The Implementations Special Third-Order Ordinary Differential Equations ( ODE ) for 5 th-order 3 rd-stage Diagonally Implicit Type Runge-Kutta Method ( DITRKM )

The derivation of 5 order diagonal implicit type Runge Kutta methods (DITRKM5) for solving 3 special order ordinary differential equations (ODEs) is introduced in the present study. The DITRKM5 techniques are the name of the approach. This approach has three equivalent non-zero diagonal elements. To investigate the current study, a variety of tests for five various initial value problems (IVPs) with different step sizes h were implemented. Then, a comparison was made with the methods indicated in the other literature of the implicit RK techniques. The numerical techniques are elucidated as the qualification regarding the efficiency and number of function evaluations compared with another literature of the implicit RK approaches from the result of the computations. In addition, the stability polynomial for DITRK method is derived and analyzed.


Introduction
Third-order ODEs are used in neural network engineering and applied sciences, the dynamics of fluid flow, the ship's motion, and electric circuits, among other fields [1][2][3][4][5][6]. Consider the numerical method for solving the special "initial value problems" (IVPs) for order three as the following form The implicit methods are important because they can reach high orders of accuracy at the equivalent number of stages, which can be represented as an advantage that leads to the more accurate than the explicit approaches. This manufactures it easier to exist the solution to the difficulties of the problems.
So, the implicit RK techniques play an important role for denomination the physical and mathematical problems, like a differential algebraic equation.
In addition, diagonal implicit RK (DIRK) techniques are also pointed to as semi-implicit approaches or semi explicit RK techniques since they obtained at minimum one value does not zero for the lower of the triangular diagonal matrices. Therefore, to solve Eq. (2.1), two general strategies can be employed. The elementary way is to transfer the Eq. (2.1) into a problem with first-order then apply any pattern of the RK approach to it.
As a result, numerous implicit RK approaches, such as Ismail et al. [7] and others, have been developed. The second option is to use the RK Type method to directly solve Eq. (2.1). For second-order systems, several scholars provided an efficient implicit RK technique (see [8][9][10][11][12][13]). Ghawadri et al. [14], constructed a solution to the ill-posed issue for a beam with an elastically base using special fourth-order ODEs. Moreover, [15][16][17] developed a solution of the special 3 rd order for the ODEs directly by RK technique. Finally, Senu [18] and Fawzi et al. [19] constructed the embedded the RK technique to solve 3 rd order for the ODEs.
A significant objective for current research is to show how particular third-order ODEs are solving via the DIRECT method. Additionally, while solving eq. (2.1) numerically, the algebraic order of the technique used must be taken into account, as this is the most important factor in achieving high accuracy. Section 2.2 demonstrates the basic idea of construction and derivation of the DITRK system for addressing Initial Value Problems (IVPs). The DITRK technique's order criteria are outlined in Section 2.3. Section 2.4 describes the 3 rd stage 5 th order (DITRKM5) methods. In Section 2.5, the analyses of the stability polynomial for the DITRK method are presented. In Section 2.6, mentions the DITRK approach with five IVPs. In Section 2.7, the validation of the current approach compared with those in the other literatures of the implicit RK techniques.

The Methodology of DITRK Techniques
For solving IVPs in eq. (2.1), the prevalent formula of the implicit RK approach for the stage can be expressed as follows: The parameters of diagonal implicit RK type (DITRK) methods are presumed as , , , , where , = 1, 2, 3 … , are real numbers and is referred to stage digit for the approach. This scheme is known as diagonal implicit when ≠ 0 for > . The last denomination includes the single DITRK techniques that indicate that the lower the triangular diagonal matric of have same values with ≠ 0 where = at the diagonal.
The DITRK approach proposed from the work of Butcher, as illustrated in Table 2.1 [20].

Order Conditions of the DITRK Technique
According to Mechee et al. [17], the orders of algebraic criteria for RKD approached over order 6 are as follow: Order conditions of :

Formation of the 3rd stage 5 th order (DITRKM5) Method
We implement a diagonal implicit type Runge-Kutta approach using order conditions derivations as demonstrated in section 2.3, which is developed according to Mechee work [17]. For the order DITRK approach, the local truncation error is defined as follows: for order five: Finally, coefficients of the DITRKM method for 5-order 3 stages indicated by DITRKM5 can be read as shown in

The Stability Polynomial of DITRK Method
In order to study the stability polynomial of the DITRK method, the following equation is suggested :

Test of Problems
The approaches that demonstrated in section 2.3 tested with 5 various problems in this part. The numerical results of the suggested approaches compared with those of other RK techniques at equivalent order which are already available. The numerical experiments were conducted using the following methods: (1) DITRKM5: 3 rd stage 5 th order DITRK approach computed in the present work.
(5) Radau IIA: 3 rd stage 5 th order RK approach noted in [21].  Figure (2.1) shows the efficiency of the DITRKM methods created by charting of decimal logarithm for the highest "global error" versus logarithm of function estimate. When compared the current study with another implicit RK approach for equivalent order, the DITRKM5 method requires fewer "function evaluations". The digit of equations increased three times with the problems turned to a system of 1 st order ODEs. In the comparison, the existing implicit the RK approach with the equivalent order, the "global error" and digit of "function estimate" contain the smallest maximum for the DITRKM5 method at each iteration, as shown in Figure (2.1) that obtained from Table (2.1). As shown in Figure (2.1), the fifthorder three stage results DITRK method (DITRKM5) produces more accurate findings than the other results in the literature (Radau I, Radau IA, Radau II, and Radau IIA). In this work, the logarithm of "maximum global error" is known as a logarithm function for "function evaluation" with different step size ℎ = 0.1, 0.05, 0.025, 0.0125,0.00625 for five test problems.

Conclusion
The test of the fifth-order three-stage DITRKM5 methods for the integration of ODEs obtained by testing the minimized error norm and studied the digit of the evaluations of function that described in this paper. The digit of "function evaluations" and the maximum error of the supposed technique is lower than those in implicit of the RK approaches, as indicated by numerical results in all Figure (2.1) and Table (2.1), and the introduced