In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.
Linear multistep methods (LMM) for solving first order initial value problems (ivps) is of the form
where
Conventionally, they are used to solve higher order ordinary differential equations by first reducing them to a system of first order. This approach has been extensively discussed in [
The LMM in (1) generates discrete schemes which are used to solve first order odes. Various forms of this LMM have been developed [
The introduction of continuous collocation methods as against the discrete schemes enhances better global error estimation and ability to approximate solution at all interior points [
In [
In [
where
The use power series as basis function for derivation of continuous LMM are based on the property of analytic function that given the Taylor’s polynomial of the form
The approximate function
In this study we proposed the polynomial function of the form in [
which is of Type (3) to develop a continuous LMM for the solution of initial value problem of the form:
This paper is organized as follows: Section 1 consists of introduction and background of study; Section 2, we derive a continuous approximation to
Consider the third order differential Equation (7), we proposed an approximate solution of the form:
where
The derivative of (8) up to the third order yield
And
Collocating (10) at
of equations given below
The above equations are solved to obtain the values of
The continuous polynomial obtained when the values of
Evaluating (14) at
The method (15) is a specific member of the conventional LMM which can expressed as
Following [
where
where the
In the sense of [
Using the concept above, the method (19) has order
Considering the first characteristics polynomial of the method of Equation (15) given as
Putting
Applying the boundary locus method, we have that
In the spirit of Lambert (1973),
By letting
At
Single step method can be used to solve higher order ordinary differential equations directly without the need to first reducing it to an equivalent system of first order.
Consider the initial value problem in (7). For our method of order
Then the known values of
where
Our methods of order
The following initial value problems were used as our test problems:
Exact solution:
Exact solution:
Exact solution:
X | Exact solution | New result method (P = 9) | Error in our for (P = 9) | Error in [ |
---|---|---|---|---|
0.1 | 0.904837 | 0.940837 | 0.0000+00 | 2.1760E−12 |
0.2 | 0.818731 | 0.818731 | 2.7756E−14 | 1.3935E−11 |
0.3 | 0.740818 | 0.740818 | 1.5838E−12 | 3.4443E−11 |
0.4 | 0.670320 | 0.670320 | 2.7879E−11 | 6.4477E−11 |
0.5 | 0.606531 | 0.606531 | 2.9477E−11 | 1.0316E−10 |
0.6 | 0.548812 | 0.548812 | 8.5048E−11 | 1.4979E−10 |
0.7 | 0.496585 | 0.496585 | 8.0357E−11 | 2.0486E−10 |
0.8 | 0.449329 | 0.449329 | 1.6601E−10 | 2.6756E−10 |
0.9 | 0.406570 | 0.406570 | 1.1176E−10 | 6.9382E−10 |
1.0 | 0.367879 | 0.367879 | 1.4871E−10 | 1.4224E−10 |
X | Exact solution | New result method (P = 9) | Error in our for (P = 9) |
---|---|---|---|
0.1 | 0.299819E+01 | 0.299819E+01 | 6.6218E−13 |
0.2 | 0.298681E+1 | 0.298681E+01 | 6.2238E−11 |
0.3 | 0.295939E+01 | 0.295939E+01 | 3.5134E−09 |
0.4 | 0.291189E+01 | 0.291189E+01 | 6.1100E−07 |
0.5 | 0.284197E+01 | 0.284197E+01 | 6.4183E−07 |
0.6 | 0.274843E+01 | 0.274843E+01 | 1.8082E−06 |
0.7 | 0.263083E+01 | 0.263083E+01 | 1.3511E−06 |
0.8 | 0.248921E+01 | 0.248921E+01 | 1.3367E−06 |
0.9 | 0.232389E+01 | 0.232390E+01 | 7.9041E−06 |
1.0 | 0.213534E+01 | 0.213537E+01 | 3.7360E−05 |
X | Exact solution | New result method (P = 9) | Error in our for (P = 9) |
---|---|---|---|
0.1 | 0.321517E+01 | 0.321517E+01 | 0.0000E+00 |
0.2 | 0.330140E+01 | 0.330140E+01 | 2.8422E−13 |
0.3 | 0.352980E+01 | 0.352980E+01 | 1.6729E−12 |
0.4 | 0.381182E+01 | 0.381182E+01 | 2.9983E−11 |
0.5 | 0.414872E+01 | 0.414872E+01 | 3.1673E−11 |
0.6 | 0.454212E+01 | 0.454212E+01 | 9.1899E−11 |
0.7 | 0.499375E+01 | 0.499375E+01 | 8.9531E−11 |
0.8 | 0.550554E+01 | 0.550554E+01 | 1.9168E−10 |
0.9 | 0.607960E+01 | 0.607960E+01 | 2.1110E−10 |
1.0 | 0.671828E+01 | 0.671828E+01 | 4.9398E−10 |
1.1 | 0.742417E+01 | 0.742417E+01 | 8.6728E−10 |
1.2 | 0.820012E+01 | 0.820012E+01 | 2.3764E−09 |
We have developed and implemented our methods using Taylor series of the same order as the schemes that we developed. Some special and general third order initial value problems (ivps) were used to test the efficiency of our methods. Our method was found to be zero stable, consistent and convergent. The better accuracy of our method can be shown from the numerical examples.
D. O.Awoyemi,S. J.Kayode,L. O.Adoghe, (2014) A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations. American Journal of Computational Mathematics,04,119-126. doi: 10.4236/ajcm.2014.43011