Numerical Analysis. With Emphasis on the Application of Numerical Techniques to Problems of Infinitesimal Calculus in Single Variable. By Z. Kopal. Pp. xiv, 556. 63s. 1955. (Chapman & Hall, London)

The mathematical model for many problems is arising in different industries of natural science, basically formulated using differential, integral and integro-differential equations. The investigation of these equations is conducted with the help of numerical integration theory. It is commonly known that a class of problems can be solved by applying numerical integration. The construction of the quadrature formula has a direct relation with the computation of definite integrals. The theory of definite integrals is used in geometry, physics, mechanics and in other related subjects of science. In this work, the existence and uniqueness of the solution of above-mentioned equations are investigated. By this way, the domain has been defined in which the solution of these problems is equivalent. All proposed four problems can be solved using one and the same methods. We define some domains in which the solution of one of these problems is also the solution of the other problems. Some stable methods with the degree p<=8 are constructed to solve some problems, and obtained results are compared with other known methods. In addition, symmetric methods are constructed for comparing them with other well-known methods in some symmetric and asymmetric mathematical problems. Some of our constructed methods are compared with Gauss methods. In addition, symmetric methods are constructed for comparing them with other well-known methods in some symmetric and asymmetric mathematical problems. Some of our constructed methods are compared with Gauss methods. On the intersection of multistep and hybrid methods have been constructed multistep methods and have been proved that these methods are more exact than others. And also has been shown that, hybrid methods constructed here are more exact than Gauss methods. Noted that constructed here hybrid methods preserves the properties of the Gauss method.

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Novel Symmetric Numerical Methods for Solving Symmetric Mathematical Problems

The present paper is devoted to the application of local polynomial and non-polynomial
interpolation splines of the third order of approximation for the numerical solution of the Volterra integral
equation of the second kind. Computational schemes based on the use of the splines include the ability to
calculate the integrals over the kernel multiplied by the basis function which are present in the computational
methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra
integral equations is also discussed. The results of the numerical experiments are presented.

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Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

In this note we deal with quantitative approximation with bounded linear operators defined on C(X),in particular with positive ones, where C(X)=C_R (X,d) denotes the Banach lattice of real-valued continuous functions defined on the compact metric space (X, d) with norm given by ‖f‖x={max|f(x)|, x∈X}.We also assume that X has a diameter of d(X)>0.The first such a theorem for general positive linear operators and X=[a,b] equipped with the Euclidian distance is due to Mamedov.For

On Quantitative Approximation

As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems. The Tikhonov regularization method is well known. This method is usually applied to an integral equation and a system of linear algebraic equations. The authors firstly propose to reduce the integral equation of the first kind to a system of linear algebraic equations. This system is usually extremely ill-posed. Therefore, it is necessary to carry out the Tikhonov regularization for the system of equations. In this paper, to form a system of linear algebraic equations, local polynomial and non-polynomial spline approximations of the second order of approximation are used. The results of numerical experiments are presented.

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On the Solution of Fredholm Integral Equations of the First Kind

This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

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Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

/pdf/approximation-by-the-third-order-splines-on-uniform-and-non-3dpbvi4gel.pdf

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

This paper is devoted to the construction of localapproximations of functions of one and two variables using thepolynomial, the trigonometric, and the exponential splines. Thesesplines are useful for visualizing flows of graphic information.Here, we also discuss the parallelization of computations. Someattention is paid to obtaining two-sided estimates of theapproximations using interval analysis methods. Particularattention is paid to solving the boundary value problem by usingthe polynomial splines and the trigonometric splines of the thirdand fourth order approximation. Using the considered splines,formulas for a numerical differentiation are constructed. Theseformulas are used to construct computational schemes for solvinga parabolic problem. Questions of approximation and stability ofthe obtained schemes are considered. Numerical examples arepresented.

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I. G. Burova

Papers

Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

On the Solution of Fredholm Integral Equations of the First Kind

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

Approximations with polynomial, trigonometric, exponential splines of the third order and boundary value problem