IMPLICIT INTERVAL MULTISTEP METHODS FOR SOLVING THE INITIAL VALUE PROBLEM

2002 ◽  
Vol 8 (1) ◽  
pp. 17-30 ◽  
Author(s):  
MAŁGORZATA JANKOWSKA ◽  
ANDRZEJ MARCINIAK
Keyword(s):  
Initial Value Problem ◽  
Multistep Methods ◽  
Initial Value

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).

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 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 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.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

Sign in / Sign up

Export Citation Format

Share Document