Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

A convergence rate for approximate solutions of Fredholm integral equations of the first kind

We study the convergence rate for solving Fredholm integral equations of the first kind by using the well known collocation method. By constructing an approximate interpolation neural network, we deduce the convergence rate of the approximate solution by only using continuous functions as basis functions for the Fredholm integral equations of the first kind. This convergence rate is bounded in terms of a modulus of smoothness.     

Faedo–Galerkin Approximation of Solution for a Nonlocal Neutral Fractional Differential Equation with Deviating Argument

In this work, we study a class of nonlocal neutral fractional differential equations with deviated argument in the separable Hilbert space. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We investigate the existence and uniqueness of the mild solution for every approximate integral equation by virtue of the theory of analytic semigroup theory via the technique of Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. The Faedo–Galerkin approximation of the solution is studied and demonstrated some convergence results. Finally, we give an example.

     

On nonlinear Fredholm integral equations with non-differentiable Nemystkii operator

From decomposition method for operators, we consider a Newton-Steffensen iterative scheme for approximating a solution of nonlinear Fredholm integral equations with non-differentiable Nemystkii operator. By means of a convergence study of the iterative scheme applied to this type of nonlinear Fredholm integral equations, we obtain domains of existence and uniqueness of solution for these equations. In addition, we illustrate this study with a numerical experiment.     

Convergence in the integral metric of the general projective method for solving singular integral equations of the first kind with the Cauchy kernel

We study a projective method for solving singular integral equations of the first kind with the Cauchy kernel. Depending on the index of the equation, we introduce pairs of weight spaces which represent a restriction of the space of summable functions. We prove the correctness of the stated problem. We obtain sufficient conditions for the convergence of the projective method in the integral metric.     

求解第一类Fredholm积分方程的多层迭代算法

本文先把正则化后的第二类积分方程分解为等价的一对不含积分算子K^*K、仅含积分算子K以及K^*的方程组, 再用截断投影方法离散方程组, 采用多层迭代算法求解截断后的等价方程组, 并给出了后验参数的选择方法, 确保近似解达到最优.与传统全投影方法相比, 减少了积分计算的维数, 保持了最优收敛率. 最后, 算例说明了算法的有效性.     

Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

Convergence of spline collocation for Volterra integral equations

Estimates for step-by-step interpolation projections are established. Depending on the spectrum of the transfer matrix these estimates allow to obtain the pointwise convergence of the projectors to the identity operator or, in some limit cases, to prove stable convergence of the corresponding approximate operators of integral equations. This, via general convergence theorems for operator equations, permits to get the convergence of collocation method for Volterra integral equations of the second kind in spaces of continuous or certain times continuously differentiable functions. Applications in the case of the most practical types of splines are analyzed.     

一类奇异积分方程组的样条间接近似解法

本文利用三次复插值样条函数给了定义于复平面上光滑封闭曲线上的一类奇异积分方程组（１）的一种间接近似解法，讨论了误差估计和一致收敛性。     

Convergence of the Lavrent’ev method

The convergence of the Lavrent’ev method, which is a well-known regularization method for integral equations of the first kind, is analyzed as applied to equations with arbitrary linear bounded operators. A theorem concerning necessary and sufficient conditions for this convergence is proved. It is shown that these conditions are satisfied for two classes of integral equations that do not possess the properties required by the classical Lavrent’ev method.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Parallel regularized Newton method for nonlinear ill-posed equations

We introduce a regularized Newton method coupled with the parallel splitting-up technique for solving nonlinear ill-posed equations with smooth monotone operators. We analyze the convergence of the proposed method and carry out numerical experiments for nonlinear integral equations.     

Parameter identification in nonlocal nonlinear evolution equations

We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed.     

A New defy for iteration methods

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

An SQP method for optimal control of weakly singular Hammerstein integral equations

We investigate local convergence of an SQP method for nonlinear optimal control of weakly singular Hammerstein integral equations. Sufficient conditions for local quadratic convergence of the method are discussed.     

Uniform convergence of the rectangle method for singular integral equation with the Hölder density

We study an approximate method for solving singular integral equations. It implies an approximation of a singular operator by means of a compound quadrature formula similar to the rectangle one. The corresponding systems of linear algebraic equations are solvable if so is the integral equation, while its coefficients satisfy the strong ellipticity condition. Under these restrictions we obtain a bound for the rate of convergence of solutions of systems of linear equations to the solution of the considered integral equation in the uniform vector norm.     

Spline Collocation for Cordial Volterra Integral Equations

We study the convergence and convergence speed of two versions of spline collocation methods on the uniform grids for linear Volterra integral equations of the second kind with noncompact operators.     

The renormalizing series of some integral equations

We consider integral equations for which the perturbation expansion gives a power series in a parameter h whose coefficients are divergent integrals. We eliminate the divergent integrals by introducing a renormalizing Z(t, h) series in the minimal subtraction scheme. We investigate the convergence of the formal Z series in relation to the kernels of the integral equations. We find a relation of the renormalizing series to the Lagrange inversion series and also some other relations.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.