In this paper, we developed a new continuous block method by the method of interpolation and collocation to derive new scheme. We adopted the use of power series as a basis function for approximate solution. We evaluated at off grid points to get a continuous hybrid multistep method. The continuous hybrid multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent, zero stable and convergent. The results were found to compete favorably with the existing methods in terms of accuracy and error bound. In particular, the scheme was found to have a large region of absolute stability. The new method was tested on real life problem namely: Dynamic model.
In this paper, we considered the method of approximate solution of the general second order initial value problem of the form
where
Equation (1) is of interest to researchers because of its wide application in engineering, control theory and other real life problem, hence the study of the methods of its solution. Hence, authors proposed methods with different basis functions and among them are [
Block method was later proposed. This block method has the properties of Runge-kutta method for being self-starting and does not require development of separate predictors or starting values. Among these authors are [
In this paper, we propose a new one-twelfth step continuous hybrid block method for the numerical inte- gration of second order initial value problems with constant step-size which is then implemented in block mode.
The paper is organized as followed: Section 2 considers the mathematical formulation of the method. Section 3 considers the analysis of the basic properties of the method. Section 4 considers the Region of absolute stability of our method. Section 5 considers the application of the derived method to solve some second order Ordinary Differential Equations and conclusion.
We consider the simple power series as a basis function for approximation:
where
And
Substituting (1) into (4) gives
We shall consider a step-length of
Interpolating (3) at
where
Solving (6) for the
where
Evaluating (2) at
The Predictors are expressed as follows:
The combination of Equations (9), (10), (11) and (12), yield the block of the form
Writing (17) explicitly gives
Substituting (18) into (13)-(16) gives the following Block-Predictor as follows
Let the linear operator defined on the method be
Expanding the form
Definition: The linear operator and the associated block method are said to be of order p if
error is given by
Expanding the block in Taylor series expansion gives
Comparing the coefficients of h, the order of the block is p = 5
With error constant
In numerical analysis, it is necessary that the method satisfies the necessary and sufficient conditions.
A numerical method is said to be consistent if the following conditions are satisfies
1) The order of the scheme must be greater than or equal to 1 i.e.
2)
3)
4)
where,
The general form of block method is given as
Applying (22)-(25) to (26) gives
Since no root has modulus greater than one and
According to Areo and Adeniyi [
together with the stability function
Hence, we express the block method (18) in form of
The elements of the matrices A, B, U and V are substituted and computing the stability function with Maple software yield, the stability polynomial of the method which is then plotted in MATLAB environment to produce the required absolute stability region of the methods, as shown by the figure below
The graph
In this section, we discuss the strategy for the implementation of the method. In addition, the performance of the method is tested on some modeled examples of second order initial value problems in Ordinary Differential Equations. Absolute error of the approximate solution are then compared with the existing methods. In particular, the comparison are made with those proposed by Awoyemi et al. and Ehigie et al.
Discussion of the results of the methods are also done here.
The method is tested on some numerical problems to test the accuracy of the proposed methods and our results are compared with the results obtained using existing methods.
