A Finite Difference Technique for Numerical Solution of the Boundary Value Problem in ODEs of Order Three
Download PDF
Authors: P. K. PANDEY
DOI: 10.46793/KgJMat2602.245P
Abstract:
In the article, we study the approximate numerical solution to the boundary value problem in ordinary differential equations. In the present article, a third-order two-point boundary value problem is considered for discussion. We developed a second order accurate finite difference method for the approximate numerical solution of the considered problem. We took a special boundary condition; we did not find this boundary condition in the literature. We have discussed the standard convergence analysis of the proposed method. Numerical experiments on linear, nonlinear, and obstacle problems approve the order of accuracy and efficiency of the method.
Keywords:
Boundary value problem, difference method, differential equation of order three, convergence analysis, quadratic order.
References:
[1] R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986.
[2] M. Gregus, Third Order Linear Differential Equations, Series: Mathematics and its Applications 22, Springer, Netherlands, 1987.
[3] C. P. Gupta and V. Lakshmikantham, Existence and uniqueness theorems for a third-order three point boundary value problem, Nonlinear Analysis: Theory, Methods & Applications 16(11) (1991), 949–957. https://doi.org/10.1016/0362-546X(91)90099-M
[4] K. N. Murty and Y. S. Rao, A theory for existence and uniqueness of solutions to three-point boundary value problems, J. Math. Anal. Appl. 167(1) (1992), 43–48. https://doi.org/10.1016/0022-247X(92)90232-3
[5] B. Hopkins and N. Kosmatov, Third-order boundary value problems with sign-changing solutions, Nonlinear Anal. 67 (2007), 126–137.
[6] I. A. Tirmizi, On numerical solution Of third-order boundary-value problems, Communications In Applied Numerical Methods 7 (1991), 309–313. https://doi.org/10.1002/cnm.1630070409
[7] E. A. Al-Said, Numerical solutions for system of third-order boundary value problems, Int. J. Comput. Math. 78(1) (2001), 111–121. https://doi.org/10.1080/00207160108805100
[8] P. K. Pandey, An efficient numerical method for the solution of third order boundary value problem in ordinary differential equations, Int. J. Comput. Sci. Math. 9(2) (2018), 171–180. https://doi.org/10.1504/IJCSM.2018.091732
[9] M. K. Jain, S. R. K. Iyenger and Jain, R. K. Jain, Numerical Methods for Scientific and Engineering Computation (2∕e), Willey Eastern Limited, New Delhi, 1987.
[10] R. S. Varga, Matrix Iterative Analysis, Second Revised and Expanded Edition, Springer Verlag, Heidelberg, 2000.
[11] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1990.
[12] H. N. Calgar, S. H. Calgar and E. H. Twizell, The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions, Int. J. Comput. Math. 71 (1999), 373–381. https://doi.org/10.1080/00207169908804816
[13] T. S. El-Danaf, Quartic nonpolynomial spline solutions for third order two-point boundary value problem, World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2(9) (2008), 637–640.
[14] A. Ghazala, T. Muhammad, S. S. Shahid and U. R. Hamood, Solution of a linear third order multi-point boundary value problem using RKM, British Journal of Mathematics & Computer Science 3(2) (2013), 180–194. https://doi.org/10.9734/BJMCS/2013/2362
[15] M. A. Noor and A. K. Khalifa, A numerical approach for odd-order obstacle problems, Int. J. Comput. Math. 54(1) (1994), 109–116.
Login:
