Extended Continuous Block Backward Differentiation Formula for Stiff Systems

2015 ◽  
Vol 55 (1) ◽  
pp. 5-18 ◽  
Author(s):  
O. A. Akinfenwa ◽  
S. N. Jator
Keyword(s):  
Approximate Solutions ◽  
Stiff Systems ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
First Derivative ◽  
Error Constant ◽  
Single Block ◽  
Continuous Block

Abstract We present an Extended Continuous Block Backward Differentiation Formula (ECBBDF) of order k+1 for the numerical solution of stiff ordinary differential equations. This is achieved by constructing an Extended Continuous Backward Differentiation formula (ECBDF) together with the additional methods from its first derivative and are combined to form a single block of methods that simultaneously provide the approximate solutions for the stiff Initial Value Problems (IVPs). The error constant and stability property of the (ECBBDF) is discussed. We use the specific cases k = 4 and k = 5 to illustrate the process. The performance of the method is demonstrated on some numerical examples to show the accuracy and efficiency advantages of the method.

scholarly journals Solvability of Sequence Spaces Equations of the form (Eα)Δ + Fx = Fb

2015 ◽  
Vol 55 (1) ◽  
pp. 109-131
Author(s):  
Bruno de Malafosse
Keyword(s):  
Stability Property ◽  
Approximate Solutions ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
First Derivative ◽  
Error Constant ◽  
Single Block ◽  
Continuous Block

Abstract We present an Extended Continuous Block Backward Differentiation Formula (ECBBDF) of order k+1 for the numerical solution of stiff ordinary differential equations. This is achieved by constructing an Extended Continuous Backward Differentiation formula (ECBDF) together with the additional methods from its first derivative and are combined to form a single block of methods that simultaneously provide the approximate solutions for the stiff Initial Value Problems (IVPs). The error constant and stability property of the (ECBBDF) is discussed. We use the specific cases k = 4 and k = 5 to illustrate the process. The performance of the method is demonstrated on some numerical examples to show the accuracy and efficiency advantages of the method.

scholarly journals Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equations

2021 ◽  
Vol 09 (08) ◽  
Author(s):  
Pius Tumba ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Grid Point ◽  
Stiff Systems ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
First Derivative ◽  
Backward Differentiation Formulae ◽  
Continuous Formulation

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

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 Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1342 ◽  
Author(s):  
Hazizah Mohd Ijam ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Triangular Matrix ◽  
Differentiation Formula ◽  
Stability Properties ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Optimal Stability ◽  
Best Value ◽  
Selection Of

This paper aims to select the best value of the parameter ρ from a general set of linear multistep formulae which have the potential for efficient implementation. The ρ -Diagonally Implicit Block Backward Differentiation Formula ( ρ -DIBBDF) was proposed to approximate the solution for stiff Ordinary Differential Equations (ODEs) to achieve the research objective. The selection of ρ for optimal stability properties in terms of zero stability, absolute stability, error constant and convergence are discussed. In the diagonally implicit formula that uses a lower triangular matrix with identical diagonal entries, allowing a maximum of one lower-upper (LU) decomposition per integration stage to be performed will result in substantial computing benefits to the user. The numerical results and plots of accuracy indicate that the ρ -DIBBDF method performs better than the existing fully and diagonally Block Backward Differentiation Formula (BBDF) methods.

Study of the Numerical Solving Methods of the Motion Model of Polydispersed Droplets

2017 ◽  
Author(s):  
Xiaoqiang He ◽  
Qianfeng Liu ◽  
Chenru Zhao ◽  
Hanliang Bo
Keyword(s):  
Differential Equations ◽  
Numerical Stability ◽  
Separation Efficiency ◽  
Motion Model ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Runge Kutta Method ◽  
Special Distribution ◽  
Moisture Separator

Moisture separation is one of crucial devices in PWR power plant for it plays an irreplaceable role in eliminating droplets from steam and supplying dry-saturated for turbines. It would be helpful to design and optimize the structure of moisture separator through analyzing the behaviors of droplets. This paper studies the numerical solving methods of the motion model of polydispersed droplets. The equations of model belong to the field of nonlinear stiff ordinary differential equations, thus backward differentiation formula, a kind of multi-steps methods are used which advance in solving stiff different equations. The coefficients of equations involve the velocity and rotation of flow field in separator, and these parameters are given by the output of Fluent. After solving the differential equations we can get the velocity and angle velocity of droplets in different locations and the movement track of droplets in separator, as well as the separation efficiency of moisture separator for polydispeased droplets with special distribution. Finally through the numerical solving of the motion model of polydispersed droplets in chevron-type separator, we find that multi-steps method improves the numerical stability and reduces the steps of iteration under the same calculation precision compared with classical methods such as fourth-order Runge-Kutta method. So it lays the foundation for development of moisture separator program suit for engineering computing.

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