Approximate solutions of the Fourth-Order Eigenvalue Problem

Abstract

In this paper, the differential transformation (DTM) and the Adomian decomposition (ADM) methods are proposed for solving fourth order eigenvalue problem. This fourth order eigenvalue problem has nonstrongly regular boundary conditions. This the fourth order problem has been examined for p(t) = t, B = 0, a = 0,01 where p(t) ≠ 0 is a complex valued and a ≠ 0 The differential transformation and the Adomian decomposition methods are briefly described. An approximate solution is obtained by performing seven iterations with the Adomian decomposition method. The same number of iterations have been made in the differential transformation method. The approximation results obtained by both methods have been compared with each other. These data have been presented in table. The ADM and the DTM approximation solutions have been shown by plotting in Figure 1. Here, the approaches obtained by using the two methods are found to be in high agreement. Consequently, highly accurate approximate solutions of fourth order eigenvalue problem are obtained. Such good results also revealed that the Adomian decomposition and the differential transformation methods are fast, economical and motivating. The exact solution of the fourth order eigenvalue problem for nonstrongly regular can not be found in the literature. Therefore, this study will give an important idea to determine approximate solution behavior of this fourth order problem.

Keywords

• Fourth order eigen value problem
• approximate solutions
• not strongly regular boundary conditions
• Adomian decomposition method
• differential transform method

References

1. Reference1 Kaya, U. (2020). Basis properties of root functions of a regular fourth order boundary value problem. Hacet. J. Math. Stat., 49(1), 338-351.
2. Reference2 Chanane, B. (2010). Accurate solutions of fourth order sturm-liouville problems. Journal of Computational and Applied Mathematics, 234(2010), 3064-3071.
3. Reference3 Mukhtarov, O. Sh., Yucel, M., & Aydemir, K. (2020). Treatment a new approximation method and ıts justification for Sturm–Liouville problems. Hindawi Complexity, 2020, Article ID 8019460, 8 pages.
4. Reference4 Syam, M.I., & Siyyam, H.I. (2009). An efficient technique for finding the eigenvalues of fourth-order Sturm–Liouville problems. Chaos, Solitons and Fractals, 39, 659–665.
5. Reference5 Alquran, M.T., & Al-Khaled, K. (2010). Approximations of Sturm-Liouville eigenvalues using sinc-Galerkin and differential transform methods. Applications and Applied Mathematics: An International Journal, 5, 128 – 147.
6. Reference6 Attili, B.S., & Lesnic, D. (2006). An efficient method for computing eigenelements of Sturm-Liouville fourth-order boundary value problems. Applied Mathematics and Computation, 182, 1247–1254.
7. Reference7 Alalyani, A. (2019). Eigenvalue computation of regular 4th order Sturm-Liouville Problems. Applied Mathematics, 10, 784-803.
8. Reference8 Biazar, J., Dehghan, M., & Houlari, T. (2020). An efficient method to approximate eigenvalues and eigenfunctions of high order Sturm-Liouville problems. Computational Methods for Differential Equations, 8, 389-400.

9. Reference9 Gao, W., Ismael, H.F., Husien, AM., Bulut, H., & Baskonus, HM. (2020). Optical soliton solutions of the cubic-quartic nonlinear Schrödinger and Resonant Nonlinear Schrödinger equation with the parabolic Law. Applied Sciences, 10(1), 219.
10. Reference10 Baskonus, H.M., Sulaiman, T.A., & Bulut, H. (2018). Dark, bright and other optical solitons to the decoupled nonlinear Schrödinger equation arising in dual-core optical fibers. Opt Quant Electron, 50, 165.
11. Reference11 Adomian, G., & Rach, R. (1993). Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition. Journal of Mathematical Analysis and Applications, 174, 118-137.
12. Reference12 Zhou, J.K. (1986). Differential transform and ıts application for electrical circuits. Huazhong University Press: Wuhan. Reference13 Ayaz, F. (2004). Applications of differential transform method to differential-algebraic equations. Applied Mathematics and Computation, 152, 649-657.
13. Reference14 Abdel-Halim Hassan, I.H. (2002). On solving some eigenvalue problems by using a differential transformation. Applied Mathematics and Computation, 127, 1-22.
14. Reference15 Li, W., Pang, Y. (2020). Application of Adomian decomposition method to nonlinear systems. Adv. Differ. Equ., 67. Reference16 Chakraverty, S., Mahato, N.R., Karunakar, P., & Rao, T.D. (2019). Advanced numerical and semi‐analytical methods for differential equations. Wiley Online Library, chapter 11, 2019.
15. Reference17 Adebısı, A.F., Uwaheren, O.A., Aboların, O.E., Rajı, M.T., Adedejı , & J.A., Peter, O.J., (2021). Solution of typhoid fever model by Adomian decomposition method. J. Math. Comput. Sci., 11(2), 1242-1255.
16. Reference18 Çakır, M., & Arslan, D. (2015). The Adomian decomposition method and the differential transform method for numerical solution of multi-pantograph delay differential equations. Applied Mathematics, 6, 1332-1343.
17. Reference19 Arslan, D. (2019). A novel hybrid method for singularly perturbed delay differential equations. Gazi University Journal of Science, 32, 217-223.
18. Reference20 Arslan, D. (2018). Differential transform method for singularly perturbed singular differential equations.