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

ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING STIFF INITIAL VALUE PROBLEMS

2021 ◽  
Vol 5 (2) ◽  
pp. 120-127
Author(s):  
Muhammad Abdullahi ◽  
Hamisu Musa
Keyword(s):  
Initial Value Problems ◽  
Stiff Problems ◽  
Block Method ◽  
Initial Value ◽  
Three Solutions ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Linear And Nonlinear

This paper modified an existing 3–point block method for solving stiff initial value problems.  The modification leads to the derivation of another 3 – point block method which is suitable for solving stiff initial value problems.  The method approximates three solutions values per step and its order is 5. Different sets of formula can be generated from it by varying a parameter ρ ϵ (-1, 1) in the formula. It has been shown that the method is both Zero stable and A–Stable. Some linear and nonlinear stiff problems are solved and the result shows that the method outperformed an existing method and competes with others in terms of accuracy

scholarly journals Construction of Stable High Order One-Block Methods Using Multi-Block Triple

2019 ◽  
pp. 1-13
Author(s):  
I. J. Ajie ◽  
K. Utalor ◽  
M. O. Durojaiye
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
Shift Operator ◽  
Block Method ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Backward Differentiation Formula ◽  
The Third ◽  
Block Methods

This paper deals with the construction of l-stable implicit one-block methods for the solution of stiff initial value problems. The constructions are done using three different multi-block methods. The first multi-block method is composed using Generalized Backward Differentiation Formula (GBDF) and Backward Differentiation Formula (BDF), the second is composed using Reversed Generalized Adams Moulton (RGAM) and Generalized Adams Moulton (GAM) while the third is composed using Reversed Adams Moulton (RAM) and Adams Moulton (AM). Shift operator is then applied to the combination of the three multi-block methods in such a manner that the resultant block is a one-block method and self-starting. These one-block methods are up to order six and  with at order ten. Numerical experiments show that they are good for solving stiff initial problems.

scholarly journals An improved 2–point block backward differentiation formula for solving stiff initial value problems

2013 ◽  
Author(s):  
Hamisu Musa ◽  
Mohamed Bin Suleiman ◽  
Fudziah Ismail ◽  
Norazak Senu ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Backward Differentiation Formula

scholarly journals Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Iskandar Shah Mohd Zawawi ◽  
Zarina Bibi Ibrahim ◽  
Khairil Iskandar Othman
Keyword(s):  
Initial Value Problems ◽  
Maximum Error ◽  
Computational Time ◽  
Initial Value ◽  
Stiff Initial Value Problems ◽  
Fully Implicit ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Order Four ◽  
Differentiation Formulas

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

scholarly journals Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Mohammed Al-Refai ◽  
Muhammed Syam
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial Value ◽  
The Laplace Transform ◽  
Necessary And Sufficient

In this paper, we discuss the solvability of a class of multiterm initial value problems involving the Caputo–Fabrizio fractional derivative via the Laplace transform. We derive necessary and sufficient conditions to guarantee the existence of solutions to the problem. We also obtain the solutions in closed forms. We present two examples to illustrate the validity of the obtained results.

scholarly journals Improved Two-Point Block Backward Differentiation Formulae for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

2020 ◽  
Vol 3 (2) ◽  
pp. 200-209
Author(s):  
S Adee ◽  
VO Atabo
Keyword(s):  
Initial Value Problems ◽  
Taylor Series Expansion ◽  
Kutta Method ◽  
Initial Value ◽  
Stiff Initial Value Problems ◽  
First Order ◽  
New Methods ◽  
Runge Kutta Method ◽  
Backward Differentiation Formulae ◽  
Convergence And Stability Analysis

Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.

scholarly journals Numerical solutions of nonstandard first order initial value problems

1992 ◽  
Vol 5 (1) ◽  
pp. 69-82 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
First Order ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IV P) y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NST D I V Ps.

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