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Solving coupled non-linear higher order BVPs using improved shooting method

Shumaila Javeed
Shumaila Javeed
 et 
Evren Hincal
Evren Hincal
   | 10 janv. 2024
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Article Category: Original Study
Publié en ligne: 10 janv. 2024
Pages: 165 - 178
Reçu: 10 juin 2023
Accepté: 03 sept. 2023
DOI: https://doi.org/10.2478/ijmce-2024-0013
Mots clés
© 2024 Shumaila Javeed et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Shooting Method
Shooting Method

Fig. 2

Problem 1: Comparison of Algorithm 1, 2 and 3 in approaching f (1) = 0.4, f″(1) = 0 and θ′(1) = 0 after 4 time recursive iterations with h = 0.001.
Problem 1: Comparison of Algorithm 1, 2 and 3 in approaching f (1) = 0.4, f″(1) = 0 and θ′(1) = 0 after 4 time recursive iterations with h = 0.001.

Fig. 3

Problem 1: Comparison of Algorithm 1, 2 and 3 in approaching f (1) = 0.4, f ″(1) = 0 and θ′(1) = 0 after 21 time recursive iterations with h = 0.001.
Problem 1: Comparison of Algorithm 1, 2 and 3 in approaching f (1) = 0.4, f ″(1) = 0 and θ′(1) = 0 after 21 time recursive iterations with h = 0.001.

Fig. 4

Problem 2: Comparison of Algorithm 1, 2 and 3 in approaching f′(10) = 0 after 1 time recursive iterations with h = 0.01.
Problem 2: Comparison of Algorithm 1, 2 and 3 in approaching f′(10) = 0 after 1 time recursive iterations with h = 0.01.

Fig. 5

Problem 2: Comparison of Algorithm 1, 2 and 3 in approaching θ (10) = 0 after 12 times recursive iterations with h = 0.01.
Problem 2: Comparison of Algorithm 1, 2 and 3 in approaching θ (10) = 0 after 12 times recursive iterations with h = 0.01.

Fig. 6

Problem 2:Comparison of Algorithm 1, 2 and 3 in approaching f ′(10) = 0 after 4 times recursive iterations with h = 0.01.
Problem 2:Comparison of Algorithm 1, 2 and 3 in approaching f ′(10) = 0 after 4 times recursive iterations with h = 0.01.

Fig. 7

Problem 2: Comparison of Algorithm 1, 2 and 3 in approaching θ (10) = 0 after 69 times recursive iterations with h = 0.01.
Problem 2: Comparison of Algorithm 1, 2 and 3 in approaching θ (10) = 0 after 69 times recursive iterations with h = 0.01.

Comparison of absolute percentage error of Algorithms 1, 2 and 3 in hitting right BCs f (1) = 0.4, f″(1) = 0 and θ′(1) = 0.

Algo Ab err (%) f Ab err (%) f″ Ab err (%) θ′
1 5.999999999950489 × 10−4 0.0041000 0.0168000
2 2 × 10−4 0.0014000 0.0013000
3 0.0000000000 0.0000000 0.0000000

Comparison of Algorithms 1, 2 and 3 and their absolute percentage error and number of iterations in hitting conditions θ (10) = 0.

Algo t s θ IT CPU time [s] Abs err (%) θ
1 2.414213562373132 1.350383812070473 0.0000031736 69 4.173 3.1736 ×10−4
2 2.414213562373139 1.350376073781895 0.0000001639 32 2.039 1.6390 ×10−5
3 2.414213562373149 1.350375652724329 0.0000000000 12 1.113 0.0000000000

Comparison of Algorithms 1, 2 and 3 in hitting right BCs f (1) = 0.4, f″(1) = 0 and θ′(1) = 0.

Algo t s f f″ θ′ No. of IT
1 −2.6809589589908 −3.5959727116775 0.400006 0.000041 0.000168 21
2 −2.6809636175929 −3.5959908500960 0.400002 0.000014 0.000013 10
3 −2.6809659852767 −3.5959915546254 0.400000 0.000000 0.000000 4

Comparison of Algorithms 1, 2 and 3 and their absolute percentage errors and number of iterations (IT) in hitting conditions f′(10) = 0.

Algo t s f′ IT Abs err (%) f′
1 2.414213562379738 1.350433988415789 0.0000000059 4 5.90000000×10−7
2 2.414213558978872 1.350423606035475 −0.0000030464 2 3.0464000×10−4
3 2.414213562373149 1.350401422589242 0.0000000000 1 0.0000000000
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