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This paper focuses on the development of a hybrid method with block extension for direct solution of initial value problems (IVPs) of general third-order ordinary differential equations. Power series was used as the basis function for the solution of the IVP. An approximate solution from the basis function was interpolated at some selected off-grid points while the third derivative of the approximate solution was collocated at all grid and off-grid points to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was implemented with five test problems including the Genesio equation to confirm its accuracy and usability. The rate of convergence (ROC) reveals that the method is consistent with the theoretical order of the proposed method. Comparison of the results with some existing methods shows the superiority of the accuracy of the method.

The focus of this article is to find an approximate solution on a given interval to third order initial value problems (IVP) of the type

y ‴ ( x ) = f ( x , y ( x ) , y ′ ( x ) ) , y ( x ) = α a , y ′ ( x ) = α b , y ″ ( x ) = α c (1)

where x ∈ [ a , b ] ⊂ ℝ and y ( x ) , f ( x , y ( x ) , y ′ ( x ) , y ″ ( x ) ) ∈ ℝ n . In recent time, direct numerical solution of (1) without reduction to equivalent first-order initial value problems (see: [

Recently, in order to remove the difficulties usually encountered by adopting this mode of solution, researchers ( [

The quest for numerical methods with better accuracy has also led to the introduction of hybrid linear multistep methods which have recorded high success since its introduction. These successes motivated us to propose a hybrid method with block extension for the solution of (1).

In the next section, we discuss in detail the derivation of the proposed method with its implementation in block mode, followed by analysis of the proposed method to establish the numerical stability, numerical example to demonstrate the efficiency advantages of the proposed method and subsequently. Conclusion was drawn on the performance of the proposed method when applied to solve the numerical examples.

In order to obtain a numerical formula for the approximate solution of (1), the function

y ( x ) = ∑ v ( c + i ) − 1 g v x v (2)

is considered as the basis where x is continuous within the interval [ a , b ] , c and i denote collocation and interpolation points respectively. Variable g v ’s are coefficients to be determined. The third derivative of (2) equated to (1) is given as

∑ v ( c + i ) − 1 v ( v − 1 ) ( v − 2 ) g v x v − 3 = f ( x , y , y ′ , y ″ ) (3)

Evaluating (2) at x = x n + v , v = 2 8 , 4 8 , 6 8 , (3) at x = x n + v , v = 0 ( 2 8 ) 1 using τ = x − x n + k h yield the following interpolation and collocation matrix

X A = B (4)

where

A = ( g 0 g 1 g 2 g 3 g 4 g 5 g 6 g 7 ) , B = ( y n + 2 8 y n + 4 8 y n + 6 8 f n f n + 2 8 f n + 4 8 f n + 6 8 f n + 1 ) ,

X = ( 1 1 4 1 16 1 64 1 256 1 1024 1 4096 1 16384 1 1 2 1 4 1 8 1 16 1 32 1 64 1 128 1 3 4 9 16 27 64 81 256 243 1024 729 4096 2187 16384 0 0 0 6 0 0 0 0 0 0 0 6 6 15 4 15 8 105 128 0 0 0 6 12 15 15 105 8 0 0 0 6 18 135 4 405 8 8505 128 0 0 0 6 24 60 120 210 ) .

where f n + v = f ( x n + v , y ′ n + v , y ″ n + v ) , y n + v ≈ y ( x n + v ) . Solving the matrix Equation (4) for coefficients g v ’s and substituting into (2) yields after some simplification the continuous method

y ¯ ( x ) = ζ 1 4 y n + 1 4 + ζ 1 2 y n + 1 2 + ζ 3 4 y n + 3 4 + h 3 ( Θ 0 ( x ) f n + Θ 1 4 ( x ) f n + 1 4 + Θ 1 2 ( x ) f n + 1 2 + Θ 3 4 ( x ) f n + 3 4 + Θ 1 ( x ) f n + 1 ) (5)

with the following coefficients:

ζ 1 4 = 8 τ 2 − 10 τ + 3

ζ 1 2 = − 16 τ 2 + 16 τ − 3

ζ 3 4 = 8 τ 2 − 6 τ + 1

Θ 0 = 1 322560 ( 4 τ − 1 ) ( 2 τ − 1 ) ( 4 τ − 3 ) ( 512 τ 4 − 1472 τ 3 + 1360 τ 2 − 400 τ + 7 )

Θ 1 4 = − 1 80640 ( 4 τ − 3 ) ( 2 τ − 1 ) ( 4 τ − 1 ) ( 512 τ 4 − 1248 τ 3 + 688 τ 2 + 258 τ − 203 )

