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Fermat Collocation Method for Solvıng a Class of the First Order Nonlinear Differential Equations

Dilek Taştekin, Salih Yalçinbaş*

Department of Mathematics Celal Bayar University, Muradiye, Manisa, Turkey

Abstract: In this study, we present a reliable numerical approximation of the some first order nonlinear ordinary differential equations with the mixed condition by the using a new Fermat collocation method. The solution is obtained in the form of a truncated Fermat series with easily determined components. Also, the method can be used to solve Riccati equation. The numerical results show the effectuality of the method for this type of equations. Comparing the methodology with some known techniques shows that the existing approximation is relatively easy and highly accurate.

Keywords: Nonlinear ordinary differential equations, Riccati equation, Fermat polynomials, collocation points

Received January 14, 2014; Revised February 01, 2014; Accepted February 25, 2014

Citation: Dilek Taştekin, Salih Yalçinbaş, Fermat Collocation Method for Solvıng a Class of the First Order Nonlinear Differential Equations. Journal of Mathematical Sciences and Applications, 2 (1). 4-9. January 2014. doi:10.12691/jmsa-2-1-2

1. Introduction

Nonlinear ordinary differential equations are frequently used to model a wide class of problems in many areas of scientific fields; chemical reactions, spring-mass systems bending of beams, resistor-capacitor-inductance circuits, pendulums, the motion of a rotating mass around another body and so forth [1,2]. These equations here also demonstrated their usefulness in ecology, economics, biology, astrophysics and engineering. Thus, methods of solution for these equations are of great importance to engineers and scientists [3,4].

In this paper, for our aim we consider the first order nonlinear ordinary differential equation of the form

[picture] | (1)

under the mixed conditions

[picture] | (2)

and look for the approximate solution in the form

[picture] | (3)

[5,6]

which is a Fermat polynomial of degree N, where yn(n = 0,1,…,N)are the coefficients to be determined. Here P(x), Q(x), R(x), S(x), T(x) and g(x) are the functions defined on a≤x≤b; the real coefficients α,β and λ are appropriate constants. Note that, if S(x)=T(x)=0 in Eq. (1), it is a Riccati Equation [7,8,9].

2. Fundamental Matrix Relations

Our aim is to find the matrix form of each term in the nonlinear equation given by Eq. (1). Firstly, we consider the solution y(x) defined by a truncated series (3) and then we can convert to the matrix form

[picture] | (4)

where

If we differentiate expression (4) with respect to x, we obtain

[picture] | (5)

where

On the other hand, the matrix form of expression y²(x) is obtained as

or briefly

[picture] | (6)

where

By using the expression (4), (5) and (6) we obtain

[picture] | (7)

Following a similar way to (6), we have

[picture] | (8)

where

3. Matrix Relations Based on Collocation Points

Let us use the collocation points defined by

[picture] | (9)

in order to

By putting the collocation points (9) into Eq. (1), we get the equation

[picture] | (10)

By using the relations (4), (5), (6), (7) and (8); the system (10) can be written in the matrix form

or shortly

[picture] | (11)

where

4. Method of Solution

The fundamental matrix equation (11) corresponding to Eq. (1), can be written as

or

[picture] | (12)

where

We can find the corresponding matrix equation for the condition (2), using the relation (4), as follows:

[picture] | (13)

so that

We can write the corresponding matrix form (13) for the mixed condition (2) in the augmented matrix form as

[picture] | (14)

where

To obtain the approximate solution of Eq. (1) with the mixed condition (2) in the terms of Fermat polynomials, by replacing the row matrix (14) by the last row of the matrix (11), we obtain the required augmented matrix:

or the corresponding matrix equation

[picture] | (15)

where

The unknown coefficients set {y₀,…,yn} can be determined from the nonlinear system (15). As a result, we can obtain approximate solution in the truncated series form (3).

5. Accuracy of Solution

We can check the accuracy of the solution by following procedure [11,12,13]: The truncated Fermat series in (3) have to be approximately satisfying Eq. (1); that is, for each x=xi∈[a,b], i=1,2,…

and E(xi)≤10-ki (ki is any positive integer).

