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

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 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 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 An Accurate Block Solver for Stiff Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
H. Musa ◽  
M. B. Suleiman ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Variable Step Size ◽  
Initial Value ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
Variable Step ◽  
Compute Solution ◽  
The Stability ◽  
Size Mode

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy.

scholarly journals Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Jiang Zhu ◽  
Dongmei Liu
Keyword(s):  
Time Scales ◽  
Initial Value Problems ◽  
Second Order ◽  
Dynamic Equations ◽  
Maximum Principles ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Approximation Theorems ◽  
Construction Techniques ◽  
Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

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