^{1}

^{*}

^{1}

^{*}

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.

Consider the linear seventh order differential equation

with the boundary conditions

Generally, this problem is difficult to solve analytically. Several numerical and semi-analytical methods have been developed for solving high order boundary value problems. For instance, a different approach of solving linear two-point boundary value problem has first been suggested by Bickley in 1968 [

In the present paper, the seventh order boundary value problems are solved using ninth degree spline approximation and compared with the solution obtained by eighth degree spline solution [

We divide the interval _{0}, the function y(x) in the interval

proceeding to the next interval_{n}. Thus the function y(x) is represented in the form

It can be shown that S(x) and its first six derivatives are continuous across nodes.

Method of Obtaining the Solution of Seventh Order Boundary Value Problems Using Ninth Degree Spline FunctionConsider the linear seventh order differential equation

with the boundary conditions

From (3), and taking spline approximation in (2) at

thus we get ninth degree spline approximation of_{ }is derived below. From Equation (1) we get

Substituting (1) and (4) in the differential Equation (2) at

where

Since S(x) approximates

From (5)-(12) we have (n + 8) equations, if these equations are taken in the order (7), (9), and (11) with

b, a will be an upper triangular matrix with two lower sub diagonals. The forward elimination is then simple with only two multipliers at each step, and back substitution is correspondingly easy.

In this section we consider three linear boundary value problems. Their numerical solution and absolute errors are given at different step lengths. The approximate solution, exact solutions and absolute errors at the grid points are summarized in tabular form. Further the approximate solution and exact solution have been shown graphically. The comparison of maximum absolute errors at different step lengths has been presented in tabular form.

Consider the linear non homogeneous seventh order boundary value problem with constant coefficients.

With the boundary conditions

The exact solution is

We find the solution of (13)-(14) by taking step lengths h = 0.2 and h = 0.1 at equal subintervals.

Solution with h = 0.2

The ninth degree spline S(x) which approximates y(x) is given by

where

We have 13 unknowns

Since

Differentiating (17)

and the seventh derivative is

Solving set of equations obtained from (16) we get the following values,

Substituting these values in Equation (15) we get the spline approximation

The values of

Solution with h = 0.1

Since h = 0.1 we suppose the grid points x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}, where, x_{0} = 0, x_{1} = 0.1, x_{2} = 0.2, x_{3} = 0.3, x_{4} = 0.4, x_{5} = 0.5, x_{6} = 0.6, x_{7} = 0.7, x_{8} = 0.8, x_{9} = 0.9, x_{10} = 1.

From Equation (1) ninth degree spline S(x) which approximate s u(x) becomes

From Equation (22) and the boundary conditions we get the following values

x | S(x) | u(x) | Absolute Error |
---|---|---|---|

0.0 | 0.00000000000 | 0.000000000000 | 0.000000000 |

0.2 | 0.19542437561 | 0.195424441305 | 3.74400E−09 |

0.4 | 0.35803789382 | 0.358037927433 | 3.36110E−08 |

0.6 | 0.437308436394 | 0.437308512093 | 7.56990E−08 |

0.8 | 0.356086488213 | 0.356086548558 | 6.03450E−08 |

1.0 | 0.000000000000 | 0.000000000000 | 0.000000000 |

Substituting these values in Equation (22) we get the spline approximation _{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} have been given in the

Consider non-homogeneous linear seventh order boundary value problem with variable coefficients

x | S(x) | u(x) | Absolute Error |
---|---|---|---|

0.0 | 0.000000000000 | 0.000000000000 | 0.00000000 |

0.1 | 0.099465382958 | 0.099465382626 | −3.3200E−10 |

0.2 | 0.195424444859 | 0.195424441305 | −3.5540E−09 |

0.3 | 0.283470361956 | 0.283470349590 | −1.2366E−08 |

0.4 | 0.358037954657 | 0.358037927400 | −2.7224E−08 |

0.5 | 0.412180361425 | 0.412180317675 | −4.3750E−08 |

0.6 | 0.437308566741 | 0.437308512093 | −5.4648E−08 |

0.7 | 0.428881204050 | 0.42888068568 | −5.1837E−08 |

0.8 | 0.356086581653 | 0.356086548558 | −3.3095E−08 |

0.9 | 0.221364288517 | 0.221364280004 | −8.5130E−09 |

1.0 | 0.000000000000 | 0.000000000000 | 0.00000000 |

Subject to the boundary conditions

The exact solution is

We find the solution of (23)-(24) by taking the step lengths h = 0.2 and h = 0.1 at equal sub intervals.

