Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type (MRKT) methods for solving
This work deals with exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods for solving third-order ordinary differential equations (ODEs)
This sort of problems is often found in numerous physical problems like thin film flow, gravity-driven flows, electromagnetic waves, and so on. In the past and recent years many researchers constructed exponentially fitted and trigonometrically fitted explicit Runge-Kutta methods for solving first-order and second-order ordinary differential equations. Paternoster [
In this paper we construct explicit exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods with four-stage fifth-order, called EFMRKT5 and TFMRKT5, respectively. Section
This section discusses the oscillatory and nonoscillatory properties of the third-order linear differential equation
If
A solution of (
Particularly, this paper deals with two cases based on ( where
In this section, we will determine the conditions and develop exponentially fitted and trigonometrically fitted MRKT methods. In order to construct the exponentially fitted and trigonometrically fitted MRKT methods, the extra
for
The parameters of the MRKT methods are
The MRKT method can be expressed in Butcher notation using the table of coefficients as follows (see Table
The Butcher tableau MRKT method.
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To construct the exponentially fitted Runge-Kutta type four-stage fifth-order method the functions
and six more equations corresponding to
Solving (
Referring to the following fifth-order four-stage method developed by Fawzi et al. [
we solve (
These lead to our new exponentially fitted Runge-Kutta type four-stage fifth-order explicit MRKT method denoted as EFMRKT5. The corresponding Taylor series expansion of the solution is given by
This results in the new method called EFMRKT5. As
Exponentially fitted method leads to trigonometrically fitted method when replacing
Consider the same coefficients of fifth-order four-stage method developed by Fawzi et al.[
Next, solving (
These lead to our new explicit trigonometrically fitted MRKT which is called TFMRKT5 method. The corresponding Taylor series expansion of the solution is given by
where
This results in the new method called TFMRKT5. As
In this section, we will find the principal local truncation errors for
The
Notes: from
In this section, we will apply the new explicit exponentially fitted modified Runge-Kutta type method to some
Estimated frequency
Estimated frequency
Estimated frequency
Estimated frequency
Estimated frequency
Estimated frequency
Estimated frequency
Estimated frequency
Here, we will use the suggested method to a famous problem in engineering and physics based on the thin film flow of a liquid. Many researchers in the literature explain this problem more. Momoniat and Mahomed[
In this research, we have derived exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods for solving
Numerical results for problem in Thin Film Flow (
| Exact Solution | RK5B | RKF5 | RKT5 | EMFRKT5 | TFRK | TFMRKT5 |
---|---|---|---|---|---|---|---|
0.0 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 |
0.2 | 1.221211030 | 1.2212100068 | 1.2212100097 | 1.2212100039 | 1.2212100052 | 1.2212100218 | 1.2212100052 |
0.4 | 1.488834893 | 1.4888347851 | 1.4888347895 | 1.4888347797 | 1.4888347885 | 1.4888348090 | 1.4888347885 |
0.6 | 1.807361404 | 1.8073614063 | 1.8073614114 | 1.8073613988 | 1.8073614237 | 1.8073614357 | 1.8073614237 |
0.8 | 2.179819234 | 2.1798192463 | 2.1798192513 | 2.1798192371 | 2.1798192873 | 2.1798192788 | 2.1798192873 |
1.0 | 2.608275822 | 2.6082748841 | 2.6082748883 | 2.6082748735 | 2.6082749587 | 2.6082749176 | 2.6082749587 |
Numerical results for problem in thin film flow (
| Exact Solution | RK5B | RKF5 | RKT5 | EFMRKT5 | TFRK | TFMRKT5 |
---|---|---|---|---|---|---|---|
0.0 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 |
