terms can be estimated by Fast Fourier Transform (FFT). The numerical examples and the number of operations show the advantages of this method to some other usual methods.

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 nonlinear integral equations and Λ-bounded variation

We discuss the existence or the existence and uniqueness of global and local Λ-bounded variation (ΛBV) solutions as well as continuous ΛBV-solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations formulated in terms of the Lebesgue integral. Since the space of functions of bounded variation in the sense of Jordan is a proper subspace of functions of Λ-bounded variation and for some class of functions φ, the space of functions of bounded φ-variation in the sense of Young is also a proper subspace of the space under consideration, our results extend known results in the literature.

Error estimation problem of nonlinear integral equations in the space Lp( p ⩾ 1)

The main purpose of this paper is to intoduce and study approximate solution method for nonlinear Volterra–Hammerstein integral equations with an arbitrary smooth (nonlinear) kernel function. At the end of this paper, numerical example is treated.

BVφ-solutions of nonlinear integral equations

In this paper we investigate solutions of nonlinear Hammerstein and Volterra–Hammerstein integral equations in the space of functions of bounded φ-variation in the sense of Young. We prove the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions are under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray–Schauder alternative for contractions.

On BV-Solutions of Some Nonlinear Integral Equations

In this paper we prove uniqueness theorems for bounded variation (shortly: BV) solutions and continuous BV-solutions of the Hammerstein and the Volterra–Hammerstein integral equations. We investigate real-valued functions and functions with values in a Banach space.

Numerical solution of non-stationary aero-autoelasticity integro-differential operator equations

The spectral method of G. N. Elnagar, which yields spectral convergence rate for the approximate solutions of Fredholm and Volterra-Hammerstein integral equations, is generalized in order to solve the larger class of integro-differential functional operator equations with spectral accuracy. In order to obtain spectrally accurate solutions, the grids on which the above class of problems is to be solved must also be obtained by spectrally accurate techniques. The proposed method is based on the idea of relating, spectrally constructed, grid points to the structure of projection operators which will be used to approximate the control vector and the associated state vector. These projection operators are spectrally constructed using Chebyshev-Gauss-Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. Simulation studies demonstrate computational advantages relative to other methods in the literature.

Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations

In this paper, the Chebyshev spectral (CS) method for the approximate solution of nonlinear Volterra-Hammerstein integral equations (τ)= F(τ)+ ∫ o τ K(τr)G(r,Y(r)) dr,τ∈[0,T] is investigated. The method is applied to approximate the solution not to the equation in its original form, but rather to an equivalent equation z( t)= g( t, y( t)), t ∈ [−1, 1]. The function z is approximated by the Nth degree interpolating polynomial z N , with coefficients determined by discretizing g( t, y( t)) at the Chebyshev-Gauss Labatto nodes. We then define the approximation to y to be of the form y N(t)=f(t)+ ∫ -1 1 K(t,s)z N(s) ds,τ∈[-1,1] and establish that, under suitable conditions, lim N→∞ y N ( t) = y( t) uniformly in t. Finally, a numerical experiment for a nonlinear Volterra-Hammerstein integral equation is presented, which confirms the convergence, demonstrates the applicability and the accuracy of the Chebyshev spectral (CS) method.

Asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations

Recently, Kumar and Sloan introduced a new collocation-type method for the numerical solution of Fredholm-Hammerstein integral equations. Brunner applied this method to nonlinear Volterra integral and integro-differential equations and discussed its connection with the iterated collocation method. In the present paper, the asymptotic error expansion of this method for nonlinear Volterra integral equations at mesh points is obtained. We show that when piecewise polynomials of Θin p-1 are used, the approximate solution admits an error expansion in powers of stepsize h, beginning with a term h p . For a special choice of the collocation points, the leading terms in the error expansion for the collocation solution contain only even powers of the stepsize h, beginning with a term h 2 p . Thus Richardson's extrapolation can be performed on the solution; and this will greatly increase the accuracy of the numerical solution.

On the boundedness and the asymptotic behaviour of the non-negative solutions of Volterra-Hammerstein integral equations

We study the Volterra-Hammerstein integral equation $$U(t,x) = U_O (t,x) + \mathop \smallint \limits_O^t \mathop \smallint \limits_D f(y, U (t - s,y)) h (x,y,s)dsdy,$$ t≥0, x∈D. We derive sufficient conditions for the boundedness of all non-negative solutions U. We show that, for bounded non-negative solutions U, U(t,.) is positive on D for sufficiently large t>0, if we impose appropriate positivity assumptions on f and h. If we additionally assume that, for y∈D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases, the following alternative holds for any bounded non-negative solution U: Either U(t,.) converges toward zero for t→∞, pointwise on D, or U(t,.) converges, for t→∞, toward the unique bounded positive solution of the corresponding Hammerstein integral equation, uniformly on D. We indicate conditions for the occurrence of each of the two cases.