Runge Kutta method for third order differential equation with generalized Cauchy boundary conditions
DOI:
https://doi.org/10.38152/bjtv7n4-020Keywords:
3rd ODE, Runge Kutta Method, initials conditions, Dirichlet BC, Neumann BC, Cauchy BC, newton algorithmAbstract
The paper focuses in developing a numerical method allowing to solving a third order nonlinear differential equation (DE) with generalized Cauchy boundary conditions (BC). The problem consists in transforming the third order DE, to a system of twelve nonlinear first order differential equation (FODE) with three initials conditions (IC). The resolution of this system of FODE is made by the adaptation of the Runge Kutta method of order 4. The determination of the IC is made by the resolution of an algebraic system of three nonlinear equations, of which the resolution is made simultaneously by the Newton's algorithm. For each Newton’s method iteration, a system of nonlinear algebraic equations is obtained whose solution is made by the method of Gauss. The function y(x) and these successive derivatives are obtained point by point in the interval [a1, a2]. The initial values of Newton's algorithm can be taken by the physical model since DE can model a precise physical phenomenon. The application is made by the consideration of an arbitrary example of 3rd DE by modifying only their BC. The validation is made by considering the 3th Falkner-Skan DE of the boundary layer.
References
Ahmadinia L. M., (2006) “Numerical solution of third-order boundary value problems,” Iranian Journal of Science & Technology, Transaction A, Vol. 30, N° A3, PP. 291-295.
Anderson J. D. Jr., (1982) “Modern compressible flow with historical perspective,” 2nd Edition, McGraw-Hill Book Company, New York, USA.
Bashir A., Khan R. A. and Eloe P. W., (2002) “Generalized quasi linearization method for a second order three-point boundary-value problem with nonlinear boundary conditions,” Electronic Journal of Differential Equations, Vol. 2000, N° 90, PP. 1-12
Bashir A., Khan R. A. and Sivasundaram S., (2003) “Generalized quasi linearization method for nonlinear functional differential equations,” Journal Applied Mathematical Stochastic Analysis, Vol. 16, N° 01, PP. 33-43. doi:10.1155/S1048953303000030
Butcher J. C., (1996) "A history of RK methods," Applied Numerical Mathematics, Vol. 20, N° 03, PP. 247-260. https://doi.org/10.1016/0168-9274(95)00108-5
Comolet R., (1979) “Mécanique Expérimentale des Fluides. Dynamique des fluides Réels,” Tomes 2, 3ème Edition, Masson.
Démidovitch B. and Maron I., (1987) “Eléments de calcul numérique,” Editions MIR, Moscow, Russia.
Deuflhard P., (2004) “Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms,” Springer Series in Computational Mathematics, Vol. 35, Springer, Berlin.
Duarte L., Duarte S., Mota L. D. and Skea J., (2001) “Solving second-order ordinary differential equations by extending the prelle-singer method,” Journal of Physics A: Mathematical and General, Vol. 34, N° 14
Iserles A., (1996) “A First Course in the Numerical Analysis of Differential Equations,” Cambridge University Press.
Izmailov F. and Solodov M. V., (2014) “Newton-Type Methods for Optimization and Variational Problems,” Springer Series in Operations Research and Financial Engineering, Springer.
Lambert J. D., (1991) “Numerical Methods for Ordinary Differential Systems,” John Wiley & Sons, Chichester.
Luther H. A., (1966) “Further explicit fifth-order Runge-Kutta formulas,” SIAM Review, Vol. 8, PP. 374-380.
Mukhtar N. Z, Majid Z. A., Ismail F. and Suleiman M., (2012) “Numerical solution for solving second order ordinary differential equations using block method,” International Journal of Modern Physics: Conference Series, Vol. 9, PP. 560–565
Ortega J. M. and Rheinboldt W. C., (2000) “Iterative Solution of Nonlinear Equations in Several Variables,” Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.
Polyanin A. D. and Manzhirov A.V., (2006) “Handbook of Mathematics for Engineers and Scientists,” Chapman & Hall CRC Press Boca Raton.
Polyanin A. D., Zaitsev V. F. and Moussiaux A., (2002) “Handbook of First Order Partial Differential Equations,” Taylor & Francis, London, UK.
Polyanin A. D. and Zaitsev V. F., (2003) “Handbook of Exact Solutions for Ordinary Differential Equations,” 2nd Edition, Chapman & Hall CRC Press Boca Raton.
Raltson A. and Rabinowitz A., (1985) “A First Course in Numerical Analysis,” McGraw Hill Book Company.
Stoer J. and Burlisch B., (2002) “Introduction to Numerical Analysis,” text in Applied Mathematics, 12, Springer, 3rd Edition.
Strauss W., (2008) “Partial Differential Equations. An Introduction,” John Wiley & Sons, Ltd, 3ème Edition, Masson.
Verner J. H., (2014) ‘‘Explicit Runge-Kutta pairs with lower stage order, Numerical Algorithms,’’ Vol. 65, N° 03, PP. 555-577.
Yahiaoui T., (2024) “Numerical method for solving second order nonlinear differential equation with arbitrary boundary conditions,” Brazilian Journal of Technology, Accepted for publication, 2024
Zwillinger D., (1997) “Handbook of Differential Equations,” 3rd Edition, Academic Press Boston