Θ 1 2 = 1 53760 ( 4 τ − 3 ) ( 2 τ − 1 ) ( 4 τ − 1 ) ( 512 τ 4 − 1024 τ 3 + 240 τ 2 + 272 τ + 147 )

Θ 3 4 = − 1 80640 ( 4 τ − 3 ) ( 2 τ − 1 ) ( 4 τ − 1 ) ( 512 τ 4 − 800 τ 3 + 16 τ 2 + 62 τ + 7 )

Θ 1 = 1 322560 ( 4 τ − 1 ) ( 2 τ − 1 ) ( 4 τ − 3 ) ( 512 τ 4 − 576 τ 3 + 16 τ 2 + 48 τ + 7 )

Evaluating the continuous scheme (5) at τ = 0 , 1 and its first and second derivatives at τ = 0 yield two discrete, one first and second derivatives schemes. These can be represented in a block matrix finite difference form as

ϒ 0 Y m , 0 = ϒ 1 Y m , 1 + h ϒ 2 Y m , 2 + h 2 ϒ 3 Y m , 3 + h 3 H ¯ F m , 0 + h 3 H F m , 1 (6)

where T denotes the transpose,

H = ( 107 64512 − 103 107520 43 107520 − 47 645120 83 5040 − 1 168 13 5040 − 19 40320 1863 35840 − 243 35840 45 7168 − 81 71680 34 315 1 210 2 105 − 1 504 ) ;

H ¯ = ( 0 0 0 113 71680 0 0 0 331 40320 0 0 0 331 40320 0 0 0 31 840 ) ; ϒ 0 = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ;

ϒ 1 = ( 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ) ; ϒ 2 = ( 0 0 0 1 4 0 0 0 1 2 0 0 0 3 4 0 0 0 1 ) ; ϒ 3 = ( 0 0 0 1 32 0 0 0 1 8 0 0 0 9 32 0 0 0 1 2 )

Vectors Y m ,0 , Y m ,1 , Y m ,2 , Y m ,3 , F m ,0 and F m ,1 are

Y m , 0 = ( y n + 1 4 , y n + 1 2 , y n + 3 4 , y n + 1 ) T , Y m , 1 = ( y n − 1 4 , y n − 1 2 , y n − 3 4 , y n ) T , Y m , 2 = ( y n − 1 4 , y n − 1 2 , y n − 3 4 , y ′ n ) T , Y m , 3 = ( y n − 1 4 , y n − 1 2 , y n − 3 4 , y ″ n ) T , F m , 0 = ( f n − 1 4 , f n − 1 2 , f n − 3 4 , f n ) T , F m , 1 = ( f n + 1 4 , f n + 1 2 , f n + 3 4 , f n + 1 ) T .

This section presents analysis of the proposed method (6) vis-a-vis the order, consistency, zero-stability and convergence.

The linear operator associated with the block method (6) is

L [ y ( x ) ; h ] = ϒ 0 Y m , 0 − ( ϒ 1 Y m , 1 + h ϒ 2 Y m , 2 + h 2 ϒ 3 Y m , 3 + h 3 H ¯ F m , 0 + h 3 H F m , 1 ) (7)

where y ( x ) is an arbitrary function which is continuously differentiable on [ a , b ] . Following Lambert [

L [ y ( x ) ; h ] = c ¯ 0 y ( x ) + c ¯ 1 h y ′ ( x ) + c ¯ 2 h 2 y ″ ( x ) + ⋯ + c ¯ p h p y p ( x ) + ⋯ , (8)

where the constant coefficients c ¯ p , p = 0 , 1 , 2 , ⋯ are given as follows:

c ¯ 0 = ζ k + ζ u + ζ v + ζ w

c ¯ 1 = k ζ k + u ζ u + v ζ v + w ζ w

c ¯ 2 = 1 2 ! ( k 2 ζ k + u 2 ζ u + v 2 ζ v + w 2 ζ w )

c ¯ 3 = 1 3 ! ( k 3 ζ k + u 3 ζ u + v 3 ζ v + w 3 ζ w ) − 1 0 ! ( Θ 0 + Θ k + Θ u + Θ v + Θ w )

c ¯ 4 = 1 p ! ( k 4 ζ k + u 4 ζ u + v 4 ζ v + w 4 ζ w ) − 1 ( p − 3 ) ! ( k Θ k + u Θ u + v Θ v + w Θ w )

⋮

c ¯ p = 1 p ! ( k p ζ k + u p ζ u + v p ζ v + w p ζ w ) − 1 ( p − 3 ) ! ( k p − 3 Θ k + u p − 3 Θ u + v p − 3 Θ v + w p − 3 Θ w ) , p = 4,5, ⋯

Going by Lambert [

L [ y ( x ) ; h ] = 0 ( h p + 1 ) , c ¯ 0 = c ¯ 1 = ⋯ = c ¯ p = 0 , c ¯ p + 3 ≠ 0

Therefore c ¯ p + 3 is the error constant and c ¯ p + 3 h p + 3 y p + 3 ( x n ) is the principal local truncation error at point x n . The order of the proposed method (6) and the corresponding error constant are as reported in

Definition 1 (consistency).

