WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 19, 2020
The Number of Solutions to the Boundary Value Problem With Linear-quintic and Linear-cubic-quintic Nonlinearity
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Abstract: The nonlinear oscillators describing by differential equations of the form x00 = ¡ax + cx5 and x00 = ¡ax + bx3 + cx5 are studied. Multiplicity results for both types of equations, given with the Neumann boundary conditions are obtained. It is shown that the number of solutions depend on the coefficient a only. The exact estimates of the number of solutions are obtained. Practical issues, such as the representation of solutions in terms of Jacobian elliptic functions and calculation of the initial values for solutions of boundary value problems, are considered also. The illustrative examples are provided. Outlines of future research conclude the article.
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Keywords: Boundary value problem, quintic nonlinearity, cubic-quintic nonlinearity, phase trajectory, multiplicity of solutions, Jacobian elliptic function.
Pages: 589-597
DOI: 10.37394/23206.2020.19.64