Abstract: This study presents the application of collocation methods for the solution of second order nonlinear ordinary differential equations. Newtion’s linearization scheme is used to linearize the non-linear problems and the linearized problem leads to the use of iterations. Two collocation methods used in this study are standard and perturbed collocation by chebyshev polynomials. Numerical computations are carried out to illustrate the application of the methods and also, the methods are compared in terms of accuracy and computational cost.
INTRODUCTION
Nonlinear differential equations are used in modeling many real life problems in science and Engineering. Nonlinear ordinary differential equations mostly defy closed form solutions because the actual elegant theory valid for their linear counterparts often fails for them. Newton’s linearization procedures leading to the use of iterations are commonly employed to facilitate provision of analytic solution.
Collocation methods involve the determination of an approximate solution in a suitable set of functions sometimes called trials. Taiwo (1991) was the first to advocate the use of collocation by polynomials rather than at equidistant points.
For the purpose of our discussion, let us consider a nonlinear second order ordinary differential equation of the form:
| (1) |
Together with the linear boundary conditions:
| (2) |
and
| (3) |
Here, a,b, α and β are constants:
The Newton’s scheme from Taylor’s Series expansion given by:
| (4) |
is used throughout this study.
Hence, from Eq. 1, we obtain the following:
| (5) |
Thus, substituting Eq. 5 into 4, the Newton’s Scheme becomes:
| (6) |
where,
The Newton’s linearization leads to the use of the following iteration:
| (7) |
Together with the boundary conditions
| (8) |
and
| (9) |
NUMERICAL SOLUTION TECHNIQUES
Method 1: Standard collocation techniques: The linear Eq. 7 together with the boundary conditions Eq. 8 and 9 can be treated by constructing a Chebyshev polynomial (Taiwo, 1986) solution in the interval a ≤x≤b.
In order to apply this techniques to Eq. 7, we assume an approximate solution of the form:
| (10) |
where, ai (i = 0(1)N) are constants to be determined and Ti(x) are the Chebyshev polynomials valid in a ≤x≤b and defined by:
| (11) |
To determine ai in Eq. 10, we substitute 10 into 7, we obtain:
| (12) |
together with the boundary conditions:
| (13) |
and
| (14) |
Thus, collocating Eq. 12 at points x = xi, yields:
| (15) |
where, for some obvious practical reasons, we choose the collocation points x = xi to be:
| (16) |
Thus, we have (N-1) collocation equations in (N + 1) unknowns. Two additional equations are obtained using Eq. 13 and 14. Altogether, we have (N+1) collocation equations which give the unique values of the (N + 1) constants αi(i = 0(1)N).
Method 2: Perturbed collocation method: The perturbed collocation method is an attempt to improve the accuracy and efficiency of the standard collocation method 1 (Lanczos, 1938; Chen, 1981).
In order to apply this method, Eq. 10 is substituted into a slightly perturbed Eq. 12, we obtain:
| (17) |
where, PN(x) is an orthogonal polynomial of degree N (Chebyshev polynomial ) for a single polynomial approximation over [a, b] or Legendre Polynomial for piecewise polynomial over (a, b) and τi(i = 1,2) are free τ parameters (Chen, 1981). Thus, collocating Eq. 17 at points x = xi, yields:
| (18) |
| (19) |
where, for some obvious practical reasons, we choose the collocation points x = xi to be:
| (20) |
Hence, we have (N+1) collocation equations in (N+3) unknowns. Now, two extra equations are obtained from Eq. 13 and 16.
Thus, altogether we now have (N+3) collocation equations which give the unique values of the (N+3) constants a1, a2. a3, . . . aN, τ1 and τ2 to obtain a single polynomial approximation.
NUMERICAL EXAMPLES AND DISCUSSION OF RESULTS
Example 1:
| (21) |
With the following conditions:
y(0) = 4 |
and
y(1) = l |
The analytical solution of example 1 is given as:
The linearized example 1 is given as:
together with the boundary conditions
yK+1(0) = 4 |
and
yK+1(1) = 1 |
For this example, y0(x) = x is used
Remarks: All the results are obtained at the fifth iterations.
Example 2:
and
The analytical solution is:
The Newton Scheme is given by:
For k = 0, the following initial approximation is used.
RESULTS
Table 1-3 show the numerical solutions in terms of the approximate solutions obtained for example 1 for both the standard and perturbed methods. The results show that as the values of N increases, the approximate solutions obtained for perturbed collocation method converge faster to the exact solution than the standard method. Though, the perturbed method involves large matrix system of equations of the same degree, it is interesting to compare the accuracy and the computational cost involved.
| Table 1: | Results of approximate solution of methods 1 and 2, for N = 2, example 1 |
| Table 2: | Results of approximate solution of Methods 1 and 2, for N = 3, example 1 |
| Table 3: | Results of approximate solution of methods 1 and 2, for N = 4, example 1 |
For example, for the case N = 4, the perturbed method involve 7x7 systems of algebraic equations with approximate solution y4(x) = 1.108108655 compare to the standard method that only involve 5x5 with approximate solution y4(x) = 1.1044376279, more work were involved in perturbed collocation method than the standard method, but the closeness of the results obtained compensated for this when compared with the exact solution y(x) =1.108033241. Also, these were true for Table 4-6 for example 2.
| Table 4: | Results of approximate solution of Methods 1 and 2, for N = 2, example 2 |
| Table 5: | Results of approximate solution of methods 1 and 2, for N = 3, example 2 |
| Table 6: | Results of approximate solution of methods 1 and 2, for N = 4, example 2 |
CONCLUSION
The results obtained for the two examples considered were presented. The tables show the numerical solution in terms of the approximation solutions of the two methods considered for examples 1 and 2. We observed that the approximate solutions of the perturbed collocation method converges to the exact solution faster than the Standard method for the two examples considered.