The proposed method (6) is said to be consistent if the order of method is greater than or equal to one, that is if p ≥ 1 . In addition to

1) ρ ( 1 ) = 0 and

Scheme | Order | Error constant | |
---|---|---|---|

y n + 1 4 | 5 | 1.55 E − 06 | |

(6) | y n + 1 2 | 5 | 3.39 E − 07 |

y n + 3 4 | 5 | 5.26 E − 08 | |

y n + 1 | 5 | 8.28 E − 07 |

2) ρ ′ ( 1 ) = σ ( 1 ) where ρ ( z ) and σ ( z ) are 1st and 2nd characteristics polynomial respectively.

Definition 2 (Zero-stability).

The block method (6) is said to be zero-stable if the roots

ρ ( z ) = det [ ∑ i = 0 k ϒ ( i ) z k − i ] (9)

satisfies | z i | ≤ 1 , i = 1 , ⋯ , k and the roots with | z i | = 1 , the multiplicity must not exceed one. Applying (9) to the proposed method (6) yields the following

ρ ( z ) = | [ z 0 0 0 0 z 0 0 0 0 z 0 0 0 0 z ] − [ 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ] | = | [ z 0 0 − 1 0 z 0 − 1 0 0 z − 1 0 0 0 z − 1 ] | = z 3 ( z − 1 ) (10)

This result shows that the method is zero-stable.

Definition 3 (convergence).

The necessary and sufficient condition for the proposed method (6) to be convergent are that it must be consistent and zero-stable according to Dahlquist see [

In order to study the stability domain of the proposed method (6), the test equations

y ′ = λ y (11)

y ″ = λ 2 y (12)

and

y ‴ = λ 3 y (13)

are applied to the block method (6) with z = λ y and η represents the roots of the first characteristic polynomial of the block method (6). This is then reformulated as a general linear method as discussed in [

[ Y ⋮ Y i + 1 ] = [ A V ⋮ ⋱ ⋮ B U ] [ h 3 f ( y ) ⋮ Y i − 1 ] (14)

where

A = [ 0 0 0 0 0 113 71680 107 64512 − 103 107520 43 107520 − 47 645120 331 40320 83 5040 − 1 168 13 5040 − 19 40320 331 40320 1863 35840 − 243 35840 45 7168 − 81 71680 31 840 34 315 1 210 2 105 − 1 504 ] ,

B = [ 113 71680 107 64512 − 103 107520 43 107520 − 47 645120 31 840 34 315 1 210 2 105 − 1 504 ]

U = [ 0 1 0 1 ] , V = [ 0 0 0 0 0 1 1 1 1 1 ] T , f ( y ) = [ f n f n + 1 4 f n + 1 2 f n + 3 4 f n + 1 ] T ,

Y i + 1 = [ y n + 1 4 y n + 1 ] , Y i − 1 = [ y n + 1 4 y n ]

By solving the stability function

p ( η , z ) = | η I − ( v + z B m ( I − z A m ) − 1 U m ) | (15)

yields the polynomial

p ( η , z ) = η 315 [ 2835 η z 4 + 4737600 η z 3 + 50591 z 4 + 417312000 η z 2 − 134158720 z 3 − 145871596800 z 2 + 124853944320000 η − 20808990720000 z − 124853944320000 9 z 4 + 15040 z 3 + 1324800 z 2 + 396361728000 ] (16)

(16) and its derivatives are then plotted in the MATLAB environment given the stability region displayed in

Definition 4 (Lambert and Watson [

Method (6) is P-stable if its interval of periodicity is ( 0, ∞ ) . It is clearly

shown in

Example I

The first numerical example to be considered is the oscillatory problem

y ‴ = 3 sin x , y ( 0 ) = 1 , y ′ ( 0 ) = 0 , y ″ ( 0 ) = − 2 , h = 0.1

with the theoretical solution

y ( x ) = 3 cos x + x 2 2 − 2.

