Geodesic
A Study of a Mathematical Model with a Movable Singular Point in a Fourth-Order Nonlinear Differential Equation
Russian journal of nonlinear dynamics, Tome 19 (2023) no. 4, pp. 575-584.

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This article introduces a mathematical model that utilizes a nonlinear differential equation to study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this equation is challenging due to the presence of moving singular points. The article addresses two main problems: first, it establishes the existence and uniqueness of the solution of the equation and, second, it provides precise criteria for determining the existence of a moving singular point. Additionally, the article presents estimates of the error in the analytical approximate solution and validates the results through a numerical experiment.
Keywords: nonlinear differential equations, movable singular point, exact criteria of exis- tence, necessary and sufficient conditions, Cauchy problem 
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M. V. Gasanov; A. G. Gulkanov. A Study of a Mathematical Model with a Movable Singular Point in a Fourth-Order Nonlinear Differential Equation. Russian journal of nonlinear dynamics, Tome 19 (2023) no. 4, pp. 575-584. http://geodesic.mathdoc.fr/item/ND_2023_19_4_a7/

[1] Newman, K. B., Buckland, S. T., Morgan, B. J. T., King, R., Borchers, D. L., Cole, D. J., Besbeas, P., Gimenez, O., and Thomas, L., Modelling Population Dynamics: Model Formulation, Fitting and Assessment Using State-Space Methods, Methods in Statistical Ecology, Springer, New York, 2014, xii, 215 pp. | DOI

[2] Smart, W. M., Celestial Mechanics, Wiley, New York, 1961, 381 pp.

[3] Robinson, J. C., An Introduction to Ordinary Differential Equations, Cambridge Univ. Press, Cambridge, 2004, xiv, 399 pp. | Zbl

[4] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations: Runge – Kutta and General Linear Methods, Wiley, Hoboken, N.J., 1987, 528 pp. | Zbl

[5] Golubev, V. V., Lectures on Analytical Theory of Differential Equations, Gostekhizdat, Moscow, 1950, 436 pp. (Russian)

[6] Riccati, J. F., “Animadversationes in aequationes differentiales secundi gradus”, Acta eruditorum, Suppl. 8, Grossium Gletitschium, Lipsiae, 1724, 66–73

[7] Márquez Lozano, A. P., Garrido Letrán, T. M., Recio Rodríguez, E., and de la Rosa Silva, R., “Lie Symmetries and Exact Solutions for a Fourth-Order Nonlinear Diffusion Equation”, Math. Meth. Appl. Sci., 45:17 (2022), 10614–10627 | DOI

[8] Kudryashov, N. A., “Exact Solutions of Equation for Description of Embedded Solitons”, Optik, 268 (2022), 169801 | DOI

[9] Kudryashov, N. A., “Exact Solutions of the Complex Ginzburg – Landau Equation with Law of Four Powers of Nonlinearity”, Optik, 265 (2022), 169548 | DOI

[10] Kudryashov, N. A., “One Method for Finding Exact Solutions of Nonlinear Differential Equations”, Commun. Nonlinear Sci. Numer. Simul., 17:6 (2012), 2248–2253 | DOI | MR | Zbl

[11] Ahmed, S. A., Qazza, A., and Saadeh, R., “Exact Solutions of Nonlinear Partial Differential Equations via the New Double Integral Transform Combined with Iterative Method”, Axioms, 11 (2022), 247, 16 pp. | DOI

[12] Astashova, I., Bartušek, M., Došlá, Z., and Marini, M., “Asymptotic Proximity to Higher Order Nonlinear Differential Equations”, Adv. Nonlinear Anal., 11:1 (2022), 1598–1613 | DOI | MR | Zbl

[13] Astashova, I., “On Quasi-Periodic Solutions to a Higher-Order Emden – Fowler Type Differential Equation”, Bound. Value Probl., 2014 (2014), 174 | DOI | MR | Zbl

[14] Glassey, R. T., “On the Asymptotic Behavior of Nonlinear Wave Equations”, Trans. Amer. Math. Soc., 182 (1973), 187–200 | DOI | MR | Zbl

[15] Alfimov, G. L., Barashenkov, I. V., Fedotov, A. P., Smirnov, V. V., and Zezyulin, D. A., “Global Search for Localised Modes in Scalar and Vector Nonlinear Schrödinger-Type Equations”, Phys. D, 397 (2019), 39–53 | DOI | MR

