Algorithm of modified variable step block backward differentiation formulae for solving first order stiff ODEs

2018 ◽  
Author(s):  
Asma Izzati Asnor ◽  
Siti Ainor Mohd Yatim ◽  
Zarina Bibi Ibrahim
Keyword(s):  
First Order ◽  
Stiff Odes ◽  
Variable Step ◽  
Backward Differentiation Formulae

A variable-step variable-order algorithm for systems of stiff odes

1997 ◽  
Vol 63 (1-2) ◽  
pp. 149-157 ◽  
Author(s):  
T. Van Hecke ◽  
G. Vanden G Berghe ◽  
M. Van Daele ◽  
H De Meyer
Keyword(s):  
Variable Order ◽  
Stiff Odes ◽  
Variable Step ◽  
Order Algorithm

scholarly journals A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Nazreen Waeleh ◽  
Zanariah Abdul Majid
Keyword(s):  
Initial Value Problems ◽  
Adaptive Strategy ◽  
Variable Step Size ◽  
Block Method ◽  
Initial Value ◽  
Step Size ◽  
First Order ◽  
Variable Step ◽  
Improved Performance ◽  
Fifth Order

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs.

Computational Treatment of First Order Delay Differential Equations Using Hybrid Extended Second Derivative Block Backward Differentiation Formulae

2020 ◽  
Vol 1 (1) ◽  
Author(s):  
C. Chibuisi ◽  
Bright Okore Osu ◽  
C. Olunkwa ◽  
S. A. Ihedioha ◽  
S. Amaraihu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Derivative ◽  
Delay Term ◽  
First Order ◽  
Block Form ◽  
Collocation Approach ◽  
Backward Differentiation Formulae ◽  
Delay Differential ◽  
Step Number

This paper considers the computational solution of first order delay differential equations (DDEs) using hybrid extended second derivative backward differentiation formulae method in block form without the implementation of interpolation techniques in estimating the delay term. By matrix inversion approach, the discrete schemes were obtained through the linear multistep collocation approach from the continuous form of each step number which after implementation strongly revealed the convergence and region of absolute stability of the proposed method. Computational results are presented and compared to the exact solutions and other existing method to demonstrate its efficiency and accuracy.

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 THE GOURSAT PROBLEM FOR HYPERBOLIC LINEAR THIRD ORDER EQUATIONS

2001 ◽  
Vol 6 (2) ◽  
pp. 270-279 ◽  
Author(s):  
V. I. Korzyuk
Keyword(s):  
Existence And Uniqueness ◽  
Energy Inequality ◽  
Goursat Problem ◽  
Differentiation Operator ◽  
Cylindrical Domain ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Variable Step ◽  
Linear Differential

The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.

Three-stage Hermite–Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs

CALCOLO ◽  
2014 ◽  
Vol 52 (3) ◽  
pp. 371-405 ◽  
Author(s):  
Truong Nguyen-Ba ◽  
Thierry Giordano ◽  
Rémi Vaillancourt
Keyword(s):  
Variable Step Size ◽  
Step Size ◽  
Stiff Odes ◽  
Variable Step
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