If max(10-ki)=10-k (k is any positive integer) is prescribed, then the truncation limit N is increased until the difference E(xi) at each of the points xi becomes smaller than the prescribed 10-k.

6. Numerical Examples

In this section, two numerical examples are given to illustrate the accuracy and efficiency of the presented method.

Example 6.1. [14] Let us first consider the first-order nonlinear differential equation

[picture] | (16)

with condition

and the approximate solution y(x) by the truncated Taylor polynomial

where

For N=2 the collocation points become

From the fundamental matrix equations for the given equation and condition respectively are obtained as

and

so that

The augmented matrix for this fundamental matrix equation is calculated

From the obtained system, the coefficients y₀,y₁ and y₂ are found as y₀=0, y₁=1/3 and y₂=0.

Hence we have the Fermat polynomial solution

Example 6.2. [14] Consider the following nonlinear differential equation given by

[picture] | (17)

with the initial condition

So that

For N=2 the collocation points become

From the fundamental matrix equations for the given equation and condition respectively are obtained as

and

so that

The augmented matrix for this fundamental matrix equation is calculated

From the obtained system, the coefficients y₀,y₁ and y₂ are found as y₀=0, y₁=1/3 and y₂=0.

Hence we have the Fermat polynomial solution

Example 6.3. Consider the following nonlinear differential equation given by

[picture] | (18)

with the initial condition

So that

The solutions obtained for N=2,5,7 are compared with the exact solution is ex, which are given in Figure 1.We compare the numerical solution and absolute errors for N=2,5,7 in Table 1.

Table 1. Comparison of the numerical errors of Example 6.3

Figure 1. Numerical and exact solution of Example 6.3 for N=2,5,7

Example 6.4. Let us first consider the first-order nonlinear differential equation

[picture] | (19)

with condition

Taking N=3,5,7 we obtain the approximate solution of this nonlinear differential equation.

The values of this solution are compared with the exact solution is x+sinx and given errors in Table 2.

Figure 2. Numerical and exact solution of Example 6.4 for N=3,5,7

Table 2. Comparison of the absolute errors of Example 6.4

7. Conclusion

In this study, a new Fermat approximation method for the solution of a class of first order nonlinear differential equations has been presented. The principal advantage of this method is the capability to succeed in the solution up to all term of Fermat expansion. It is seen from Example 6.3 that Fermat collocation method gives well results for the different values N. To obtained numerical results show that accuracy improves when N is increased as shown from Example 6.4. Examples, tables and figures indicate that the present method is convenient, reliable and effective. Also it is important to note that Fermat coefficients of the solution are found very simply by using the computer programs [15].

References

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[2]   | L. M. Kells, Elementary Differential Equations, Mcgraw-Hill Book Company, Newyork, 1965.
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[4]   | S. Yalçınbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integrodifferential equations in terms of Taylor polynomials, Applied Mathematics and Computation 112, 291-308, 2000.
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[10]   | R. A. Adams, Calculus, Addison-Wesley Publihers Limited pp.562, 1990.
[11]   | M. Gulsu, M. Sezer, On the solution of the Riccati Equation by the Taylor Matrix Method, Applied Mathematics and Computation 176 pp. 414-421, 2006.
[12]   | K. Erdem, S.Yalçınbaş, Bernoulli Polynomial Approach to High-Order Linear Differential-Difference Equations, AIP Conference Proceedings, 1479, 360-364, 2012.
[13]   | S.Yalçınbaş, M.Aynigül, T.Akkaya, Legendre series solutions of Fredholm integral equations, Mathematical and Computational Applications, 15(3), 371-381, 2010.
[14]   | E. Küçükşahin, Taylor polynomial solutions of some first order nonlinear differantial equations, M. Sc. Thesis, Mugla University, 2008.
[15]   | D. Taştekin, S. Yalçınbaş, M. Sezer, Taylor collocation method for solving a class of the first order nonlinear differential equations, Mathematical and Computational Applications, vol.18, no.3, pp.383-391, 2013.