Solution when h = 0.2

Since h = 0.2 we suppose the grid points

From Equation (1) ninth degree spline S(x) which approximates u(x).

From S(x) and boundary conditions we get the following values.

a = 1, b = 0, c = −0.5, d = −0.333333 Equation (25) reduces to the form

From equation

we get j = −0.00119047619047 and from the remaining conditions we have the following values

Substituting these values in Equation (25) we get the spline approximation S(x) of u(x). The values of S(x), u(x) and the corresponding absolute errors at x_{1}, x_{2}, x_{3} and x_{4} has been given in

x | S(x) | u(x) | Absolute Error |
---|---|---|---|

0.0 | 1.000000000000 | 1.000000000000 | 0.00000000 |

0.2 | 0.977122197869 | 0.977122206528 | 8.6590E−09 |

0.4 | 0.895094735449 | 0.895094818584 | 8.3135E−08 |

0.6 | 0.728847331127 | 0.728847520156 | 1.8903E−07 |

0.8 | 0.445108018183 | 0.445108185698 | 1.6752E−07 |

1.0 | 0.00000000000 | 0.000000000000 | 0.00000000 |

Solution when h = 0.1

Since h = 0.1 we suppose the grid points x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} where

x_{0} = 0, x_{1} = 0.1, x_{2} = 0.2, x_{3} = 0.3, x_{4} = 0.4, x_{5}=0.5, x_{6} = 0.6, x_{7} = 0.7, x_{8} = 0.8, x_{9} = 0.9, x_{10} = 1

From Equation (1) ninth degree spline S(x) which approximates u(x) becomes

From S(x) and boundary conditions we get the following values.

a = 1, b = 0, c= −0.5, d = −0.333333, with these values (27) reduces to the form

proceeding as in the above, we get the following values

The values of S(x), u(x) and the corresponding absolute errors at x_{1}, x_{2}, _{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} has been given in

Consider the linear non-homogeneous seventh order boundary value problem with constant coefficients.

subject to the boundary conditions

The exact solution is

We find the solution of (29) by taking the step lengths h = 0.2 and h = 0.1 at equal sub intervals.

The values of S(x), u(x) and the corresponding absolute errors at x_{1}, x_{2}, x_{3} and x_{4} have been given in _{1}, x_{2}, _{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9} and x_{10} have been given in

x | S(x) | u(x) | Absolute Error |
---|---|---|---|

0.0 | 1.000000000000 | 1.000000000000 | 0.000000000 |

0.1 | 0.994653825368 | 0.994653826268 | 9.00000E−10 |

0.2 | 0.977122206669 | 0.977122206528 | 2.98590E−10 |

0.3 | 0.944901160432 | 0.944901165303 | 4.87100E−09 |

0.4 | 0.895094814647 | 0.895094818584 | 3.93700E−09 |

0.5 | 0.824360636278 | 0.824360635350 | −9.28000E−10 |

0.6 | 0.728847525865 | 0.728847520156 | −5.70900E−09 |

0.7 | 0.604125874632 | 0.604125812241 | −6.23910E−08 |

0.8 | 0.445108198680 | 0.445108185698 | −1.29820E−08 |

0.9 | 0.245960300626 | 0.245960311115 | 1.04890E−08 |

1.0 | 0.000000000000 | 0.000000000000 | 0.000000000 |

x | S(x) | u(x) | Absolute Error |
---|---|---|---|

0.0 | 1.000000000000 | 1.000000000000 | 0.000000000 |

0.2 | 0.977122202945 | 0.977122206528 | 3.58300E−09 |

0.4 | 0.895094784979 | 0.895094818584 | 3.36050E−08 |

0.6 | 0.728847097431 | 0.728847520156 | 4.22725E−07 |

0.8 | 0.44510798551 | 0.445108185698 | 1.71470E−07 |

1.0 | 0.00000000000 | 0.000000000000 | 0.000000000 |

x | S(x) | u(x) | Absolute Error |
---|---|---|---|

0.0 | 1.000000000000 | 1.000000000000 | 0.0000000000 |

0.1 | 0.994653826178 | 0.994653826268 | 9.000000E−11 |

0.2 | 0.977122205542 | 0.977122206528 | 9.860000E−10 |

0.3 | 0.944901150816 | 0.944901165303 | 1.448700E−08 |

0.4 | 0.895094780191 | 0.895094818584 | 3.839200E−08 |

0.5 | 0.824360562443 | 0.824360635350 | 7.290630E−08 |

0.6 | 0.728847490727 | 0.728847520156 | 2.942900E−08 |

0.7 | 0.604125798479 | 0.604125812241 | 1.376200E−08 |

0.8 | 0.445108155996 | 0.445108185698 | 2.970200E−08 |

0.9 | 0.245960126066 | 0.24596011115 | −1.491600E−08 |

1.0 | 0.000000000000 | 0.00000000000 | 0.0000000000 |

The numerical results obtained by ninth degree spline approximation are compared with the numerical results obtained by eighth degree spline approximation [

