Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas (DI2BBDF) for solving first-order initial value problem (IVP) with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing methods.

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SIMPLE ALGORITHM'S FOR THE CALCULATION OF HEAT TRANSPORT PROBLEM IN PLATE

Mathematical Modelling and Analysis ◽

10.3846/13926292.2001.9637148 ◽

2001 ◽

Vol 6 (1) ◽

pp. 85-96

Author(s):

H. Kalis ◽

I. Kangro

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Heat Transport ◽

Initial Value Problem ◽

Transport Problem ◽

Two Dimensional ◽

Initial Value ◽

Equation Solution ◽

Averaging Methods ◽

First Order

The approximations of some heat transport problem in a thin plate are based on the finite volume and conservative averaging methods [1,2]. These procedures allow one to reduce the two dimensional heat transport problem described by a partial differential equation to an initial‐value problem for a system of two ordinary differential equations (ODEs) of the first order or to an initial‐value problem for one ordinary differential equations of the first order with one algebraic equation. Solution of the corresponding problems is obtained by using MAPLE routines “gear”, “mgear” and “lsode”.

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On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six

Symmetry ◽

10.3390/sym12060952 ◽

2020 ◽

Vol 12 (6) ◽

pp. 952

Author(s):

Amiratul Ashikin Nasarudin ◽

Zarina Bibi Ibrahim ◽

Haliza Rosali

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Similar Type ◽

Differentiation Formula ◽

First Order ◽

Backward Differentiation Formula ◽

Fully Implicit ◽

Stiff Odes ◽

Backward Differentiation Formulas ◽

Differentiation Formulas

In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.

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Continuous One Step Linear Multi-Step Hybrid Block Method for the Solution of First Order Linear and Nonlinear Initial Value Problem of Ordinary Differential Equations

10.5772/intechopen.95619 ◽

2021 ◽

Author(s):

Kamoh Nathaniel ◽

Kumleng Geoffrey ◽

Sunday Joshua

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Value Problem ◽

Initial Value Problems ◽

Algebraic Equations ◽

Initial Value ◽

Closed Form Solutions ◽

First Order ◽

One Step ◽

Collocation Approach

In this paper, a collocation approach for solving initial value problem of ordinary differential equations (ODEs) of the first order is presented. This approach consists of reducing the problem to a set of linear multi-step algebraic equations by approximating the ODE with a shifted Legendre polynomial basis function to determine the unknown constants. The proposed method is simple and efficient; it approximates the solutions very closely to the closed form solutions. Some problems were considered using Maple Software to illustrate the simplicity, efficiency and accuracy of the method. The results obtained revealed that the hybrid method can be suitable candidate for all forms of first order initial value problems of ordinary differential equations.

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A Class of Hybrid Multistep Block Methods with A–Stability for the Numerical Solution of Stiff Ordinary Differential Equations

Mathematics ◽

10.3390/math8060914 ◽

2020 ◽

Vol 8 (6) ◽

pp. 914

Author(s):

Zarina Bibi Ibrahim ◽

Amiratul Ashikin Nasarudin

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Ordinary Differential Equations ◽

Stability And Convergence ◽

Stiff Differential Equations ◽

Stiff Ordinary Differential Equations ◽

Block Methods ◽

Backward Differentiation Formulas ◽

The Stability ◽

Differentiation Formulas

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

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On Nagumo's Condition

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-109-2 ◽

1972 ◽

Vol 15 (4) ◽

pp. 609-611 ◽

Cited By ~ 8

Author(s):

Thomas Rogers

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Uniqueness Theorem ◽

Initial Value Problem ◽

Initial Value ◽

Image Position

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and ifthen there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.

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