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 On the Comparison of Gauss and Hybrid Methods and their Application to Calculation of Definite Integrals

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Necessary Conditions ◽  
Hybrid Methods ◽  
Definite Integral ◽  
Implicit Methods ◽  
Definite Integrals ◽  
Advantages And Disadvantages ◽  
Chebyshev Method ◽  
Gauss Method ◽  
Gauss Quadrature Formula ◽  
To Receive

As is known there is the wide class of methods for calculation of the definite integrals constructed by the well-known scientists as Newton, Gauss, Chebyshev, Cotes, Simpson, Krylov and etc. It seems that to receive a new result in this area is impossible. The aim of this work is the applied some general form of hybrid methods to computation of definite integral and compares that with the Gauss method. The generalization of the Gauss quadrature formula have been fulfilled in two directions. One of these directions is the using of the implicit methods and the other is the using of the advanced (forward-jumping) methods. Here have compared these methods by shown its advantages and disadvantages in the results of which have recommended to use the implicit method with the special structure. And also are constructed methods, which have applied to calculation of the definite integral with the symmetric bounders. As is known, one of the popular methods for calculation of the definite integrals with the symmetric bounders is the Chebyshev method. Therefore, here have defined some relations between of the above mentioned methods. For the application constructed, here methods are defined the necessary conditions for its convergence. The receive results have illustrated by calculation the values for some model integral using the methods with the degree p  8.

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.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

SENSITIVITY ANALYSIS AND IDENTIFIABILITY OF A MATHEMATICAL MODEL OF A CHEMICAL REACTION

2021 ◽  
pp. 14-18
Author(s):  
Л.Ф. Сафиуллина
Keyword(s):  
Mathematical Model ◽  
Inverse Problem ◽  
Sensitivity Analysis ◽  
Chemical Reaction ◽  
Reaction Model ◽  
Numerical Calculations ◽  
Specific Reaction ◽  
Uniqueness Of The Solution ◽  
The Mathematical Model ◽  
Kinetics Of

В статье рассмотрен вопрос идентифицируемости математической модели кинетики химической реакции. В процессе решения обратной задачи по оценке параметров модели, характеризующих процесс, нередко возникает вопрос неединственности решения. На примере конкретной реакции продемонстрирована необходимость проводить анализ идентифицируемости модели перед проведением численных расчетов по определению параметров модели химической реакции. The identifiability of the mathematical model of the kinetics of a chemical reaction is investigated in the article. In the process of solving the inverse problem of estimating the parameters of the model, the question arises of the non-uniqueness of the solution. On the example of a specific reaction, the need to analyze the identifiability of the model before carrying out numerical calculations to determine the parameters of the reaction model was demonstrated.

scholarly journals Free boundary problems in biology

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140368 ◽  
Author(s):  
Avner Friedman
Keyword(s):  
Mathematical Model ◽  
Wound Healing ◽  
Free Boundary ◽  
Existence And Uniqueness ◽  
Free Boundary Problems ◽  
Uniqueness Theorems ◽  
Biological Processes ◽  
Boundary Problems ◽  
Mathematical Problems ◽  
The Mathematical Model

In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.

Formulae for Numerical Integration

1939 ◽  
Vol 23 (256) ◽  
pp. 352-359 ◽  
Author(s):  
W. G. Bickley
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Present Writer ◽  
Order Of Accuracy ◽  
Definite Integrals ◽  
Leading Term ◽  
High Order Of Accuracy ◽  
Systematic List ◽  
Forward Integration ◽  
Very High

Numerical integration, including the numerical integration of differential equations, is becoming increasingly necessary in technical applications. For such purposes, when a very high order of accuracy is not required, formulae which use successive values of the function to be integrated are more convenient than those using differences. Individual authors have given formulae of this type from time to time but a systematic list is not known to the present writer, and would seem to be of service. The present list gives formulae of four types : (I) formulae for evaluating definite integrals ; (II) formulae for forward integration ; (III) formulae for integrating over a subrange ; and (IV) formulae involving values of both the function and its integral. The leading term in the error series is given. Finally, some comments are made upon the formulae and their uses.

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