Existence of Solutions of a Convolution Integral Equation

In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball Br belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set M∈ B r the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.

EXTRAPOLATION FOR COLLOCATION METHOD OF THE FIRST KIND VOLTERRA INTEGRAL EQUATIONS

1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])

NUMERICAL ANALYSIS FOR VOLTERRA INTEGRAL EQUATION WITH TWO KINDS OF DELAY

In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1,1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L~∞ and L_(ω~c)~2-norm. In the end, we give numerical test to confirm the conclusion.

NEW NUMERICAL METHOD FOR VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND IN PIEZOELASTIC DYNAMIC PROBLEMS

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.

Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind

Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.

Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions

The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.

LEGENDRE SPECTRAL COLLOCATION METHOD FOR VOLTERRA-HAMMERSTEIN INTEGRAL EQUATION OF THE SECOND KIND

This paper is concerned with obtaining the approximate solution for Volterra- Hammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function w（x） = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L2 norm and L^∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.

ON THE REGULARIZATION METHOD OF THE FIRST KIND OFFREDHOLM INTEGRAL EQUATION WITH A COMPLEX KERNEL AND ITS APPLICATION

The regularized integrodifferential equation for the first kind of Fredholm, integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-dimensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given.

Numerical approach for a system of second kind Volterra integral equations in magneto-electro-elastic dynamic problems

The elastodynamic problems of magneto-electro-elastic hollow cylinders in the state of axisymmetric plane strain case can be transformed into two Volterra integral equations of the second kind about two functions with respect to time. Interpolation functions were introduced to approximate two unknown functions in each time subinterval and two new recursive formulae are derived. By using the recursive formulae, numerical results were obtained step by step. Under the same time step, the accuracy of the numerical results by the present method is much higher than that by the traditional quadrature method.

Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

Volterra Integral Equations and Some Nonlinear Integral Equations with Variable Limit of Integration as Generalized Moment Problems

In this paper we will see that, under certain conditions, the techniques of generalized moment problem will apply to numerically solve an Volterra integral equation of first kind or second kind. Volterra integral equation is transformed into a one-dimensional generalized moment problem, and shall apply the moment problem techniques to find a numerical approximation of the solution. Specifically you will see that solving the Volterra integral equation of first kind f（t） = {a^t K（t, s）x（s）ds a ≤ t ≤ b or solve the Volterra integral equation of the second kind x（t） =f（t）＋{a^t K（t,s）x（s）ds a ≤ t ≤ b is equivalent to solving a generalized moment problem of the form un = {a^b gn（s）x（s）ds n = 0,1,2… This shall apply for to find the solution of an integrodifferential equation of the form x＇（t） = f（t） ＋ {a^t K（t,s）x（s）ds for a ≤ t ≤ b and x（a） = a0 Also considering the nonlinear integral equation： f（x）= {fa^x y（x-t）y（t）dt This integral equation is transformed a two-dimensional generalized moment problem. In all cases, we will find an approximated solution and bounds for the error of the estimated solution using the techniques ofgeneralized moment problem.

Existence of Nontrivial Solutions of A Nonlinear Volterra Integral Equation

Some new results on the existence of nonnegative nontrivial solutions of the nonlinear Volterra integral equation,u(x)= k(x,s)g(u(s))ds,(x≥0),are given and their applications are shown.

GLOBAL SOLUTIONS OF SYSTEMS OF NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained, which extend the related results for this class of equations on a finite interval with a finite. number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.

Function-valued partial Padé-type approximation method for estimating eigenvalues of Fredholm integral equation of the second kind

Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.

USING FREDHOLM INTEGRAL EQUATION OF THE SECOND KINDTO SOLVE THE VERTICAL VIBRATION OF ELASTICPLATE ON AN ELASTIC HALF SPACE

The dual integral equations of vertical forced vibration of elastic plate on an elastic half space subject to harmonic uniform distribution loading are established according to the mixed boundary-value condition. By applying Abel transformation the dual integral equations are reduced to Fredholm integral equation of the second kind which is solved numerically.

Numerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method

Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.

SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.

The Successive Approximation Method for Solving Nonlinear Fredholm Integral Equation of the Second Kind Using Maple

In this paper, we will use the successive approximation method for solving Fredholm integral equation of the second kind using Maple18. By means of this method, an algorithm is successfully established for solving the non-linear Fredholm integral equation of the second kind. Finally, several examples are presented to illustrate the application of the algorithm and results appear that this method is very effective and convenient to solve these equations.