#### Abstract

Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type (MRKT) methods for solving are derived in this paper. These methods are constructed which exactly integrate initial value problems whose solutions are linear combinations of the set functions and for exponentially fitted and and for trigonometrically fitted with being the principal frequency of the problem and the frequency will be used to raise the accuracy of the methods. The new four-stage fifth-order exponentially fitted and trigonometrically fitted explicit MRKT methods are called EFMRKT5 and TFMRKT5, respectively, for solving initial value problems whose solutions involve exponential or trigonometric functions. The numerical results indicate that the new exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods are more efficient than existing methods in the literature.

#### 1. Introduction

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 [1] developed Runge-Kutta-NystrÃ¶m methods for ODEs with periodic solutions based on trigonometric polynomials. Vanden Berghe et al. [2] developed exponentially fitted Runge-Kutta methods. Simos [3] extended exponentially fitted Runge-Kutta methods for the numerical solution of the Schrodinger equation and related problems. Kalogiratou et al. [[4, 5]] constructed trigonometrically and exponentially fitted Runge-Kutta-NystrÃ¶m methods for the numerical solution of the Schrodinger equation and related problems which is eighth algebraic order. Next Simos et al. [6] constructed exponentially fitted Runge-Kutta-NystrÃ¶m method for the numerical solution of initial value problems with oscillating solutions. Sakas et al. [7] developed a fifth algebraic order trigonometrically fitted modified Runge-Kutta Zonneveld method for the numerical solution of orbital problems. Van de Vyver [8] in 2005 constructed Runge-Kutta-NystrÃ¶m pair for the numerical integration of perturbed oscillators. Then Yang et al. [9] constructed trigonometrically fitted adapted Runge-Kutta-NystrÃ¶m methods for perturbed oscillators. Recently, Demba et al. [10] constructed an explicit trigonometrically fitted Runge-Kutta-NystrÃ¶m method using Simos technique.

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 2 discussed the oscillatory and nonoscillatory properties of the third-order linear differential equation. In Section 3, the necessary conditions and the derivation for exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods for solving third-order ODEs are given. The error analysis of the new EFMRKT5 and TFMRKT5 methods was discussed in Section 4, respectively. The effectiveness of the new methods when compared with existing methods is given in Section 5. The thin film flow problem is discussed in Section 6.

#### 2. Third-Order Linear Differential Equation with Oscillating and Nonoscillating Solutions

This section discusses the oscillatory and nonoscillatory properties of the third-order linear differential equationA solution of (2) will be said to be oscillatory if it changes signs for arbitrarily large values of . The other solutions will be said to be nonoscillatory.

If and are constants, then it is easy to show that if (2) has an oscillatory solution, then there are two linearly independent oscillatory solutions of (2) whose zeroes separate and such that any oscillatory solution of (2) is a linear combination of them. Assuming that , and are continuous on the following will be established (see [11â€“14]).

*Definition 1. *A solution of (2) will be called oscillatory iff it has an infinity of zeroes in and nonoscillatory iff it has but a finite number of zeroes in this interval. Equation (2) is said to be oscillatory iff it has at least one (nontrivial) oscillatory solution and nonoscillatory iff all of its (nontrivial) solutions are nonoscillatory.

Particularly, this paper deals with two cases based on (2) when , as follows:(i); it is clear that the characteristic roots equations are real and one of them is zero; then solutions will consist of exponential functions.(ii); one of the characteristic roots equations is zero and another two are conjugate roots and the solutions are in oscillatory form,â€‰where is constant.

#### 3. Exponentially Fitted and Trigonometrically Fitted MRKT Methods

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 and are absolutely necessary to insert at each stage and the MRKT methods is given as follows:where

for

The parameters of the MRKT methods are , and for and are assumed to be real. If and for , it is an explicit method and otherwise implicit method.

The MRKT method can be expressed in Butcher notation using the table of coefficients as follows (see Table 1).

##### 3.1. Exponentially Fitted MRKT Method

To construct the exponentially fitted Runge-Kutta type four-stage fifth-order method the functions and need to integrate exactly at each stage; therefore the following four equations are obtained:

and six more equations corresponding to , , and :where , . The relations and will be used in the derivation process. The following order conditions are obtained: and six equations corresponding to , , and :

Solving (13) to (16), we find , ,, and .

Referring to the following fifth-order four-stage method developed by Fawzi et al. [15]:

we solve (23) to (26) and let , , , , , , , , , , and be free parameters and yields.Next, we solve (17) to (22) and use the above coefficients to find , , , , , and .

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 where

This results in the new method called EFMRKT5. As , the coefficients , and of the new method EFMRKT5 reduce to the coefficients of the original method RKT5. That is to say, , , and are identical to , and of RKT5 method. Other than that, , as EFMRKT5 method will have the same error constant as RKT5 method.

##### 3.2. Trigonometrically Fitted MRKT Method

Exponentially fitted method leads to trigonometrically fitted method when replacing with and solving (8) to (9) to find , ,, and .

Consider the same coefficients of fifth-order four-stage method developed by Fawzi et al.[15] as in Section 3.1. Solving the (31) to (34) and letting , , , , , , , , , , and be free parameters will give

Next, solving (10) to (12), and using the above Fawzi coefficients to find , ,, , , and ,

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 , the coefficients , and of the new method TFMRKT5 reduce to the coefficients of the original method RKT5. That is to say, , , and are identical to , , and of RKT5 method. Other than that, , as TFMRKT5 method will have the same error constant as RKT5 method.

#### 4. Error Analysis

In this section, we will find the principal local truncation errors for , and (i.e., ) of the new exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods, respectively. We first find the Taylor series expansion of the actual solution , the first derivative of the actual solution , and the second derivative of the actual solution , the approximate solution , the first derivative of the approximate solution , and the second derivative of the approximate solution . The local truncation errors of , , and are given as

The