Abstract
The derivation of 5th order diagonal implicit type Runge Kutta methods (DITRKM5) for solving 3rd 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.
Highlights
Third-order ordinary differential equations (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]
The implicit RK techniques play an important role for denomination the physical and mathematical problems, like a differential algebraic equation
The parameters of diagonal implicit RK type (DITRK) methods are presumed as ci, aij, di, bi, gi where i, j = 1, 2, 3 ... , s are real numbers and m is referred to stage digit for the approach
Summary
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 y′′′(x) = f(x, y(x)). 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. 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.
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