Fourth order Runge-Kutta method in 2d for linear boundary value problems

1991 ◽  
Vol 32 (6) ◽  
pp. 1229-1245 ◽  
Author(s):  
Zoltan Demendy
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Linear Boundary ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

scholarly journals A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

2021 ◽  
Vol 3 (1) ◽  
pp. 8-21 ◽  
Author(s):  
Mst. Sharmin Banu ◽  
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Exact Result ◽  
Sixth Order ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fourth Order Method

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

PERFORMANCE OF MODIFIED NON-LINEAR SHOOTING METHOD FOR SIMULATION OF 2ND ORDER TWO-POINT BVPS

2016 ◽  
Vol 78 (8-2) ◽  
Author(s):  
Abdul Manaf ◽  
Norma Alias ◽  
Mustafa Habib
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Shooting Method ◽  
Boundary Value ◽  
Point Boundary ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Research Article ◽  
Predictor Corrector

In this research article, numerical solution of nonlinear 2nd order two-point boundary value problems (TPBVPs) is discussed by the help of nonlinear shooting method (NLSM), and through the modified nonlinear shooting method (MNLSM). In MNLSM, fourth order Runge-Kutta method for systems is replaced by Adams Bashforth Moulton method which is a predictor-corrector scheme. Results acquired numerically through NLSM and MNLSM of TPBVPs are discussed and analyzed. Results of the tested problems obtained numerically indicate that the performance of MNLSM is rapid and provided desirable results of TPBVPs, meanwhile MNLSM required less time to implement as comparable to the NLSM for the solution of TPBVPs.

scholarly journals A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

2021 ◽  
pp. 8-21
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Exact Result ◽  
Sixth Order ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fourth Order Method

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

Elastohydrodynamic Lubrication in Rolling and Sliding Contacts

1972 ◽  
Vol 94 (4) ◽  
pp. 324-329 ◽  
Author(s):  
C. M. Rodkiewicz ◽  
V. Srinivasan
Keyword(s):  
Film Thickness ◽  
Quadrature Formula ◽  
Fourth Order ◽  
Elastohydrodynamic Lubrication ◽  
Experimental Film ◽  
Kutta Method ◽  
Sliding Contacts ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Good Agreement

A solution to the elastohydrodynamic lubrication problem for the case of two rolling cylinders, at different speeds, is presented. The lubricant is assumed compressible throughout the region. The fourth-order Runge-Kutta method for the lubricant and an improved quadrature formula for the elastic calculations are used. Pressure and film-thickness profiles are presented for different rolling velocities. There is a good agreement with the experimental film thickness data, available in literature.

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