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Preface


This section provides an idea of the basic methods behind numerical methods of solving or, more precisely, providing approximations to ordinary differential equations (ODEs for short). In no way is this description to be considered complete.

Instead of continuous functions defined on some finite or semi-infinite interval, the finite difference method (FDM for short) treats them as a set of discrete points, often referred to as a “grid” or “mesh,” and their derivatives are approximated by appropriate “finite differences” involving these values. Such approach transfers an initial or oundary value problem for an ordinary differential equation (ODE for short) into a difference equation or a system of equations. A resulting finite difference equation (FDE) usually has a much reacher structure than a corresponding ODE for continuous functions. Although a solution of FDE mimics solution of ODE, it may exhibite properties that are not related to the continuous solution of ODE. Therefore, restoration of a continuous solution from its discrete counterpart is usually an ill-posed problem.

This section discusses differential equations involed in modeling fluid mechnics. we consider two such problems---Blasius equation and Jeffery--Hamel flow.

Finite Difference Schemes


A starting point of a finite difference method or scheme is utilization of Taylor's series approximation. Therefore, all functions to be considered are assumed to satisfy conditions of Taylor's series approximation. In the following equation, x0 is a reference point, and ∆x > 0. Assuming that the required derivatives exist, we have, for n ≥ 1,
\[ f\left( x_0 + \Delat x \right) = f\left( x_0 \right) + f'\left( x_0 \right) \Delat x + \frac{1}{2}\, f''\left( x_0 \right) \left( \Delat x \right)^2 + \cdots + \frac{1}{n!}\, f^{(n)} \left( x_0 \right) \left( \Delat x \right)^n + R_n \]
where the remainder term Rn is given by
\[ R_n = \frac{1}{(n+1)!}\, f^{(n+1)} \left( \xi \right) \left( \Delat x \right)^{n+1} , \]
and ξ lies between x0 and x0 + Δx. As usual, prime indicates the derivative (in Lagrange notation).

When n = 1, we use Taylor's series for approximation of the first derivative

\[ f' \left( x_0 \right) \approx \frac{f\left( x_0 + \Delta x \right) - f\left( x_0 \right)}{\Delta x} . \]
This formula that should be familiar to you provides a finite difference approximation to the derivative at point x0. It is an example of forward difference approximation that is usually assume that Δx is positive. However, there is another finite difference approximation for the derivative, known as backward approximation:
\[ f' \left( x_0 \right) \approx \frac{f\left( x_0 \right) - f\left( x_0 - \Delta x \right)}{\Delta x} . \]

If we add both sides of these taylor's approximations and divide by 2, we obtain another approximation for the derivative:

\[ f' \left( x_0 \right) \approx \frac{f\left( x_0 + \Delta x \right) - f\left( x_0 - \Delta x \right)}{2\Delta x} . \]

 

  1. Anguelov, R. and Lubuma, J. M. S. 2001. Contributions of the Mathematics of the Non- Standard Finite Method with Applications to Certain Discrete Schemes, Journal of Computational and Applied Mathematics, 17:518-543.
  2. Anguelov, R. and Lubuma, J. M. S. 2003. Non- Standard Finite Difference Method by Non-Local Approximation, Mathematics and Computer in Simulation, 2003, 61:465-475.
  3. Ciślinski, J.L., On the exact discretization of theclassical harmonic oscillator equation, On the exact discretization of the classical harmonic oscillator equation, 2011,
  4. Ibijola, E. A. and Sunday, J. 2010. A Comparative Study of Standard and Exact Finite-Difference Schemes for Numerical Solution of Ordinary Differential Equations Emanating from the Radioactive Decay of Substance, Australian Journal of Basic and Applied Sciences, 4(4): 624-632
  5. Mickens, R. E. 1981. Non-Linear Oscillations, Cambridge University Press, New York.
  6. Mickens, R. E. 1999. Applications of Non-Standard Method for Initial Value Problem, World Scientific, Singapore.
  7. Sunday, J., Ibijola, E.A., and Skwame, Y., On The Theory and Applications of Nonstandard Finite Difference Method for Singular Ordinary Differential Equations, Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (4): 643-647.

 

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