Recently many approximate techniques [1,2,3,4,5,6] have been developed for handling complicated physical problems. One of such complications arises for solving the singular initial value problems. The study of this topic has concerned the interest of many mathematicians and physicists. A first order singular initial value problem is encountered in ecology in the computation of avalanche run-up [7]. Several authors evaluated the singular initial value problems by both analytical and numerical techniques. Koch and Weinmuller [8] discussed the existence of an analytic solution of the first order singular initial value problems. Auzinger et al. [9] and Koch et al. [10–11] applied well-known acceleration technique Iterated Defect Correction (IDeC) based on implicit Euler method. Recently, Hasan et al. [12,13,14,15,16] derived some implicit formulae for solving first and second order singular initial value problems based on the integral formulae derived in [17–18]. These implicit methods give more accurate results than those obtained by the implicit Euler, second, third and fourth- order implicit Runge-Kutta (RK) methods. In this article, we develop three new (i.e. third, fourth and fifth-order) implicit formulae for solving singular initial value problems. Romberg scheme have been applied at the initial point for obtaining the improved results. Some suitable examples have been presented to illustrate these methods.
Earlier singular integrals were evaluated by Fox [19] based on extrapolation approach and then several Romberg schemes were used to improve the results. But Huq et al. [17] derived a straightforward formula for evaluating
By considering fourth order Lagrange's interpolation, Hasan et al. [18] derived another formula for evaluating integral Eq. (1) as
Then the following implicit formula
In this article, we have derived three implicit formulae which follow a class together with formula Eq. (3) [17]. The error of this formula is
By utilization of fourth order Lagrange's interpolation formula we have derived a third order integral formula as
By utilization of fifth order Lagrange's interpolation formula we obtain the fourth order integral formula as
Based on formula (10), the fourth order implicit method has been proposed as
Similarly, by using sixth order Lagrange's interpolation formula we have derived a fifth order integral formula as
The error of this formula is 1.73578 × 10−11
Based on formula (12), the fifth order implicit method has been proposed as
According to [19], the first Romberg's formula for evaluating
Similarly, the proposed Romberg scheme for the fourth order implicit formula becomes
And the Romberg scheme for the fifth order implicit formula Eq. (13) can be proposed as
In this section, the method has been illustrated with the following examples.
First we have solved this equation by our third order (i.e. Eq. (9)), fourth order (i.e. Eq. (11)) and fifth order ((i.e. Eq. (13)) formulae, and then Hasan's [13, 16] formulae. All the absolute errors have been calculated and presented respectively in Table 4.1(a) and Table 4.1(b). Finally we have obtained the solution of the Eq. (21) using the first Romberg's scheme of third order (i.e. Eq. (18)), fourth order (i.e. Eq. (19)) and fifth order ((i.e. Eq. (20)) formulae (together with Hasan's formulae) at initial point and then solved the rest again by third, fourth and fifth order formulae. The absolute errors of all the solutions have been presented in Tables 4.1(c), Tables 4.1(d) and Table 4.1(e). The exact solution of Eq. (21) is
Absolute errors of rhe solution of Eq. (21) at
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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 rhe solution of Eq. (21) at
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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 |
Absolute errors of the solution of Eq. (21) at
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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.
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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 the solution of Eq. (22) at x=0.0000000001 by Hasan's [13, 16] methods and new methods with Romberg scheme.
h | Errors | ||||
---|---|---|---|---|---|
Hasan 2 |
Hasan 3 |
New 3 |
New 4 |
New 5 |
|
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 |
Three implicit formulae and their Romberg's scheme have been presented for solving singular initial value problems. The approximate solutions of some first order linear and non-linear equations have been compared with their exact solutions. For the linear equation, the approximate solution of Eq. (21) has been obtained by the new formulae and the errors have been presented in Tables 4.1(a) and 4.1(b) together with corresponding errors of Hasan's methods. Tables 4.1(a) and 4.1(b) show that, the new methods give better results than those obtained by Hasan's methods. For Romberg scheme, when
From the above observation it may conclude that, our third-, fourth- and fifth order formulae provide better results than other existing formulae for both linear and non-linear equations. The utilization of Romberg scheme at initial stage provides more and more accurate results. However, the fifth order formula provides the best results among all the formulae presented in this article. It is clear that the six order formula must provide better results than our presented fifth order formulae since its order of error is