[16] Alfimov, G. L., Fedotov, A. P., and Sinelshchikov, D. I., “Determination of the Blow Up Point for Complex Nonautonomous ODE with Cubic Nonlinearity”, Phys. D, 402 (2020), 132245, 8 pp. | DOI | MR | Zbl

[17] Ufimsk. Mat. Zh., 7:2 (2015), 3–18 (Russian) | DOI | MR

[18] Zezyulin, D. A., Slobodyanyuk, A. O., and Alfimov, G. L., “On Nonexistence of Continuous Families of Stationary Nonlinear Modes for a Class of Complex Potentials”, Stud. Appl. Math., 148:1 (2022), 99–124 | DOI | MR

[19] Alfimov, G. L. and Avramenko, A. I., “Coding of Nonlinear States for the Gross – Pitaevskii Equation with Periodic Potential”, Phys. D, 254 (2013), 29–45 | DOI | MR | Zbl

[20] Orlov, V. N., “Investigation of the Approximate Solution of the Abel Differential Equation in the Vicinity of a Moving Singular Point”, Vestn. MGTU. Ser. Estest. Nauki, 4(35) (2009), 23–32 (Russian)

[21] Orlov, V. N., “The Exact Application Area Borders of Abel Differential Equation Approximate Solution in the Area of the Movable Special Point Approximate Meaning”, Vestn. Voronezhsk. Gos. Tekhn. Univ., 5:10 (2009), 192–195 (Russian)

[22] Orlov, V. N., “About Approximate Solution of the First Painlevé Equation”, Vestn. KGTU, 2 (2008), 42–46 (Russian)

[23] Orlov, V. and Gasanov, M., “Technology for Obtaining the Approximate Value of Moving Singular Points for a Class of Nonlinear Differential Equations in a Complex Domain”, Mathematics, 10:21 (2022), 3984, 7 pp. | DOI

[24] Orlov, V. and Gasanov, M., “Research of a Third-Order Nonlinear Differential Equation in the Vicinity of a Moving Singular Point for a Complex Plane”, E3S Web Conf., 263 (2021), 03019, 7 pp. | DOI

[25] Orlov, V. and Gasanov, M., “Analytic Approximate Solution in the Neighborhood of a Moving Singular Point of a Class of Nonlinear Equations”, Axioms, 11:11 (2022), 637, 8 pp. | DOI

[26] Orlov, V. and Gasanov, M., “Study of Wave Processes in Elastic Beams and Nonlinear Differential Equations with Moving Singular Points”, IOP Conf. Ser.: Mater. Sci. Eng., 1030 (2021), 012081, 9 pp. | DOI

[27] Differ. Uravn., 40:6 (2004), 756–762, 861 (Russian) | DOI | MR | Zbl

[28] Veresovich, P. P. and Yablonskii, A. I., “Moving Singular Points of Third Order Systems of Differential Equations”, Differ. Uravn., 13:11 (1977), 1932–1939 (Russian) | MR | Zbl

[29] Differ. Uravn., 40:6 (2004), 756–762 (Russian) | DOI | MR | Zbl

[30] Orlov, V. and Chichurin, A., “Comparative Analysis for Mathematical Models of a Cantilever Type Structure”, AIP Conf. Proc., 2497:1 (2023), 030003 | DOI

[31] Orlov, V. and Chichurin, A., “The Influence of the Perturbation of the Initial Data on the Analytic Approximate Solution of the Van der Pol Equation in the Complex Domain”, Symmetry, 15:6 (2023), 1200, 9 pp. | DOI

[32] Orlov, V., “Moving Singular Points and the Van der Pol Equation, As Well As the Uniqueness of Its Solution”, Mathematics, 11:4 (2023), 873, 7 pp. | DOI

[33] Ostaszewska, U., Schmeidel, E., and Zdanowicz, M., “Existence of Solutions to Nonlinear Fourth-Order Beam Equation”, Qual. Theory Dyn. Syst., 22:3 (2023), 92, 17 pp. | DOI | MR | Zbl

[34] Li, Y., “Existence of Positive Solutions for the Cantilever Beam Equations with Fully Nonlinear Terms”, Nonlinear Anal. Real World Appl., 27 (2016), 221–237 | DOI | MR | Zbl