Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4) = f(x, u, u′, u″)

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.

Implicit Two-Point Block Method for Solving Fourth-Order Initial Value Problem Directly with Application

“…e system in Problem 6 and Problem 7 has integrated in the interval t ∈ [0, 10] and t ∈ [0, 3], respectively. A comparison of numerical outcomes is made with the solution of the same group of systems when they are solved by using RKTF5 [13] and RKF5 [19] methods. We have solved these systems to consider the maximum errors and the number of function evaluations at different step 0.00 + 00 0.000000 + 00 0.8 0.00 + 00 0.000000 + 00 0.9 0.00 + 00 0.000000 + 00 1.0 0.00 + 00 1.719064 − 19 Overall, the proposed method I2PBDO6 of order 6 has shown remarkable convergence since the approximate solution is very close to the exact solution.…”

“…e classical way of solving them is by reducing the equation into the system of firstorder ODEs, but this process is too rigorous compared with the direct methods [4,5]. In addition, many researchers have presented direct methods to avoid the reduction effort [6][7][8][9][10][11][12][13][14][15]. To enhance the efficacy of numerical methods, the block method is introduced with the idea of producing simultaneously r-point of the approximate solutions at one time step.…”

Derivation of Embedded Diagonally Implicit Methods for Directly Solving Fourth-order ODEs

“…However, it would be more efficient if numerical methods could be used to solve the problem accurately and quickly. In [4][5][6][7][8][9][10][11][12] contain such works. Multistep strategies for solving ODEs require initial values.…”

Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

“…Considering the impact, a new perspective to compare the potentials of both methods should be investigated as well as existing comparative studies. First of all, it is well known that the highest order of an A-stable multi-step method is two, so lots of research [12][13][14][15][16][17][18][19][20][21][22][23][24] developing higher order methods have focused on either multi-step methods satisfying some less restrictive stability condition or multi-stage methods which combine A-stability with high-order accuracy [2,[25][26][27][28][29]. In addition, multi-stage methods such as Runge-Kutta (RK) type methods do not require any additional memory for function values at previous steps since it does not use any previously computed values [30][31][32].…”