scholarly journals On Some Comparison of Computing Indefinite Integrals with the Solution of the Initial-value Problem for ODE

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Initial Value Problem ◽  
Multistep Methods ◽  
Quadrature Formulas ◽  
Definite Integral ◽  
Initial Value ◽  
Definite Integrals ◽  
Quadrature Methods ◽  
Mathematical Problems ◽  
Interpolation Polynomials ◽  
Special Case

The initial-value problem for the ODE is one of the classical mathematical problems, which was fundamentally investigated by many authors. This problem has been basically studied by using the quadrature formulas. Note that in the construction of quadrature formulas are used interpolation polynomials with different properties. Here, has been established some connection between the ODE and definite integrals, by using of which have constructed effective methods for computing of definite integrals. By using some multistep methods have demonstrated the advantage of the multistep methods. And also demonstrated the advantages of the proposed here methods in the construction of which didn’t use the theory of interpolation polynomials. Quadrature methods are studied as the special case of the multistep methods. And also have determined the maximal order of the quadrature method. Here received the apriori estimation for the errors of quadrature methods. Proposed concrete methods some of which have applied to the computing of the model definite integral.

scholarly journals On approximation of definite integrals by compound quadrature formulas using derivatives

2021 ◽  
pp. 88-104
Author(s):  
В.В. Шустов
Keyword(s):  
Numerical Methods ◽  
Integration Method ◽  
Quadrature Formulas ◽  
Definite Integral ◽  
Definite Integrals ◽  
Nodal Points ◽  
Point Integration ◽  
The Given ◽  
Maclaurin Formula

Рассмотрена задача вычисления определенного интеграла функции, для которой известны значения ее самой и набора производных до заданного порядка в точках отрезка интегрирования. Построены составные квадратурные формулы, которые используют значения функции и ее производных до m-го порядка включительно. Получено представление остаточного члена, выраженное через производную соответствующего порядка и число узловых точек. Приведены примеры интегрирования заданных функций с исследованием погрешности и ее оценки. Дано сравнение с известными численными методами и формулой Эйлера-Маклорена, которое показало повышенную точность и лучшую сходимость метода двухточечного интегрирования The problem of computing a definite integral of a function for which the values of itself and the set of derivatives up to a given order at the points of the interval of integration are known is considered. Composite quadrature formulas are constructed that use the values of the function and its derivatives up to the m-th order inclusive. A representation of the remainder is obtained, expressed in terms of the derivative of the corresponding order and the number of nodal points. Examples of integration of the given functions with the study of the error and its estimation are given. A comparison is made with the known numerical methods and the Euler-Maclaurin formula, which showed increased accuracy and better convergence of the two-point integration method.

The asymptotic discretization error of a class of methods for solving ordinary differential equations

1967 ◽  
Vol 63 (2) ◽  
pp. 461-472 ◽  
Author(s):  
J. M. Watt
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Initial Value Problem ◽  
General Class ◽  
Asymptotic Form ◽  
Multistep Methods ◽  
Discretization Error ◽  
Linear Multistep Methods ◽  
Initial Value

AbstractThe order and asymptotic form of the error of a general class of numerical method for solving the initial value problem for systems of ordinary differential equations is considered. Previously only the convergence of the methods, which include Runge-Kutta and linear multistep methods, has been discussed.

scholarly journals Novel Symmetric Numerical Methods for Solving Symmetric Mathematical Problems

2021 ◽  
Vol 15 ◽  
pp. 1545-1557
Author(s):  
V. R. Ibrahimov ◽  
G.Yu. Mehdiyeva ◽  
Xiao-Guang Yue ◽  
Mohammed K.A. Kaabar ◽  
Samad Noeiaghdam ◽  
...  
Keyword(s):  
Numerical Integration ◽  
Hybrid Methods ◽  
Multistep Methods ◽  
Integration Theory ◽  
Symmetric Methods ◽  
Definite Integrals ◽  
Gauss Method ◽  
Mathematical Problems ◽  
Uniqueness Of The Solution ◽  
The Mathematical Model

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.

scholarly journals A Class of Adams-like Implicit Collocation Methods of Higher Orders for the solutions of Initial Value Problems

2011 ◽  
Vol 8 (1) ◽  
pp. 47-51
Author(s):  
J. O. Fatokun ◽  
Tsaku. Nuhu ◽  
I. K. O. Ajibola
Keyword(s):  
Numerical Experiment ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Multistep Methods ◽  
Research Work ◽  
Quadrature Rule ◽  
Collocation Methods ◽  
Initial Value ◽  
Adams Methods ◽  
Predictor Corrector

The focus of this research work is the derivation of a class of Adams-like collocation multistep methods of orders not exceeding p=9. Numerical quadrature rule is used to derive steps k= 3,...,8 of the Adams methods. Convergence of each formula derived is established in this paper. As a numerical experiment, the step six pair of the Adams method so derived was used as predictor-corrector pair to solve a non-stiff initial value problem. The absolute errors show an accuracy of o(h7).

scholarly journals Cubature method for the numerical solution of the characteristic initial value problem

1968 ◽  
Vol 8 (2) ◽  
pp. 355-368 ◽  
Author(s):  
M. K. Jain ◽  
K. D. Sharma
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Goursat Problem ◽  
Euler Method ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Value ◽  
Runge Kutta Method ◽  
Quadrature Methods ◽  
Adams Methods ◽  
Image Position

The resemblance of the Goursat problem for the hyperbolic partial differential equations to the initial value problem for the ordinary differential equations has suggested the extension of many well known numerical methods existing for (1.2) to the numerical treatment of (1.1). Day [2] discusses the quadrature methods while Diaz [3] generalizes the simple Euler-method. Moore [6] gives an analogue to the fourth order Runge-Kutta-method and Tornig [7] generalizes the explicit and implicit Adams-methods.

scholarly journals Multistep Methods of the Hybrid Type and Their Application to Solve the Second Kind Volterra Integral Equation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1087
Author(s):  
Vagif Ibrahimov ◽  
Mehriban Imanova
Keyword(s):  
Initial Value Problem ◽  
Multistep Methods ◽  
Sufficient Conditions ◽  
Algebraic Equations ◽  
Exact Methods ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Advantages And Disadvantages ◽  
Construction Methods ◽  
Second Derivative Method

There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from them. For this, we have used multistep methods of advanced and hybrid types for the construction methods, with the best properties of the intersection of them. We also show some connection of the methods constructed here with the methods which are using solving of the initial-value problem for ODEs of the first order. Some of the constructed methods have been applied to solve model problems. A formula is proposed to determine the maximal values of the order of accuracy for the stable and unstable methods, constructed here. Note that to construct the new methods, here we propose to use the system of algebraic equations which allows us to construct methods with the best properties by using the minimal volume of the computational works at each step. For the construction of more exact methods, here we have proposed to use the multistep second derivative method, which has comparisons with the known methods. We have constructed some formulas to determine the maximal order of accuracy, and also determined the necessary and sufficient conditions for the convergence of the methods constructed here. One can proved by multistep methods, which are usually applied to solve the initial-value problem for ODE, demonstrating the applications of these methods to solve Volterra integro-differential equations. For the illustration of the results, we have constructed some concrete methods, and one of them has been applied to solve a model equation.

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