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Solving nonlinear ordinary differential equations with variable coefficients by elastic transformation method

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Abstract

Aiming at the difficult problem of solving nonlinear ordinary differential equation with variable coefficients, In this paper, the definition of elastic transformation is introduced, and the elastic transformation method for solving differential equations with variable coefficients is proposed. By this method, a class of nonhomogeneous nonlinear first-order and a class of nonlinear third-order ordinary differential equations with variable coefficients can be transformed into the homogeneous linear special equations (Associated Legendre equation, Gegenbauer equation, Hypersphere equation) which the general solutions can be found. Then, according to the definition of elasticity and the general solution of the special equation, the general solution of the original differential equations are obtained. The elastic transformation method not only expands the solvable classes of ordinary differential equation, but also promotes the application of special equation. More importantly, the elastic transformation method provides a new idea for solving low-order and high-order nonlinear ordinary differential equation with variable coefficients.

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References

  1. Bhrawy, A.H., Alghamdi, M.A.: Approximate solutions of Fisher’s type equations with variable coefficients[J]. Abst. Appl.Anal. (2013). https://doi.org/10.1155/2013/176730

    Article  MATH  Google Scholar 

  2. Fatoorehchi, H., Abolghasemi, H.: Series solution of nonlinear differential equations by a novel extension of the Laplace transform method[J]. Int. J. Comput. Math. 93(8), 1299–1319 (2015). https://doi.org/10.1080/00207160.2015.1045421

    Article  MATH  Google Scholar 

  3. Dzhumabaev, D.S., