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Implicit Methods for Numerical Solution of Singular Initial Value Problems

Habibur Rahaman

Habibur Rahaman

,

M. Kamrul Hasan

M. Kamrul Hasan

,

und

M. Shamsul Alam

M. Shamsul Alam

| 15. Apr. 2020

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Online veröffentlicht: 15. Apr. 2020

Seitenbereich: 1 - 8

Eingereicht: 24. Aug. 2019

Akzeptiert: 25. Dez. 2019

DOI: https://doi.org/10.2478/amns.2020.2.00001

Schlüsselwörter
Singular Integrals, Romberg scheme, Singular initial value problems, Implicit Runge-Kutta method

© 2020 Md. Habibur Rahaman et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Absolute errors of rhe solution of Eq. (21) at x = 0.01 by Hasan's [13, 16] methods and new methods.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻³	1.16741×10⁻²	7.722254×10⁻³	7.36938×10⁻³	5.33270×10⁻³	4.12995×10⁻³
10⁻⁴	3.70766×10⁻³	2.49450×10⁻³	2.38698×10⁻³	1.76377×10⁻³	1.39456×10⁻³
10⁻⁵	1.17390×10⁻³	7.94790×10⁻⁴	7.60955×10⁻⁴	5.65208×10⁻⁴	4.49539×10⁻⁴
10⁻⁶	3.71357×10⁻⁴	2.51870×10⁻⁴	2.41183×10⁻⁴	1.79468×10⁻⁴	1.42998×10⁻⁴

Absolute errors of the solution of Eq. (22) at x=0.0001 by Hasan's [13, 16] methods and new methods.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻⁵	1.37739×10⁻³	9.30410×10⁻⁴	8.90420×10⁻⁴	6.59620×10⁻⁴	5.23240×10⁻⁴
10⁻⁶	4.35887×10⁻⁴	2.95420×10⁻⁴	2.82850×10⁻⁴	2.10290×10⁻⁴	1.67420×10⁻⁴
10⁻⁷	1.37870×10⁻⁴	9.35400×10⁻⁵	8.95700×10⁻⁵	6.66700×10⁻⁵	5.31400×10⁻⁵
10⁻⁸	4.36000×10⁻⁵	2.95900×10⁻⁵	2.83400×10⁻⁵	2.11000×10⁻⁵	1.68200×10⁻⁵

Absolute errors of the solution of Eq. (22) at x=0.0000000001 by Hasan's [13, 16] methods and new methods with Romberg scheme.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻¹⁰	2.47256×10⁻⁸	9.85800×10⁻¹⁰	5.20713×10⁻¹⁰	1.02940×10⁻¹²	5.06390×10⁻¹²
10⁻¹¹	7.83301×10⁻⁹	3.14843×10⁻¹⁰	1.67741×10⁻¹⁰	2.72390×10⁻¹²	4.40500×10⁻¹³
10⁻¹²	2.47734×10⁻⁹	9.98726×10⁻¹¹	5.33553×10⁻¹¹	1.16940×10⁻¹²	2.0100×10⁻¹⁴
10⁻¹³	7.83433×10⁻¹⁰	3.16100×10⁻¹¹	1.68998×10⁻¹¹	3.96500×10⁻¹³	1.40000×10⁻¹⁵

Absolute errors of the solution of Eq. (21) at x = 0.0001 by Hasan's [13, 16] methods and new methods with Romberg scheme.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻⁵	1.70708×10⁻⁵	6.05027×10⁻⁶	6.49296×10⁻⁶	6.94702×10⁻⁶	6.93944×10⁻⁶
10⁻⁶	6.96009×10⁻⁶	3.94596×10⁻⁷	5.37472×10⁻⁷	6.93500×10⁻⁷	6.95451×10⁻⁷
10⁻⁷	2.35661×10⁻⁶	2.76102×10⁻⁸	1.78571×10⁻⁸	6.84388×10⁻⁸	6.94899×10⁻⁸
10⁻⁸	7.60778×10⁻⁷	2.39618×10⁻⁸	9.55524×10⁻⁹	6.56421×10⁻⁹	6.93995×10⁻⁹