This example was solved by [

Example II

The second example considered is the special third order problem

y ‴ = e x , y ( 0 ) = 3 , y ′ ( 0 ) = 1 , y ″ ( 0 ) = 5 , h = 0.1

with the theoretical solution

y ( x ) = y ( x ) = 2 + 2 x 2 + e x

Source: [

Example III

Another example considered is a general third order problem

y ‴ + 2 y ″ − 9 y ′ − 18 y = − 18 x 2 − 18 x + 22 , y ( 0 ) = − 2 , y ′ ( 0 ) = − 8 , y ″ ( 0 ) = − 12 , h = 0.1

with the theoretical solution

X-value | Exact Result | Computed | Errors | Error [ |
---|---|---|---|---|

0.1000 | 0.990012495834077020 | 0.990012495834030842125648 | 4.645616E(−14) | 1.743050e−14 |

0.2000 | 0.960199733523725120 | 0.960199733523539455437233 | 1.854379E(−13) | 1.082467e−13 |

0.3000 | 0.911009467376818090 | 0.911009467376402269866494 | 4.157891E(−13) | 2.711165e−13 |

0.4000 | 0.843182982008654940 | 0.843182982007919653122556 | 7.355953E(−13) | 5.079270e−13 |

0.5000 | 0.757747685671117830 | 0.757747685669975944913026 | 1.142203E(−12) | 8.164580e−13 |

0.6000 | 0.656006844729034370 | 0.65600684472903489172286 | 1.632248E(−12) | 1.199707e−12 |

0.7000 | 0.539526561853464590 | 0.539526561851263593901655 | 2.201685E(−12) | 1.654343e−12 |

0.8000 | 0.410120128041495670 | 0.410120128038650431437839 | 2.845831E(−12) | 1.674639e−10 |

0.9000 | 0.269829904811992540 | 0.269829904808433956143307 | 3.559413E(−12) | 3.336392e−10 |

1.0000 | 0.120906917604417960 | 0.120906917600082534334178 | 4.336618E(−12) | 5.001723e−10 |

y ( x ) = − 2 e 3 x + e − 2 x + x 2 − 1.

The theoretical solution at x = 1 is y ( 1 ) ≊ − 40.0357385631387227899630 . The errors were obtained at x = 1 using our method at a fixed step-size h = 0.1 ; 0.05 ; 0.025 ; 0.0125 ; 0.00625 . The numerical results are compared with those of [

Example IV

General nonlinear third order equation

y ‴ = y ′ ( 2 x y ″ + y ′ ) , y ( 0 ) = 1 , y ′ ( 0 ) = 1 / 2 , y ″ ( 0 ) = 0.1 , h = 0.1

with the theoretical solution

y ( x ) = 1 + 1 2 log ( 2 + x 2 − x )

is also considered. Source: [

Application to solve nonlinear Genesio equation

The chaotic Genesio equation reported in [

y ‴ = − α y ″ − β y ′ + f ( y (x))

with

f ( y ( x ) ) = − γ y ( x ) + y 2 (x)

y ( 0 ) = 0.2 , y ′ ( 0 ) = − 0.3 , y ″ ( 0 ) = 0.1 , x ∈ [ a , b ]

where α = 1.2 , β = 2.92 and γ = 6 are the positive constants that satisfied

α β < γ

b | h | Method | Step | Maximum Error |
---|---|---|---|---|

Proposed method | 30 | 9.6015e(−17) | ||

BHCM | 34 | 7.48e(−17) | ||

0.01 | Adams | 100 | 6.40e(−10) | |

Olabode | 34 | 8.89e(−13) | ||

Adesanya | 25 | 1.75e(−14) |

h | y | Max. Error | Error in [ | ROC |
---|---|---|---|---|

0.1 | −40.0357384989252357390316 | 6.421349E(−8) | 1:340886(−03) | - |

0.05 | −40.0357385621393199701503 | 9.994028E(−10) | 9:258900(−05) | 6.00 |

0.025 | −40.0357385631231226524929 | 1.560014E(−11) | 6.075364(−06) | 6.00 |

0.0125 | −40.0357385631384791804478 | 2.436095E(−13) | 3.889526(−07) | 6.00 |

0.00625 | −40.0357385631387191829967 | 3.606966E(−15) | 2.460220(−08) | 6.08 |

for the solution to exist. The solution of this problem is presented in

In this work, hybrid method with block extension for the direct solution of third order ordinary differential equations has been proposed. Numerical examples are considered to demonstrate the efficiency advantage of the method especially the Genesio equation which is chaotic in nature. The analysis, stability and numerical examples revealed that the proposed method is efficient for direct solution of third order ordinary differential equations.

The authors declare no conflicts of interest regarding the publication of this paper.

Duromola, M.K. and Momoh, A.L. (2019) Hybrid Numerical Method with Block Extension for Direct Solution of Third Order Ordinary Differential Equations. American Journal of Computational Mathematics, 9, 68-80. https://doi.org/10.4236/ajcm.2019.92006