Computation of eigenfunctions of nonlinear boundary-value -transmission problems by developing some approximate techniques
Resumo
In this study, we investigate a boundary value problem for nonlinear Sturm-Liouville equations with additional transmission conditions at one interior singular point. Known numerical methods are intended for solving initial and boundary value problems without transmission conditions. By modifying the Adomian decomposition method and the differential transform method, we present a new numerical algorithm to compute the eigenvalues and eigenfunctions of the considered boundary-value-transmission problem. Some graphic illustrations of the approximate eigenfunctions are also presented.
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Referências
Adomian, G., A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Model. 13 (7), 17-43, (1992). DOI: https://doi.org/10.1016/0895-7177(90)90125-7
Asil, V., Bulut, H., Evans, D. J., The Adomian decomposition method for the approximate solution of homogeneous differential equations with dual variable and dual coefficient, International Journal of Computer Mathematics 82, No. 8, 977–986, (2005). DOI: https://doi.org/10.1080/0020716042000261450
Attili, B. S., The Adomian decomposition method for computing eigenelements of Sturm-Liouville two point boundary value problems, Appl. Math. Comput. 168, 1306–1316, (2005). DOI: https://doi.org/10.1016/j.amc.2004.10.020
Aydemir, K., Mukhtarov, O. Sh., Class of Sturm-Liouville problems with eigenparameter dependent transmission conditions, Numerical Functional Analysis and Optimization 38:10, 1260-1275, (2017). DOI: https://doi.org/10.1080/01630563.2017.1316995
Bulut, H., Evans, D. J., On the solution of the Riccati equation by the Decomposition method, Intern. J. Computer Math. 79(1), 103-109, (2002). DOI: https://doi.org/10.1080/00207160211917
Chen, C. K., Ho, S.H., Application of Differential Transformation to Eigenvalue problems, Appl. Math. Comput. 79, 173–188, (1996). DOI: https://doi.org/10.1016/0096-3003(95)00253-7
Somali, S., Gokmen, G., Adomian Decomposition Method for Nonlinear Sturm-Liouville Problems, Surv. Math. Its Appl. 2, 11–20, (2007).
Hassan, I. A. H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, Solitons Fractals 36(1), 53-65, (2008). DOI: https://doi.org/10.1016/j.chaos.2006.06.040
Hassan, I. H. A., Erturk, V. S., Solutions of Different Types of the linear and Non-linear Higher-Order Boundary Value Problems by Differential Transformation Method, European Journal of Pure and Applied Mathematics Vol.2, No.3, 426-447, (2009).
Mukhtarov, O. Sh., Kandemir, M., Asymptotic behaviour of eigenvalues for the discontinuous boundary-value problem with functional-transmission conditions, Acta Mathematica Scientia 22B(3), 335-345, (2002). DOI: https://doi.org/10.1016/S0252-9602(17)30303-X
Mukhtarov, O. Sh., Aydemir, K., Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scientia 35(3), 639-649, (2015). DOI: https://doi.org/10.1016/S0252-9602(15)30010-2
Mukhtarov, O. Sh., Olgar, H., Aydemir, K., Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat 29:7 , 1671-1680, (2015). DOI: https://doi.org/10.2298/FIL1507671M
Ol˘gar, H., Mukhtarov, O. Sh., Aydemir, K., Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem, Filomat 32:3, 911-920, (2018). DOI: https://doi.org/10.2298/FIL1803911O
Sheikholeslami, M., Ganji, D. D., Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM, Computer Methods in Applied Mechanics and Engineering 283, 651-663, (2015). DOI: https://doi.org/10.1016/j.cma.2014.09.038
Wazwaz, A. M., The Decomposition Method for Approximate Solution of the Goursat Problem, Applied Mathematics and Computation 69, 299-311, (1995). DOI: https://doi.org/10.1016/0096-3003(94)00137-S
Wazwaz, A. M., A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method, Journal of Computational and Applied Mathematics 235, 1914–1924, (2011). DOI: https://doi.org/10.1016/j.cam.2010.09.007
Wazwaz, A. M., Computing Eigenelements of Sturm-Liouville Problems of Fractional order via Fractional Differential Transform Method, Mathematical and Computational Applications 16, No.3, 712-720, (2011). DOI: https://doi.org/10.3390/mca16030712
Yucel, M., Mukhtarov, O. Sh., A New Treatment of the Decomposition Method for Nonclassical Boundary Value Problems, Journal of Advanced Physics 7.2, 161-166, (2018). DOI: https://doi.org/10.1166/jap.2018.1412
Yucel, M., Mukhtarov, O. Sh., Application of differential transform method and Adomian decomposition method for solving of one nonlinear boundary-value transmission problem, AIP Conference Proceedings 2183, 090011, (2019). DOI: https://doi.org/10.1063/1.5136211
Zhou, J. K., Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, (1986) (in Chinese).
Titchmarsh E.C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, Second edn. Oxford Univ. Press. London (1962).
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