scholarly journals Solving Nonstiff Higher Order Odes Using Variable Order Step Size Backward Difference Directly

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohamed bin Suleiman ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Variable Order ◽  
Numerical Techniques ◽  
Step Size ◽  
The Relationship

The current numerical techniques for solving a system of higher order ordinary differential equations (ODEs) directly calculate the integration coefficients at every step. Here, we propose a method to solve higher order ODEs directly by calculating the integration coefficients only once at the beginning of the integration and if required once more at the end. The formulae will be derived in terms of backward difference in a constant step size formulation. The method developed will be validated by solving some higher order ODEs directly using variable order step size. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The results presented confirmed our hypothesis.

scholarly journals Two-Point Block variable order step size Multistep Method for Solving Higher Order Ordinary Differential Equations Directly

2021 ◽  
pp. 101376
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak ◽  
Wong Tze Jin ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Multistep Method ◽  
Variable Order ◽  
Step Size

scholarly journals Solution of Higher-Order ODEs Using Backward Difference Method

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Mohamed Bin Suleiman ◽  
Zarina Bibi Binti Ibrahim ◽  
Ahmad Fadly Nurullah Bin Rasedee
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Numerical Technique ◽  
Higher Order ◽  
First Order ◽  
The Relationship

The current numerical technique for solving a system of higher-order ordinary differential equations (ODEs) is to reduce it to a system of first-order equations then solving it using first-order ODE methods. Here, we propose a method to solve higher-order ODEs directly. The formulae will be derived in terms of backward difference in a constant stepsize formulation. The method developed will be validated by solving some higher-order ODEs directly with constant stepsize. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The result presented confirmed our hypothesis.

scholarly journals A Numerical Algorithm for Solving Stiff Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computation Time ◽  
Test Problems ◽  
Variable Order ◽  
Step Size ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
The Stability

An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1289
Author(s):  
Asnor ◽  
Mohd Yatim ◽  
Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Computational Effort ◽  
Variable Order ◽  
Numerical Efficiency ◽  
Stiff Odes ◽  
Backward Differentiation Formulae ◽  
Bdf Method ◽  
Selection Of

Variable order block backward differentiation formulae (VOHOBBDF) method is employedfor treating numerically higher order Ordinary Differential Equations (ODEs). In this respect, the purpose of this research is to treat initial value problem (IVP) of higher order stiff ODEs directly. BBDF method is symmetrical to BDF method but it has the advantage of producing more than one solutions simultaneously. Order three, four, and five of VOHOBBDF are developed and implemented as a single code by applying adaptive order approach to enhance the computational efficiency. This approach enables the selection of the least computed LTE among the three orders of VOHOBBDF and switch the code to the method that produces the least LTE for the next step. A few numerical experiments on the focused problem were performed to investigate the numerical efficiency of implementing VOHOBBDF methods in a single code. The analysis of the experimental results reveals the numerical efficiency of this approach as it yielded better performances with less computational effort when compared with built-in stiff Matlab codes. The superior performances demonstrated by the application of adaptive orders selection in a single code thus indicate its reliability as a direct solver for higher order stiff ODEs.

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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