x | Exact Solution | Absolute Error [^{th} Degree | Absolute Error |
---|---|---|---|

0.0 | 0.0000000 | 0.00000000 | 0.00000000 |

0.1 | 0.09946538 | 7.9999E−08 | −3.3200E−10 |

0.2 | 0.19542444 | 9.8000E−07 | −3.5540E−09 |

0.3 | 0.28387034 | 2.3654E−05 | −1.2366E−08 |

0.4 | 0.35803792 | 7.4400E−06 | −2.7224E−08 |

0.5 | 0.41218031 | 1.1289E−05 | −4.3750E−08 |

0.6 | 0.43730851 | 1.3459E−05 | −5.4648E−08 |

0.7 | 0.42288806 | 1.4430E−05 | −5.1837E−08 |

0.8 | 0.35608654 | 2.0369E−05 | −3.3095E−08 |

0.9 | 0.22136428 | 4.7770E−05 | −8.5130E−09 |

1.0 | 0.00000000 | 0.00000000 | 0.00000000 |

x | Exact Solution | Absolute Error [^{th} Degree | Absolute Error |
---|---|---|---|

0.0 | 1.0000000000 | 0.00000000 | 0.00000000 |

0.1 | 0.994653826 | −7.9999E−09 | 9.0000E−10 |

0.2 | 0.9771222065 | −7.8799E−08 | −1.4100E−10 |

0.3 | 0.8950948185 | −2.32699E07 | 4.8710E−09 |

0.4 | 0.8950948185 | −3.85500E07 | 3.9370E−09 |

0.5 | 0.8243606350 | −4.1000E−07 | −9.2800E−10 |

0.6 | 0.7288475201 | −1.3900E−07 | −5.7090E−09 |

0.7 | 0.6041258122 | 3.87499E−07 | −6.2391E−08 |

0.8 | 0.4451081856 | 8.12699E−07 | −1.29820E08 |

0.9 | 0.2459603111 | 7.56099E−07 | 1.04890E−08 |

1.0 | 0.0000000000 | 0.000000000 | 0.000000000 |

x | Exact Solution | Absolute Error [^{th}^{ }Degree | Absolute Error |
---|---|---|---|

0.0 | 1.00000000000 | 0.0000000000 | 0.0000000 |

0.1 | 0.99465382627 | −3.048609E−09 | 9.000E−11 |

0.2 | 0.97712220653 | −2.273750E−08 | 9.860E−10 |

0.3 | 0.94490116530 | −7.243841E−08 | 1.449E−08 |

0.4 | 0.89509481858 | −1.655640E−08 | 3.839E−08 |

0.5 | 0.82436063535 | −3.192236E−07 | 7.291E−08 |

0..6 | 0.72884752016 | −5.570166E−07 | 2.943E−08 |

0.7 | 0.60412581224 | −9.098626E−07 | 1.376E−08 |

0.8 | 0.44510818570 | −1.413797E−06 | 2.970E−08 |

0.9 | 0.24596011120 | −2.103475E−06 | −1.492E−08 |

1.0 | 0.00000000000 | 0.0000000000 | 0.00000000 |

A ninth degree spline solution has been employed of example 1, 2 and 3 at step lengths h = 0.2 and h = 0.1. Numerical solutions are summarized in the tables and the comparison has been shown in figures. The maximum absolute errors at the given step length are −2.90100 × 10^{−9} and 7.2899 × 10^{−10}, 8.6950 × 10^{−9}, 9.00000 × 10^{−10}, 2.2970 × 10^{−9} and 9.0000 × 10^{−11} respectively. These values show that the agreement between approximate solution and exact solution is good. It is observed that the solution is more accurate when step length is small. We also compare our results with the results obtained using eighth degree spline solution [

ParchaKalyani,Mihretu NigatuLemma, (2016) Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions. Journal of Applied Mathematics and Physics,04,249-261. doi: 10.4236/jamp.2016.42032