The following problems are taken as test problems:
A 10-kg mass is attached to a spring having a spring constant of 140 N/m. The mass is started in motion from the equilibrium position with an initial velocity of 1 m/sec in the upward direction and with an applied external force
It follows from Newton’s second law
or
If the system starts at t = 0 with an initial velocity
Now if
Applying the initial conditions
We get the exact solution
Note that the exponential terms, which come from the homogeneous solution represent an associated free overdamped motion, quickly die out. These terms are the transient part of the solution. The terms coming from the particular solution however, do not die out at
Exact solution:
Exact solution:
Exact solution:
Exact solution:
Exact solution:
We have proposed a new one-twelfth step hybrid block method for the numerical solution of second order initial value problems of ordinary differential equations in this paper. The method is consistent, convergent and zero stable. The method derived efficiently solved second order Initial Value Problems as can be seen in the low error constant and hence better approximation than the existing methods as can be seen in Tables 1-6. We have
X-value | Exact Result | Computed Result | Error in our Method |
---|---|---|---|
0.10000000 | −0.064362051546 | −0.064362064290 | 1.274442e−08 |
0.20000000 | −0.084307205226 | −0.084307235669 | 3.044226e−08 |
0.30000000 | −0.084052253134 | −0.084052294635 | 4.150135e−08 |
0.40000000 | −0.075293042133 | −0.075293087518 | 4.538448e−08 |
0.50000000 | −0.063570639604 | −0.063570683902 | 4.429806e−08 |
0.60000000 | −0.051421170694 | −0.051421211160 | 4.046609e−08 |
0.70000000 | −0.039930529564 | −0.039930565039 | 3.547450e−08 |
0.80000000 | −0.029498658628 | −0.029498688913 | 3.028463e−08 |
0.90000000 | −0.020212691313 | −0.020212716720 | 2.540758e−08 |
1.00000000 | −0.012026994254 | −0.012027015325 | 2.107144e−08 |
X-value | Exact Result | Computed Result | Error in our Method | Error in Ehigie [ |
---|---|---|---|---|
0.00000000 | 0.000000000000 | 0.000000000000 | 0.0000000000 | 0.00E+00 |
0.10000000 | −0.105170918076 | −0.105170914197 | 3.878669e−09 | 3.38E−04 |
0.20000000 | −0.221402758160 | −0.221402741657 | 1.650316e−08 | 7.95E−04 |
0.30000000 | −0.349858807576 | −0.349858767673 | 3.990346e−08 | 1.40E−03 |
0.40000000 | −0.491824697641 | −0.491824622047 | 7.559449e−08 | 2.16E−03 |
0.50000000 | −0.648721270700 | −0.648721144076 | 1.266245e−07 | 3.13E−03 |
0.60000000 | −0.822118800391 | −0.822118604536 | 1.958546e−07 | 4.33E−03 |
0.70000000 | −1.013752707470 | −1.013752421828 | 2.856427e−07 | 5.79E−03 |
0.80000000 | −1.225540928492 | −1.225540528183 | 4.003098e−07 | 7.56E−03 |
0.90000000 | −1.459603111157 | −1.459602566974 | 5.441831e−07 | 9.70E−03 |
1.00000000 | −1.718281828459 | −1.718281106834 | 7.216251e−07 | 1.22E−02 |
X-value | Exact Result | Computed Result | Error in our Method | Error in Ehigie Ey(7/4) [ |
---|---|---|---|---|
0.00000000 | 0.0000000000000000 | 0.0000000000000000 | 0.000000+00 | 0.00E+00 |
0.01000000 | 0.9048374180359596 | 0.9048374180377620 | 1.802336e−12 | 4.08E−06 |
0.02000000 | 0.8187307530779818 | 0.8187307530850245 | 7.042700e−12 | 8.21E−06 |
0.03000000 | 0.7408182206817173 | 0.7408182206971640 | 1.544664e−11 | 1.24E−05 |
0.04000000 | 0.6703200460356393 | 0.6703200460624931 | 2.685374e−11 | 1.67E−05 |
0.05000000 | 0.6065306597126341 | 0.6065306597537941 | 4.115996e−11 | 2.12E−05 |
