This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.
The subject of Ordinary Differential Equation (ODE) is an important aspect of mathematics. It is useful in modeling a wide variety of physical phenomena―chemical reactions, satellite orbit, electrical networks, and so on. In many cases, the independent variable represents time so that the differential equation describes changes, with respect to time, in the system being modeled. The solution of the equation will be a representation of the state of the system. Consequently, the problem of finding the solution of a differential equation plays a significant role in scientific research, particularly, in the stimulation of physical phenomena. However, it is usually impossible to obtain direct solution of differential equations for systems to be modeled, especially complex ones encountered in real world problems. Since most of these equations are, or can be approximated by ordinary differential equations, a fast, accurate and efficient ODE solver is much needed. The Tau method was introduced by [
The method takes advantage of the special properties of Chebychev polynomials. The main idea is to obtain an approximate solution of a given problem by solving an approximate problem. To further enhance the desired simplicity Lanczos introduced the systematic use of the canonical polynomials in the Tau method. The difficulties presented by the construction of such polynomials limited its application to first order ODE with the polynomial coefficient. The said difficulties were resolved by [
In this paper, we apply the Tau-collocation approximation method for the solution of linear initial value problems of the first and second order ODE in its differential and canonical form. We perform some numerical stimulation on some selected problems and compare the performance/effectiveness of the method with the analytic solutions given.
Lanczos [
where
and determines the coefficient
Then
where m is the order of the differential equation, s is the number of over-determination,
is the rth degree shifted Chebychev polynomial valid in the interval
The free parameters in Equation (4) and the coefficient ar,
Considering the mth order linear differential Equation ( [
with y(x) as the exact solution in
We seek an approximate solution of the differential solution by the Tau method using the nth degree polynomial function
which satisfies the perturbed problem
We equate the corresponding coefficient of x in (8) and using the initial conditions
We then solve the system of equation by Gaussian elimination method.
The Lanczos Tau method in [
Consider an approximation to the residual
Then by the Tau method, if
we have
where L is a linear differential operator of order n.
We collocate (12) at
The parameter
Let us in this section consider and obtain the error estimator for the approximate solution of (1) and (9). Let
and
where
To obtain the perturbation term
and
We then proceed to find an approximate
Thus, the error function,
and
which satisfies the conditions prescribed.
In this section, two initial value problems are considered to show the efficiency of the method.
Example 1
Consider linear initial value problem in second order ordinary differential equation
We solve [
The analytic solution is
By the Tau method we obtain the linear differential operator as
The associated canonical polynomials are obtained as follows:
The canonical polynomials,
For
These polynomials are substituted into Equation (12) to give
Using Equation (5),
Since
Now,
Using initial conditions on Equation (23) and simplifying further we get the approximate solution as
Considering the Tau-collocation method we have:
Let
Substituting into (13) we have,
Now collocating at
Example 2
Consider the first order IVP
The exact solution is
For the given IVP, we can deduce that
The differential formulation is as follows:
Let
Taking
where
hence
but
Using (28) and (30) in (29) we obtain,
Expanding and equating coefficients of powers of x, the resulting linear equations together with the equations obtained using the initial conditions is written in the form,
where
Using Equation (5), we obtain the following values,
Using these values in the matrix and solving by Gaussian elimination method, we have,
The approximate solution is:
The results obtained above show that the Tau-collocation method is appropriate for the solution of linear initial value problems of first and second kind ordinary differential equations. From the tables (
X | Exact Solution | Approximate solution, n = 2 | Error |
---|---|---|---|
0.1 | 1.2868265 | 1.2960000 | 9.1735e−03 |
0.2 | 1.5607954 | 1.5706667 | 9.8712e−03 |
0.3 | 1.8191694 | 1.8240000 | 4.8306e−03 |
0.4 | 2.0593669 | 2.0560000 | 3.3669e−03 |
0.5 | 2.2789879 | 2.2666667 | 1.2321e−02 |
0.6 | 2.4758379 | 2.4560000 | 1.9838e−02 |
0.7 | 2.6479502 | 2.6240000 | 2.3950e−02 |
0.8 | 2.7936051 | 2.7706667 | 2.2938e−02 |
0.9 | 2.9113471 | 2.8960000 | 1.5347e−02 |
1.0 | 3.0000000 | 3.0000000 | 0.0000e+00 |
X | Exact Solution | Approximate solution, n = 2 | Error |
---|---|---|---|
0.1 | 0.9090909 | 0.9090418 | 4.19133e−05 |
0.2 | 0.8333333 | 0.8332214 | 1.1195e−04 |
0.3 | 0.7692308 | 0.7691735 | 5.7276e−05 |
0.4 | 0.7142857 | 0.7143198 | 3.4081e−05 |
0.5 | 0.6666667 | 0.6667378 | 7.1164e−05 |
0.6 | 0.6250000 | 0.6250298 | 2.9821e−05 |
0.7 | 0.5882353 | 0.5881915 | 4.3800e−05 |
0.8 | 0.5555556 | 0.5554809 | 7.4633e−05 |
0.9 | 1.5263158 | 0.5265873 | 2.8445e−05 |
0.10 | 0.5000000 | 0.5000000 | 0.0000e+00 |
tion approximation method, which is very close to the minimax polynomial which minimizes the maximum error in approximation. Thus, the approximate solution will match the analytic solution as n increases.
This paper has considered Tau-collocation approximation approach for solving particular first and second order ordinary differential equations. The method offers several advantages which include, among others:
1) It takes advantages of the special properties of Chebychev polynomials which can be easily generated recursively;
2) Elements of canonical polynomials sequences by means of a simple re-cursive relation which is self starting and explicit; and
3) It can easily be programmed for experimentation.
Tau-Collocation method can be extended to higher order ordinary differential equations and stochastic differential equations. It can also be used to solve integro-differential and stochastic integro-differential equations.
James E.Mamadu,Ignatius N.Njoseh, (2016) Tau-Collocation Approximation Approach for Solving First and Second Order Ordinary Differential Equations. Journal of Applied Mathematics and Physics,04,383-390. doi: 10.4236/jamp.2016.42045