Wolfram Mathematica application to determination of the number of solutions for certain nonlinear boundary value problems

References

• Beléndez, A., Arribas, E., Beléndez, T., Pascual, C., Gimeno, E., & Álvarez, M. L. (2017). Closed-form exact solutions for the unforced quintic nonlinear oscillator. In Advances in mathematical physics. Hindawi.
   Google Scholar
• Beléndez, A., Beléndez, T., Martinez, F. J., Pascual, C., Alvarez, M. L., & Arribas, E. (2016). Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities. Nonlinear Dynamics, 86(3), 1687–24. https://doi.org/https://doi.org/10.1007/s11071-016-2986-8
   Web of Science ®Google Scholar
• Beléndez, A., Bernabeu, G., Francés, J., Méndez, D. I., & Marini, S. (2010). An accurate closed-form approximate solution for the quintic Duffing oscillator equation. Mathematical and Computer Modelling, 52(3–4), 637–641. https://doi.org/https://doi.org/10.1016/j.mcm.2010.04.010
   Google Scholar
• Chicone, C. (1987).The monotonicity of the period function for planar Hamiltonian vector fields. Journal of Differential Equations, 69(3), 310321. MR903390;. https://doi.org/https://doi.org/10.1016/0022-0396(87)90122-7
   Google Scholar
• Chicone, C. (1988). Geometric Methods for Two-Point Nonlinear Boundary Value Problems. Journal of Differential Equations, 72(2), 360–407. https://doi.org/https://doi.org/10.1016/0022-0396(88)90160-X
   Web of Science ®Google Scholar
• Elías-Zúñiga, A. (2013). Exact solution of the cubic-quintic Duffing oscillator. Applied Mathematical Modelling, 37(4), 2574–2579. https://doi.org/https://doi.org/10.1016/j.apm.2012.04.005
   Web of Science ®Google Scholar
• Ellero, E., & Zanolin, F. (2013). Homoclinic and heteroclinic solutions for a class of second-order non-autonomous ordinary differential equations: Multiplicity results for stepwise potentials. Boundary Value Problems, 2013(1), 167. https://doi.org/https://doi.org/10.1186/1687-2770-2013-167
   Google Scholar
• Gradshteyn, I. S., & Ryzhik, I. M. (2000). Table of Integrals. Series and Products, Academic Press, San Diego, Calif, USA. 6th edition. https://booksite.elsevier.com/samplechapters/9780123736376/Sample_Chapters/01~Front_Matter.pdf
   Google Scholar
• Gritsans, A., & Sadyrbaev, F. (2015). Extension of the example by Moore-Nehari . Tatra Mt. Math. Publ., 63, 115-127. https://doi.org/https://doi.org/10.1515/tmmp-2015-0024
   Google Scholar
• Kirichuka, A. (2013). Multiple solutions for nonlinear boundary value problems of ODE depending on two parameters. Proceedings of IMCS of University of Latvia, 13, 83–97. https://protect-us.mimecast.com/s/vOlrCERZP5flWgPkXIPw49I?domain=lumii.lv
   Google Scholar
• Kirichuka, A. (2016). On the Dirichlet boundary value problem for a cubic on two outer intervals and linear in the internal interval differential equation. Proceedings of IMCS of University of Latvia, 16, 54–66. https://lumii.lv/uploads/sadirbajevs_2016/Sbornik-2016english.htm
   Google Scholar
• Kirichuka, A. (2017). The number of solutions to the Neumann problem for the second order differential equation with cubic nonlinearity. Proceedings of IMCS of University of Latvia, 17, 44–51. https://lumii.lv/uploads/sadirbajevs_2018/Sbornik2018english.html
   Google Scholar
• Kirichuka, A. (2018). The number of solutions to the Dirichlet and mixed problem for the second order differential equation with cubic nonlinearity. Proceedings of IMCS of University of Latvia, 18, 63–72. https://lumii.lv/uploads/sadirbajevs_2017/Sbornik2017english.html
   Google Scholar
• Kirichuka, A. (2019). The number of solutions to the boundary value problem for the second order differential equation with cubic nonlinearity. WSEAS Transactions on Mathematics, 18(31), 230–236. https://lumii.lv/uploads/sadirbajevs_2019/Sbornik2019english.html
   Google Scholar
