SOLVING TROESCH’S PROBLEM BY USING MODIFIED NONLINEAR SHOOTING METHOD
DOI:
https://doi.org/10.11113/jt.v78.8295Keywords:
BVPs, ODEs, predictor-corrector scheme, shooting method, Troesch’s problemAbstract
In this research article, the non-linear shooting method is modified (MNLSM) and is considered to simulate Troesch’s sensitive problem (TSP) numerically. TSP is a 2nd order non-linear BVP with Dirichlet boundary conditions. In MNLSM, classical 4th order Runge-Kutta method is replaced by Adams-Bashforth-Moulton method, both for systems of ODEs. MNLSM showed to be efficient and is easy for implementation. Numerical results are given to show the performance of MNLSM, compared to the exact solution and to the results by He’s polynomials. Also, discussion of results and the comparison with other applied techniques from the literature are given for TSP. Â
References
Alias, N., M.R. Islam, T. Ahmad, and M.A Razzaque. 2013. Sequential Analysis of Drug Encapsulated Nanoparticle Transport and Drug Release Using Multicore Shared-memory Environment. Fourth International Conference and Workshops on Basic and Applied Sciences (4th ICOWOBAS) and Regional Annual Fundamental Science Symposium 2013 (11th RAFSS). Johor, Malaysia. 3 September 2013. 1-6.
Alias, N., M.R. Islam, and N.S. Rosly. 2009. A Dynamic PDE Solver for Breasts’ Cancerous Cell Visualization on Distributed Parallel Computing Systems. 8th International Conference on Advances in Computer Science and Engineering (ACSE 2009). Phuket, Thailand. 16-18, March 2009.
Alias, N., H.F.S. Saipol, and A.C.A. Ghani. 2014. Chronology of DIC Technique Based on the Fundamental Mathematical Modeling and Dehydration Impact. Journal of Food Science and Technology. 51(12): 3647-3657.
Alias, N., R. Shahril, M.R. Islam, N. Satam, and R. Darwis. 2008. 3D Parallel Algorithm Parabolic Equation for Simulation of the Laser Glass Cutting Using Parallel Computing Platform. The Pacific Rim Applications and Grid Middleware Assembly (PRAGMA15). Penang, Malaysia, Oct 21-24, 2008.
Troesch, B. 1976. A Simple Approach to a Sensitive Two-Point Boundary Value Problem. Journal of Computational Physics. 21(3): 279-290.
Hashemia, M., and S. Abbasbandyb. 2014. A Geometric Approach for Solving Troesch’s Problem. Bulletin of the Malaysian Mathematical Sciences Society.
Roberts, S., and J. Shipman. 1972. Solution of Troesch's Two-Point Boundary Value Problem By A Combination Of Techniques. Journal of Computational Physics. 10(2): 232-241.
Miele, A., A. Aggarwal, and J. Tietze. 1974. Solution Of Two-Point Boundary-Value Problems with Jacobian Matrix Characterized by Large Positive Eigenvalues. Journal of Computational Physics. 15(2): 117-133.
Chiou, J., and T. Y. Na. 1975. On the Solution of Troesch's Nonlinear Two-Point Boundary Value Problem using an Initial Value Method. Journal of Computational Physics. 19(3): 311-316.
Scott, M. R. 1974. Conversion of Boundary-Value Problems into Stable Initial-Value Problems via Several Invariant Imbedding Algorithms. Sandia Labs. Albuquerque, N. Mex. (USA).
Deeba, E., S. Khuri, and S. Xie. 2000. An Algorithm for Solving Boundary Value Problems. Journal of Computational Physics. 159(2): 125-138.
Khuri, S. 2003. A Numerical Algorithm for Solving Troesch's Problem. International Journal of Computer Mathematics. 80(4): 493-498.
Momani, S., S. Abuasad, and Z. Odibat. 2006. Variational Iteration Method for Solving Nonlinear Boundary Value Problems. Applied Mathematics and Computation. 183(2): 1351-1358.
Chang, S.H. 2010. A Variational Iteration Method for Solving Troesch’s Problem. Journal of Computational and Applied Mathematics. 234(10): 3043-3047.
Snyman, J. 1979. Continuous and Discontinuous Numerical Solutions to The Troesch Problem. Journal of Computational and Applied Mathematics. 5(3): 171-175.
Khuri, S., and A. Sayfy. 2011. Troesch’s Problem: A B-Spline Collocation Approach. Mathematical and Computer Modelling. 54(9): 1907-1918.
Zarebnia, M., and M. Sajjadian. 2012. The Sinc–Galerkin Method for Solving Troesch’s Problem. Mathematical and Computer Modelling. 56(9): 218-228.
Mohyud-Din, S. T. 2011. Solution of Troesch’s Problem using He’s Polynomials. Rev. Un. Mat. 52: 1.
Feng, X., L. Mei, and G. He. 2007. An Efficient Algorithm for Solving Troesch’s Problem. Applied Mathematics and Computation. 189(1): 500-507.
Chang, S.H., and I.L. Chang. 2008. A New Algorithm for Calculating One-Dimensional Differential Transform of Nonlinear Functions. Applied Mathematics and Computation. 195(2): 799-808.
El-Gamel, M., and M. Sameeh. 2013. A Chebychev Collocation Method for Solving Troesch’s Problem. Int. J. Math. Comput. Appl. Res. 3: 23-32.
El-Gamel, M. 2013. Numerical Solution of Troesch’s Problem by Sinc-Collocation Method. Applied Mathematics. 4(04): 707.
Manaf, A., M. Habib, and M. Ahmad. 2015. Review of Numerical Schemes for Two Point Second Order Non-Linear Boundary Value Problems. Proceedings of the Pakistan Academy of Sciences. 52 (2): 151-158.
Alias, N., and M. Islam, M. 2010. A Review of The Parallel Algorithms for Solving Multidimensional PDE Problems. Journal of Applied Sciences. 10(19): 2187-2197
Alias, N., H.F.S. Saipol, and .A.C.A. Ghani. 2012. Numerical method for Solving Multipoints Elliptic-Parabolic Equation for Dehydration Process. World Applied Science Journal. 21:130-135.
Alias, N., M.N. Mustaffa, H.F.S. Saipol, and A.C.A Ghani. 2014. High performance large sparse PDEs with parabolic and elliptic types using AGE method
Downloads
Published
Issue
Section
License
Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.