0.06000000 | 0.5488116360940276 | 0.5488116361518505 | 5.782286e−11 | 2.59E−05 |
0.07000000 | 0.4965853037914099 | 0.4965853038687461 | 7.733625e−11 | 3.09E−05 |
0.08000000 | 0.4493289641172210 | 0.4493289642167549 | 9.953394e−11 | 3.62E−05 |
0.09000000 | 0.4065696597405976 | 0.4065696598649566 | 1.243591e−10 | 4.18E−05 |
0.10000000 | 0.3678794411714401 | 0.3678794413234651 | 1.520249e−10 | 4.79E−05 |
X-value | Exact Result | Computed Result | Error in our Method | Error in Ehigie Ey(7/4) [ |
---|---|---|---|---|
0.000000 | 0.000000000000 | 0.000000000000 | 0.000000e−00 | 0.00e−00 |
0.100000 | 1.094837581925 | 1.094837581922 | 3.099965e−12 | 4.25e−06 |
0.200000 | 1.178735908636 | 1.178735908611 | 2.533729e−11 | 8.46e−06 |
0.300000 | 1.250856695787 | 1.250856695772 | 1.497313e−11 | 1.26e−05 |
0.400000 | 1.310479336312 | 1.310479336286 | 2.533840e−11 | 1.66e−05 |
0.500000 | 1.357008100495 | 1.357008100457 | 3.748513e−11 | 2.04e−05 |
0.600000 | 1.389978088305 | 1.389978088254 | 5.069922e−11 | 2.40e−05 |
0.700000 | 1.409059874522 | 1.409059874458 | 6.442646e−11 | 2.74e−05 |
0.800000 | 1.414062800247 | 1.414062800169 | 7.796741e−11 | 3.06e−05 |
0.900000 | 1.404936877898 | 1.404936877807 | 9.066370e−11 | 3.34e−05 |
1.000000 | 1.381773290676 | 1.381773290574 | 1.018543e−10 | 3.59e−05 |
X-value | Exact Result | Computed Result | Error in our Method | Error in Awoyemi et al. [ |
---|---|---|---|---|
0.00000000 | 0.000000000000 | 0.000000000000 | 0.000000e−00 | 0.0000e−00 |
0.10000000 | 1.050041729278 | 1.050041729278 | 3.086420e−14 | 2.6070e−13 |
0.20000000 | 1.100335347731 | 1.100335347731 | 2.420286e−13 | 1.9816e−09 |
0.30000000 | 1.151140435936 | 1.151140435936 | 8.442136e−13 | 6.5074e−09 |
0.40000000 | 1.202732554054 | 1.202732554052 | 2.102762e−12 | 1.5592e−08 |
0.50000000 | 1.255412811883 | 1.255412811879 | 4.368728e−12 | 3.1504e−08 |
0.60000000 | 1.309519604203 | 1.309519604195 | 8.227641e−12 | 5.6374e−08 |
0.70000000 | 1.365443754271 | 1.365443754257 | 1.439715e−11 | 9.6164e−08 |
0.80000000 | 1.423648930194 | 1.423648930170 | 2.408451e−11 | 1.5686e−08 |
0.90000000 | 1.484700278594 | 1.484700278555 | 3.921441e−11 | 2.4869e−08 |
1.00000000 | 1.549306144334 | 1.549306144271 | 6.294920e−11 | 3.8798e−08 |
X-value | Exact Result | Computed Result | Error in our Method | Error in Ehigie Ey(7/4) [ |
---|---|---|---|---|
0.52401544 | 0.250360930682 | 0.250360930682 | 1.515454e−14 | 000e−00 |
0.53401544 | 0.259074696912 | 0.259074696917 | 4.695411e−12 | 1.66e−09 |
0.54401544 | 0.267884830052 | 0.267884830070 | 1.825978e−11 | 4.70e−08 |
0.55401544 | 0.276787806164 | 0.276787806205 | 4.093692e−11 | 3.09e−07 |
0.56401544 | 0.285780064178 | 0.285780064251 | 7.270995e−11 | 5.45e−07 |
0.57401544 | 0.294858007310 | 0.294858007424 | 1.141097e−10 | 8.65e−07 |
0.58401544 | 0.304018004504 | 0.304018004668 | 1.646641e−10 | 1.28e−06 |
0.59401544 | 0.313256391883 | 0.313256392108 | 2.249845e−10 | 1.79e−06 |
0.61401544 | 0.331953526392 | 0.331953526767 | 3.750489e−10 | 2.42e−06 |
0.62401544 | 0.341404794918 | 0.341404795382 | 4.646992e−10 | 3.17e−06 |
also applied our method to the dynamic problem and the result is as displayed in
Emmanuel AdegbemiroAreo,Micheal TemitopeOmojola, (2015) A New One-Twelfth Step Continuous Block Method for the Solution of Modeled Problems of Ordinary Differential Equations. American Journal of Computational Mathematics,05,447-450. doi: 10.4236/ajcm.2015.54039