Comparison between standard and nonstandard finite difference methods for solving first and second order ordinary differential equations 

A. R. Yaghoubi ^{1}*, H. Saberi Najafi ^{2} 

^{1} Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, University Campus2, Rasht, Iran, Faculty Member of Islamic Azad University, Saravan Branch, Saravan, Iran ^{2} Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, Rasht, Iran *Corresponding author Email: abyaghoobi@phd.guilan.ac.ir 
Copyright © 2015 A. R. Yaghoubi, H. Saberi Najafi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we solve some first and second order ordinary differential equations by the standard and nonstandard finite difference methods and compare results of these methods. Illustrative examples have been provided, and the results of two methods compared with the exact solutions.
Keywords: NonStandard Finite Difference Schemes.
1. Introduction
Ronald Mickens began developing numerical schemes using nonstandard finite difference (NSFD) schemes for solving physical problems. The fundamental of this method is centered on two rules [11]:
i) The discrete firstorder derivative must take a more general form than that used in standard discretization, i.e.
(1)
Where and are known, respectively, as the numerator and denominator functions, having the properties
, (2)
Where and
The secondorder derivative discrete in the following form
. (3)
Where
. (4)
ii) Both linear and nonlinear terms involving the dependent variable may require "nonlocal" discretization; for example
(5)
The full details about these procedures are given in [1116]. The nonstandard finite difference scheme has developed as an alternative method for solving a wide range of problems whose mathematical models involve algebraic, differential, biological models and chaotic systems [1], [2] and [410].
In this work, we compare nonstandard finite difference (NSFD) and standard finite difference (FD) schemes for solving ordinary differential equations. Some famous equations such as Dynamic, Logistic, LaneEmden and Airy equations have been provided. Solution of Airy equation which is a special case of StormLiouville equation [3] can't be displayed based upon primary functions. We use the series solutions' method to find the power series solution for this secondorder linear differential equation and compare NSFD and FD methods with the power series solution.
2. Numerical examples
In this section, we apply NSFD and FD methods to obtain numerical solutions for first and second order ordinary differential equations.
2.1. First order ODE
Example 1: Consider the following first order ODE
. (6)
The exact solution of (6) is
. (7)
Standard method:
(8)
Therefore
(9)
Nonstandard method:
(10)
Where and have the properties (2). In this example, we choose the numerator and denominator functions as follows:
(11)
Therefore, we have.
(12)
In figure 1 the results of equations (9) and (12) are compared with the exact solution (8) with and
A 
B 


Fig. 1: Numerical Solutions of NSFD and FD Methods for Solving Equation (6) with (A) and (B). 
Example: 2
Consider the following first order ODE
. (13)
The exact solution of (13) is
(14)
We have the following FD and NSFD schemes respectively for solving (13)
(15)
(16)
In figure 2 the results of equations (15) and (16) are compared with the exact solution (14) with
Fig. 2: Numerical Solutions of NSFD and FD Methods for Solving Equation (13) with
Example: 3
Consider the following general nonlinear first order dynamic equation with the initial condition
(17)
Where n is a positive integer, when equation (17) becomes a logistic differential equation.
(18)
The exact solution of (18) is.
(19)
We use the following NSFD scheme for solving (18).
(20)
In figure 3, the results of the nonstandard scheme (20) is compared with the following standard scheme.
(21)
A 
B 


Fig. 3: Numerical Solutions of NSFD and FD Methods for Solving Equation (18) with (A) and (B). 
In general, we have the following nonstandard discretization equation for the equation (17)
(22)
For we have the following scheme
(23)
The exact solution of equation (18) for is
(24)
In figure 4 we compared the standard and nonstandard finite difference methods with the exact solution of equation (17) for 1.
A 
B 


Fig. 4: Numerical Solutions of NSFD and FD Methods with (A) and (B) for Solving Equation (17) with 
2.2. Second order ODE
Example: 4
Consider the following second order ODE
(25)
With the following initial conditions,
(26)
Equation (25) is called LaneEmden equation. This equation describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. The polytrophic theory of stars essentially follows out of thermodynamic considerations that deal with the issue of energy transport, through the transfer of material between different levels of the star. This equation is one of the basic equations in the theory of stellar structure and has been focused in many studies.
In equation (25) for andwith the following initial conditions we have.
(27)
For solving (28) we construct the following NSFD scheme.
(28)
After some manipulation we have
(29)
In this example the denominator functions for first and second order derivative are respectively,
(30)
Note that and satisfied relations (2) and (4). The numerator function is
In figure 5, the result of scheme (29) is compared with the following standard scheme.
(31)
Fig. 5: Numerical Solutions of NSFD and FD Methods for Solving Equation (27) with
Example: 5
Consider the following LaneEmden type equation.
(32)
The exact solution of (32) is
(33)
For solving (32) we have
(34)
Therefore
(35)
In this case, we choose and as follow:
(36)
In figure 6, the result of scheme (35) is compared with the following standard scheme.
(37)
Therefore
(38)
Fig. 6: Numerical Solutions of NSFD and FD Methods for Solving Equation (32) with
Example: 6
Consider the following second order ODE
(39)
This equation is called Airy differential equation. The solution for this equation can't be display based upon in primary functions. We use the series solution's method to find a power series solution for this secondorder linear differential equation. The general form of power series is.
(40)
Therefore, we have.
(41)
By substituting (40) and (41) in (39), after some algebraic manipulation, we obtain the following power series solution.
(42)
From the initial conditions, we obtain we will compare solutions of FD and NSFD methods with the equation (42). We have the following nonstandard scheme for solving equation (39)
(43)
Where
by solving (43) in we have.
(44)
A standard scheme for equation (39) is
(45)
In figure 7 we compared the standard and nonstandard finite difference methods for with the power series solution of equation (39), we supposed that in equation (42), n changes from 1 to 1000.
Fig. 7: Numerical Solutions of NSFD and FD Methods for Solving Equation (39) with
3. Conclusion
In this paper, we have presented the efficiency of nonstandard finite difference method in comparison with the standard finite difference method for numerical solution of first and second order ordinary differential equations. From the graphical results, clearly nonstandard method is more stable than the standard method and the domain of h for stability in the nonstandard method is larger than those of the standard method. If the denominator functions are chosen in appropriate from the nonstandard method produces better results.
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