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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 a_{r},

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