Absolute errors of the solution of Eq. (21) at x = 0.0000000001 by Hasan's [13, 16] methods and new methods with Romberg scheme.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻¹⁰	2.42985×10⁻⁸	9.92961×10⁻¹⁰	5.27850×10⁻¹⁰	6.03890×10⁻¹²	6.70960×10⁻¹²
10⁻¹¹	7.83390×10⁻⁹	3.15568×10⁻¹⁰	1.68461×10⁻¹⁰	3.43440×10⁻¹²	6.01400×10⁻¹³
10⁻¹²	2.47747×10⁻⁹	9.99449×10⁻¹¹	5.34257×10⁻¹¹	1.23850×10⁻¹²	3.83000×10⁻¹⁴
10⁻¹³	7.83460×10⁻¹⁰	3.16220×10⁻¹¹	1.69114×10⁻¹¹	4.08100×10⁻¹³	4.70000×10⁻¹⁵

Absolute errors of the solution of Eq. (22) at x = 0.0001 by Hasan's [13, 16] methods and new methods with Romberg scheme.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻⁵	9.67336×10⁻⁶	1.28488×10⁻⁵	1.32698×10⁻⁵	1.36583×10⁻⁵	5.29373×10⁻⁶
10⁻⁶	6.11857×10⁻⁶	1.07858×10⁻⁶	1.21738×10⁻⁶	1.36474×10⁻⁶	5.29644×10⁻⁷
10⁻⁷	2.24030×10⁻⁶	4.20101×10⁻⁸	8.64651×10⁻⁸	1.35502×10⁻⁷	5.27899×10⁻⁸
10⁻⁸	7.38957×10⁻⁷	1.65915×10⁻⁸	2.47758×10⁻⁹	1.32726×10⁻⁸	5.26552×10⁻⁹

Absolute errors of rhe solution of Eq. (21) at x = 0.0001 by Hasan's [13, 16] methods and new methods.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻⁵	1.40542×10⁻³	9.51680×10⁻⁴	9.11070×10⁻⁴	6.76719×10⁻⁴	5.38238×10⁻⁴
10⁻⁶	4.44595×10⁻⁴	3.01520×10⁻⁴	2.88749×10⁻⁴	2.14863×10⁻⁴	1.71200×10⁻⁴
10⁻⁷	1.40609×10⁻⁴	9.54200×10⁻⁵	9.13754×10⁻⁵	6.80328×10⁻⁵	5.42385×10⁻⁵
10⁻⁸	4.44661×10⁻⁵	3.01800×10⁻⁵	2.89019×10⁻⁵	2.15226×10⁻⁵	1.71617×10⁻⁵

Absolute errors of the solution of Eq. (21) at x = 0.000001 by Hasan's [13, 16] methods and new methods with Romberg scheme.

h	Errors
h	Hasan 2^nd	Hasan 3^rd	New 3^rd	New 4^th	New 5^th
10⁻⁷	9.50211×10⁻⁵	2.85167×10⁻⁸	1.77693×10⁻⁸	6.92689×10⁻⁸	3.69929×10⁻⁸
10⁻⁸	9.18055×10⁻⁶	2.44011×10⁻⁸	9.73290×10⁻⁹	6.67930×10⁻⁹	7.06188×10⁻⁹
10⁻⁹	2.46532×10⁻⁷	9.26524×10⁻⁹	6.62384×10⁻⁹	5.78785×10⁻¹⁰	7.04187×10⁻¹⁰
10⁻¹⁰	7.81185×10⁻⁸	3.08492×10⁻⁹	1.61689×10⁻⁹	2.95883×10⁻¹¹	6.96841×10⁻¹¹

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