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.

ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

2021 ◽  
Vol 5 (2) ◽  
pp. 442-446
Author(s):  
Muhammad Abdullahi ◽  
Hamisu Musa
Keyword(s):  
Initial Value Problem ◽  
Initial Value Problems ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
First Order ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Necessary And Sufficient

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

scholarly journals Existence and uniqueness of analytic solutions of the Shabat equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 855-862 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Upper Bound ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Initial Condition ◽  
Initial Value ◽  
First Derivative

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely forz∈ℂ:|z|<T,T>0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, andT. Furthermore, from these conditions, one can obtain an upper bound forT. Our results are in consistence with some recently found results.

scholarly journals Existence, uniqueness and continuability of solutions of impulsive differential-difference equations

1999 ◽  
Vol 12 (3) ◽  
pp. 293-300 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value

We consider an initial value problem for impulsive differential-difference equations, and obtain sufficient conditions for the existence, uniqueness, and continuability of solutions of such problem.

scholarly journals Methods for solving the initial value problem with a two-sided estimate of the local error

2021 ◽  
pp. 88-92
Author(s):  
Yaroslav Pelekh ◽  
Andrii Kunynets ◽  
Halyna Beregova ◽  
Tatiana Magerovska
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Local Error ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Embedded Methods ◽  
The Right

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

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.

Bounded solutions of functional differential equations of the neutral type with infinite delays

1982 ◽  
Vol 93 (1-2) ◽  
pp. 33-39 ◽  
Author(s):  
V. G. Angelov ◽  
D. D. Bainov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Functional Differential Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Bounded Solutions ◽  
Initial Value ◽  
Infinite Delays ◽  
Functional Differential

SynopsisIn this paper the authors obtain sufficient conditions for the existence and uniqueness of the initial value problem of functional differential equations of neutral type with infinite delays, making use of some earlier results of the present authors.

scholarly journals Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Josef Diblík ◽  
Josef Rebenda ◽  
Zdeněk Šmarda
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
Singular Initial Value Problem ◽  
The Right

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. The method of functions defined implicitly and the topological method (Ważewski's method) are used in the proofs. The results generalize some previous ones on singular initial value problems for differential equations.

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