scholarly journals A Neural Network Technique for the Derivation of Runge-Kutta Pairs Adjusted for Scalar Autonomous Problems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1842
Author(s):  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Yuri A. Khakhalev ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Neural Network ◽  
Differential Evolution ◽  
Initial Value Problem ◽  
Fitness Function ◽  
Initial Value ◽  
Runge Kutta

We consider the scalar autonomous initial value problem as solved by an explicit Runge-Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge-Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.

scholarly journals PENYELESAIAN NUMERIS DAN PERMODELAN MENGGUNAKAN PROGRAM VISUAL BASIC 6.0 PADA BIODEGRADASI ZAT ORGANIK DI SUNGAI

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Ummi Habibah ◽  
Salahuddin Salahuddin
Keyword(s):  
Initial Value Problem ◽  
Visual Basic ◽  
Initial Value ◽  
Runge Kutta ◽  
Visual Basic 6.0

Biodegrasi zat organik di sungai merupakan permasalahan dalam teknik kimia, yang diselesaikan dengan proses matematika termasuk persamaan differensial ordiner jenis initial value problem (jenis persamaan differensial ordiner 1 simultan 2 baris).Selanjutnya disusun sebuah program penyelesaian perhitungan dengan proses komputasi menggunakan Bahasa Program Visual Basic 6.0.Kata Kunci : Metode Runge – Kutta, Program Visual Basic 6.0, Neraca Massa,

A New Efficient Ode Solver For Solving Initial Value Problem

1998 ◽  
pp. 47-56
Author(s):  
Nazeeruddin Yaacob ◽  
Bahrom Sanugi
Keyword(s):  
Stability Analysis ◽  
Explicit Formula ◽  
Fourth Order ◽  
Initial Value Problem ◽  
Harmonic Mean ◽  
Kutta Method ◽  
Initial Value ◽  
Order Accuracy ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper we develop a new three-stage,fourth order explicit formula of Runge-Kutta type based on Arithmetic and Harmonic means.The error and stability analyses of this method indicate that the method is stable and efficient for nonstiff problems.Two examples are given which illustrate the fcurth order accuracy of the method. Keywords: Runge-Kutta method, Harmonic Mean, three-stage, fourth-order, covergence and stability analysis.

THREE POINTS BLOCK EMBEDDED DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING ODES

2021 ◽  
Vol 10 (11) ◽  
pp. 3449-3460
Author(s):  
Y.F. Rahim ◽  
M.E.H. Hafidzuddin
Keyword(s):  
Initial Value Problem ◽  
Numerical Results ◽  
Kutta Method ◽  
Initial Value ◽  
Stiff System ◽  
The Third ◽  
Runge Kutta Method ◽  
One Step ◽  
Runge Kutta

Block Embedded Diagonally Implicit Runge-Kutta (BEDIRK4(3)) me- thod derived using Butcher analysis and equi-distribution of error approach is outperformed standard Runge-Kutta (RK) formulae. BEDIRK4(3) method produces approximation to the solution of initial value problem (IVP) at a block of three points simultaneously. The standard one step RK3(2) method is used to approximate the solution at the first point of the block. At the second points the solution is approximated using RK4(2) method which is generated by the previous research. The same approach is used to obtain the solution at the third point. The code for this method was built and the algorithm developed is suitable for solving stiff system. The efficiency of the method is supported by some numerical results.

Trio-Geometric mean-based three-stage Runge–Kutta algorithm to solve initial value problem arising in autonomous systems

2018 ◽  
Vol 09 (04) ◽  
pp. 1850026 ◽  
Author(s):  
Vijeyata Chauhan ◽  
Pankaj Kumar Srivastava
Keyword(s):  
Error Estimation ◽  
Initial Value Problem ◽  
Autonomous System ◽  
Geometric Mean ◽  
Autonomous Systems ◽  
Initial Value ◽  
Science And Engineering ◽  
Independent Variable ◽  
Runge Kutta ◽  
Analysis Of Error

In the present worldwide scenario, plenty of problems arising in science and engineering which can be modeled as differential equations and out of these, autonomous system has become a subject of great interest. Several laws of physics in which time is considered as an independent variable are expressed as autonomous systems. In this paper, Runge–Kutta (RK) three-stage geometric mean method is used to solve the initial value problem arises in autonomous systems. The method is discussed in detail, convergence of method is discussed, the accuracy and efficiency of the method are proved by considering a numerical example. The result is compared to some other methods and proposed method is found to be more efficient. The detailed analysis of error estimation confirms that proposed method is more efficient as compared to other methods.

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