K-Step Block Hybrid Nyström-Type Method for the solution of Bratu problem with Impedance Boundary Condition.
Keywords:
Bratu Problem; Hybrid (Bhaskara) points; Block Nyström-type method; Nonlinear ODEs; Power Series Polynomials, Bhaskara Cosine-Approximation formula; Impedance boundary conditions
Abstract
Using a carefully selected step and off-step points produced from the Bhaskara Cosine approximation formula and Gauss-Lobato grid points with constant step size, this numerical study presents the Block Hybrid Nyström-type method of order seven for solving the second order Bratu problem with impedance boundary conditions at two-point concurrently. Without the use of a combination of predictor-corrector mode and or the shooting approach, the solution is achieved immediately without having to convert it to a system of first order ordinary differential equations. The diffusion of heat produced by the application that imposed the impedance conditions is the focus of a numerically tested problem. The present findings showed that the proposed method produce an efficient performance in terms of accuracy, and error when compared with the existing methods. The convergence and stability properties of the BHNTM are discussed.
References
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24. Jator, S. N., & Manathunga, V. (2018). Block Nyström type integrator for Bratu’ s equation. Journal of Computational and Applied Mathematics. https://doi.org/10.1016/j.cam.2017.06.025
2. Torkaman, S., Loghmani, G. B., Heydari, M., & Rashidi, M. M. (2019). Novel numerical solutions of nonlinear heat transfer problems using the linear barycentric rational interpolation. Heat Transfer—Asian Research, 48(4), 1318-1344.
3. Sunday, J., Shokri, A., Kwanamu, J. A., & Nonlaopon, K. (2022). Numerical integration of stiff differential systems using non-fixed step-size strategy. Symmetry, 14(8), 1575.
4. Mishra, S. K., Rajković, P., Samei, M. E., Chakraborty, S. K., Ram, B., & Kaabar, M. K. (2021). A q-gradient descent algorithm with quasi-fejér convergence for unconstrained optimization problems. Fractal and Fractional, 5(3), 110.
5. Ogunniran, M. O., Haruna, Y., & Adeniyi, R. B. Nigerian Journal of Mathematics and Applications Volume A, 28, (2019), 1− 17. Printed by Unilorin press c Nig. J. Math. Appl. http://www. njmaman. com.
6. Singh, G., Garg, A., Kanwar, V., & Ramos, H. (2019). An efficient optimized adaptive step-size hybrid block method for integrating differential systems. Applied Mathematics and Computation, 362, 124567.
7. Sunday, J., Odekunle, M. R., James, A. A., & Adesanya, A. O. (2014). Numerical solution of stiff and oscillatory differential equations using a block integrator. British Journal of Mathematics & Computer Science, 4(17), 2471-2481.
8. Ogunniran, M. O., Olaleye, G. C., Taiwo, O. A., Shokri, A., & Nonlaopon, K. (2023). Generalization of a class of uniformly optimized k-step hybrid block method for solving two-point boundary value problems. Results in Physics, 44, 106147.
9. Ajinuhi, J. O., Mohammed, U., Enagi, A. I., & Razaq, J. O. (2023). A Novel Seventh-Order Implicit Block Hybrid Nyström-Type Method for Second-Order Boundary Value Problems. International Journal of Research and Scientific Innovation, 10(11), 26-44.
10. Noor, M. A., & Noor, K. I. (2021). Some new classes of strongly generalized preinvex functions. TWMS J. Pure Appl. Math, 12(2), 181-192.
11. Nasir, N.M., AbdulMajid, Z., Ismail, F., & Bachok, N. (2018). Diagonal Block Method for Solving Two-Point Boundary Value Problems with Robin Boundary Conditions. Mathematical Problems in Engineering, 2018, 1–12. http://doi:10.1155/2018/2056834
12. T.T.Akano & O.A.Fakinlede (2015). Numerical computation of Sturm-liouville problem with Robin boundary condition, in Proceedings of the World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 9(11), 684–688.
13. Duan, J.-S., Rach, R., Wazwaz, A.-M., Chaolu, T., & Wang, Z. (2013). A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions. Journal of Applied Mathematical Modelling, 37(20-21), 8687–8708. http://doi:10.1016/j.apm.2013.02.002
14. R. Rach, J.-S. Duan, and A.-M. Wazwaz (2016), Solution of higher order, multipoint, nonlinear boundary value problems with high-order robin-type boundary conditions by the adomian decomposition method. Journal of Applied Mathematics & Information Sciences, 10(4), 1231–1242.
15. S. K. Bhatta & K. S. Sastri, (1993). Symmetric spline procedures for boundary value problems with mixed boundary conditions. Journal of Computational and Applied Mathematics, 45(3), 237–250.
16. Lang, F.G., & Xu, X.P. (2012). Quintic B-spline collocation method for second order mixed boundary value problem. Journal of Computer Physics Communications, 183(4), 913–921.
17. S. O. Fatunla, (1995). A class of block methods for second order IVPs, International Journal of Computer Mathematics, 55 (1-2), 119–133.
18. Phang,Z.A.Majid, & M.Suleiman. (2012) On the solution of two-point boundary value problems with two-point direct method. International Journal of Modern Physics: Conference Series, 9(), 566–573.
19. Z.A.Majid,M.Mughti Hasni, & N. Senu, (2013). Solving second order linear Dirichlet and Neumann boundary value problems by block method. IAENG International Journal of Applied Mathematics, 43(2), 71–76.
20. Z. Omar and O. Adeyeye (2016). Solving two-point second order boundary value problems using two-step block method with starting and non-starting values. International Journal of Applied Engineering Research, 11(4), 2407–2410
21. Z. A. Majid, (2004). Parallel block methods for solving ordinary differential equations [Ph.D. thesis], Universiti Putra Malaysia.
22. I.S.M.Zawawi, Z.B.Ibrahim, F.Ismail ,& Z.A.Majid (2012). Diagonally Implicit Block Backward Differentiations Formulas for Solving Ordinary Differential Equations, International Journal of Mathematics and Mathematical Sciences, vol.2012, Article ID 767328, 8 pages.
23. N. Zainuddin, Z. B. Ibrahim, and K. I. Othman (2014), Diagonally implicit block backward differentiation formula for solving linear second order ordinary differential equations,” in Proceedings of the 3rd international conference on fundamental and applied sciences (icfas 2014): innovative research in applied sciences for a sustainable future,.69–75, KualaLumpur,Malaysia.
24. Jator, S. N., & Manathunga, V. (2018). Block Nyström type integrator for Bratu’ s equation. Journal of Computational and Applied Mathematics. https://doi.org/10.1016/j.cam.2017.06.025
Published
2024-01-16
How to Cite
Theophilus, G. O., Olusegun, A. J., & Umaru, M. (2024). K-Step Block Hybrid Nyström-Type Method for the solution of Bratu problem with Impedance Boundary Condition. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 5(1), 34-46. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/595
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