0.2 | 1.221211030 | 1.2212100045 | 1.2212100045 | 1.2212100045 | 1.2212100045 | 1.2212100045 | 1.2212100045 |
0.4 | 1.488834893 | 1.4888347799 | 1.4888347799 | 1.4888347799 | 1.4888347799 | 1.4888347799 | 1.4888347799 |
0.6 | 1.807361404 | 1.8073613977 | 1.8073613977 | 1.8073613977 | 1.8073613977 | 1.8073613977 | 1.8073613977 |
0.8 | 2.179819234 | 2.1798192339 | 2.1798192339 | 2.1798192339 | 2.1798192340 | 2.1798192339 | 2.1798192340 |
1.0 | 2.608275822 | 2.6082748676 | 2.6082748676 | 2.6082748676 | 2.6082748677 | 2.6082748676 | 2.6082748677 |
Numerical results for problem in thin film flow (
| RK5B | RKF5 | RKT5 | EMFRKT5 | TFRK | TFMRKT5 |
---|---|---|---|---|---|---|
0.0 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 |
0.2 | 1.2211551491 | 1.2211551546 | 1.2211551394 | 1.2211551412 | 1.2211551831 | 1.2211551412 |
0.4 | 1.4881052974 | 1.4881053065 | 1.4881052807 | 1.4881052926 | 1.4881053519 | 1.4881052926 |
0.6 | 1.8042625677 | 1.8042625794 | 1.8042625459 | 1.8042625786 | 1.8042626364 | 1.8042625786 |
0.8 | 2.1715228242 | 2.1715228376 | 2.1715227987 | 2.1715228633 | 2.1715229031 | 2.1715228633 |
1.0 | 2.5909582923 | 2.5909583063 | 2.5909582638 | 2.5909583715 | 2.5909583783 | 2.5909583715 |
Numerical results for problem in thin film flow (
| RK5B | RKF5 | RKT5 | EMFRKT5 | TFRK | TFMRKT5 |
---|---|---|---|---|---|---|
0.0 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 | 1.000000000 |
0.2 | 1.2211551424 | 1.2211551424 | 1.2211551424 | 1.2211551424 | 1.2211551424 | 1.2211551424 |
0.4 | 1.4881052842 | 1.4881052842 | 1.4881052842 | 1.4881052842 | 1.4881052842 | 1.4881052842 |
0.6 | 1.8042625481 | 1.8042625481 | 1.8042625481 | 1.8042625482 | 1.8042625481 | 1.8042625482 |
0.8 | 2.1715227981 | 2.1715227981 | 2.1715227981 | 2.1715227982 | 2.1715227981 | 2.1715227982 |
1.0 | 2.5909582591 | 2.5909582591 | 2.5909582591 | 2.5909582592 | 2.5909582591 | 2.5909582592 |
Comparison of error for problem in thin film flow (
| RK5B | RKF5 | RKT5 | EFMRKT5 | TFRK | TFMRKT5 |
---|---|---|---|---|---|---|
0.0 | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) |
0.2 | 1.0230 (-6) | 1.0200(-6) | 1.2600(-6) | 1.0250(-6) | 1.0080(-6) | 1.0250(-6) |
0.4 | 1.0800(-7) | 1.0300(-7) | 1.1300(-7) | 1.0500(-7) | 8.4100(-7) | 1.0500(-7) |
0.6 | 2.0000(-9) | 7.0000(-9) | 5.0000(-8) | 2.0000(-8) | 3.2000(-8) | 2.0000(-8) |
0.8 | 1.2000(-8) | 1.7000(-8) | 3.0000(-9) | 5.300(-8) | 4.5000(-8) | 5.3000(-8) |
1.0 | 9.3800(-7) | 9.3400(-7) | 9.4800 (-7) | 8.6300 (-7) | 9.0400(-7) | 8.6300(-7) |
Comparison of error for problem in thin film flow (
| RK5B | RKF5 | RKT5 | EFMRKT5 | TFRK | TFMRKT5 |
---|---|---|---|---|---|---|
0.0 | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) |
0.2 | 1.0260(-6) | 1.0260(-6) | 1.0260(-6) | 1.0260(-6) | 1.0260(-6) | 1.0260(-6) |
0.4 | 6.0000(-7) | 6.0000(-7) | 6.0000(-7) | 6.0000(-7) | 6.0000(-7) | 6.0000(-7) |
0.6 | 9.0000(-9) | 9.0000(-9) | 9.0000(-9) | 9.0000(-9) | 9.0000(-9) | 9.0000(-9) |
0.8 | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) | 0.0000(0) |
1.0 | 9.5400(-7) | 9.5400(-7) | 9.5400(-7) | 9.5400(-7) | 9.5400(-7) | 9.5400(-7) |
The efficiency curve for EFMRKT5, RKT5, RK5B, and RKF5 for Problem
The efficiency curve for EFMRKT5, RKT5, RK5B, and RKF5 for Problem
The efficiency curve for EFMRKT5, RKT5, RK5B, and RKF5 for Problem
The efficiency curve for EFMRKT5, RKT5, RK5B, and RKF5 Problem
The efficiency curve for TFMRKT5, RKT5, RK5B, RKF5, and TFRK for Problem
The efficiency curve for TFMRKT5, RKT5, RK5B, RKF5, and TFRK for Problem
The efficiency curve for TFMRKT5, RKT5, RK5B, RKF5, and TFRK for Problem
The efficiency curve for TFMRKT5, RKT5, RK5B, RKF5, and TFRK for Problem
Plot of graph for function evaluations against step size
Plot of graph for function evaluations against step size
The principal local truncation errors for
The principal local truncation errors for
The authors declare that there are no conflicts of interest regarding the publication of this paper.