• Kirichuka, A. (2020). The number of solutions to the boundary balue broblem with linear-quintic and linear-cubic-quintic nonlinearity. WSEAS Transactions on Mathematics, 19(64), 589–597. https://doi.org/https://doi.org/10.37394/23206.2020.19.64
   Google Scholar
• Kirichuka, A., & Sadyrbaev, F. (2018a). On boundary value problem for equations with cubic nonlinearity and step-wise coefficient. Differential Equations and Applications, 10(4), 433–447. https://doi.org/https://doi.org/10.7153/dea-2018-10-29
   Web of Science ®Google Scholar
• Kirichuka, A., & Sadyrbaev, F. (2018b). Remark on boundary value problems arising in Ginzburg-Landau theory. WSEAS Transactions on Mathematics, 17, 290–295. https://www.wseas.org/multimedia/journals/mathematics/2018/a685106-1057.php
   Google Scholar
• Kirichuka, A., & Sadyrbaev, F. (2019). On the number of solutions for a certain class of nonlinear second-order boundary-value problems. Itogi Nauki I Tekhniki. Seriya “Sovremennaya Matematika I Ee Prilozheniya. Tematicheskie Obzory, 160, 32–41. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=into&paperid=422&option_lang=eng
   Google Scholar
• Kovacic, I. (2020, August). Nonlinear Oscillations. Springer International Publishing.
   Google Scholar
• Milne-Thomson, L. M. (1972). Handbook of mathematical functions, chapter 16. In M. Abramowitz & I. A. Stegun (Eds.), Jacobian elliptic functions and theta functions. Dover Publications. http://www.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf
   Google Scholar
• Moore, R., & Nehari, Z. (1959). Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc, 93(1), 30–52. https://doi.org/https://doi.org/10.1090/S0002-9947-1959-0111897-8
   Google Scholar
• Ogorodnikova, S., & Sadyrbaev, F. (2006). Multiple solutions of nonlinear boundary value problems with oscillatory solutions. Mathematical Modelling and Analysis, 11(4), 413–426. https://doi.org/https://doi.org/10.3846/13926292.2006.9637328
   Google Scholar
• Shafiq, A., & Hammouch, Z. (2020). Statistical approach of mixed convective flow of third-grade fluid towards an exponentially stretching surface with convective boundary condition. Special Functions and Analysis of Differential Equations, 307, 307–319. https://www.taylorfrancis.com/chapters/edit/ https://doi.org/10.1201/9780429320026-15 /statistical-approach-mixed-convective-flow-third-grade-fluid-towards-exponentially-stretching-surface-convective-boundary-condition-anum-shafiq-zakia-hammouch-tabassum-naz-sindhu-dumitru-baleanu
   Google Scholar
• Shafiq, A., & Khalique, C. M. (2020). Lie group analysis of upper convected Maxwell fluid flow along stretching surface. Alexandria Engineering Journal, 59(4), 2533–2541. https://doi.org/https://doi.org/10.1016/j.aej.2020.04.017
   Web of Science ®Google Scholar
• Shafiq, A., & Sindhu, T. N. (2017). Statistical study of hydromagnetic boundary layer flow of Williamson fluid regarding a radiative surface. Results in Physics, 7, 3059–3067. https://doi.org/https://doi.org/10.1016/j.rinp.2017.07.077
   Web of Science ®Google Scholar
• Shafiq, A., Sindhu, T. N., & Hammouch, Z. Characteristics of homogeneous heterogeneous reaction on flow of walters’ B liquid under the statistical paradigm. Mathematical Modelling, Applied Analysis and Computation, 272. https://www.researchgate.net/publication/334208119_Characteristics_of_Homogeneous_Heterogeneous_Reaction_on_Flow_of_Walters'_B_Liquid_Under_the_Statistical_Paradigm
   Google Scholar
• Shang, Y. (2012). A Lie algebra approach to susceptible-infected-susceptible epidemics. Electronic Journal of Differential Equations, 2012(233), 1–7. https://www.researchgate.net/publication/266859292_A_Lie_algebra_approach_to_susceptible-infected-susceptible_epidemics
   Google Scholar
• Whittaker, E. T., & Watson, G. N. (1996). A course of modern analysis. Cambridge University Press.
   Google Scholar