Elastik kiriş temeline etki eden dış kuvvetin hesaplamalı değerlendirmesi

Öz

Bu makalede, dördüncü mertebeden diferansiyel kiriş denkleminin hesaplamalı çözümü için bazı sayısal teknikler sunuyor ve kullanıyoruz. Denklem, kötü konumlu bir duruma sahip elastik bir temel üzerindeki bir kiriş sistemini tanımlar. Değerlendirme sırasında ortaya çıkan stresi azaltmak ve üstesinden gelmek için Üstel Olarak Yerleştirilmiş Sıralama Yöntemi (EFCM), Hibrit Blok Yöntemi (HBM), Homotopi Pertürbasyon Yöntemi (HPM) ve Diferansiyel Dönüşüm Yöntemi'nin (DTM) hesaplanmasına yardımcı olacak uygun algoritmalar formüle ediyoruz. Formüle edilmiş algoritmalar ayrıca sonuçların sayısal karşılaştırması için kullanılır. Sonuçlar, algoritmaların verimli olduğunu ve sayısal yöntemlerin kiriş problemlerini çözmede oldukça etkili olduğunu kanıtlamıştır.

Anahtar Kelimeler

• Kötü konumlanmış dördüncü mertebeden kiriş problemleri
• Üstel olarak yerleştirilmiş kollokasyon yöntemi (EFCM)
• Hibrit blok yöntemi (HBM)
• Homotopi pertürbasyon yöntemi (HPM)
• Diferansiyel dönüşüm yöntemi

Computational assessment of external force acting on beam elastic foundation

Abstract

In this paper, we present and employ some numerical techniques for the computational solution of fourth-order differential beam equation. The equation describes a beam system on elastic foundation with an illposed situation. We formulate suitable algorithms to aid the computation of the Exponentially Fitted Collocation Method (EFCM), Hybrid Block Method (HBM), Homotopy Perturbation Method (HPM), and Differential Transformation Method (DTM) to reduce and overcome stress involves during evaluation. The formulated algorithms are further used for numerical comparison of the results. The results show that the algorithms are efficient and numerical methods prove to be highly effective for solving beam problems.

Keywords

• Ill-Posed fourth-order beam problems
• Exponentially fitted collocation method (EFCM)
• Hybrid block method (HBM)
• Homotopy perturbation method (HPM)
• Differential transformation method (DTM)

Kaynakça

1. [1] Kelesoglu O. “The solution of fourth order boundary value problem arising out of the beam-column theory using adomian decomposition method”. Hindawi Publishing Corporation, Mathematics Problem of Engineering, 2014, 1-6, 2014.
2. [2] Saker SH, Agarwal RP, Regan DO. “Properties of solutions of fourth-order differential equations with boundary conditions”. Journal of Inequalities and Applications, 278, 1-15, 2013.
3. [3] Bougoffa L, Rach R, Wazwaz A. “On solutions of boundary value problem for fourth-order beam equations fourthorder beam equations”. Mathematical Modelling and Analysis, 21(3), 304-318, 2016.
4. [4] Ridge K. “On fourth order boundary value problems arising in beam analysis”. Differential and Integral Equations, 2(1), 91-110, 1989.
5. [5] Chen S, Ni W, and Wang C. “Positive solution of fourth order ordinary differential equation with four-point boundary conditions”. Applied Mathematics. Letter, 19, 161-168, 2006.
6. [6] Hussain K, Ismail F, and Senu N. “Solving directly special fourth-order ordinary differential equations using rungekutta type method”. Journal of Computational and Applied Mathematics, 306, 179-199, 2016.
7. [7] Talwar J, Mohanty RK. “A class of numerical methods for the solution of fourth-order ordinary differential equations in polar coordinates”. Advanced Numerical Analysis, 2012, 1-20, 2012.
8. [8] Jator S. “Numerical integrators for fourth order initial and boundary value problems”. International Journal of Pure and Applied Mathematics, 47(4), 563-576, 2008.

9. [9] Mahmoud A, Yu B, Zhang X. “Solving variable-coefficient fourth-order ODEs with polynomial nonlinearity by symmetric homotopy method”. Science Publishing Group, 7(2), 58-70, 2018.
10. [10] Boutayeb A, Abdelaziz C. “A mini-review of numerical methods for high-order problems”. International Journal of Computational Mathematics, 84(4), 563-579, 2007.
11. [11] Hetenyi M. Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering. 9th ed. Baltimore, Waverly Press, 1946.
12. [12] Hunt GW. “Reflections and symmetries in space and time”. IMA Journal of Applied Mathematics, 76(1), 2-26, 2011.
13. [13] Falade KI, Abubakar AS. “Solving Bessel differential equation of order zero using exponentially fitted collocation approximation method”. Research Journal of Mathematical and Statistical Sciences 7(2), 21-26, 2019.
14. [14] Falade KI, Baoku IG, Tiamiyu AT, Isyaku I. “On numerical computational solution of seventh order boundary value problems”. Journal of Nigerian Mathematical Society, 39(2), 255-268, 2020.
15. [15] Cole AT and Tiamiyu AT. “Hybrid block method for direct solution of general fourth order ordinary differential equations using power series”. International Conference of Mathematical Analysis, Optimization Theory and Application (ICAPTA), University of Lagos, Nigeria. 25-29 March, 2019.
16. [16] He JH. “Homotopy perturbation technique”. Computer Methods in Applied Mechanics and Engineering, 178(2), 257-262, 1999.
17. [17] He JH. “A coupling method of a homotopy technique and a perturbation technique for non-linear problems”. International Journal of Non-linear Mechanic, 35, 37-43, 2000.
18. [18] Zhou JK. Differential Transformation and its Applications for Electrical Circuits (in Chinese). Wuhan, China, Huazhong University Press, 1986.
19. [19] Chen CL and Liu YC.“Solution of two-point boundary value problems using the differential transformation method”. Journal of Optimization Theory and Application, 99, 23-35, 1998.
20. [20] Fatma A.“Applications of differential transform method to differential-algebraic equations”. Applied Mathematics and Computation, 152, 649-657, 2004.