Abstract
The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u ′ , u ′ ′ ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.
Highlights
Fourth-order ODEs can be found in several areas of neural network engineering and applied sciences [1], fluid dynamics [2], ship dynamics [3,4,5], electric circuits [6] and beam theory [7,8]
We are focusing on the algebraic theory of order conditions of RKTF method in the form of u(4) = f ( x, u, u0, u00 ) to solve ODEs of order four directly
[32,33] introduced the idea and concept of B-series theory that are dependent on algebraic order conditions
Summary
Fourth-order ODEs can be found in several areas of neural network engineering and applied sciences [1], fluid dynamics [2], ship dynamics [3,4,5], electric circuits [6] and beam theory [7,8]. We have presented the explicit formulas of RKT to solve fourth-order ODEs directly of the structure u(4) = f ( x, u, u0 , u00 ). Many researchers have used classical approaches to solve higher-order ODEs through converting them to first order system of ODEs and using appropriate numerical approach to this arrangement (see [9,10,11]). Jain et al [21] developed finite difference approach to solve ODEs of order four, all the methods discussed above are multi-step in nature. Senu et al [24] developed embedded explicit RKT method to directly solve special ODEs of order three. The main purpose of this study is using quad-colored trees theory to construct one step explicit RKT approach to solve fourth-order.
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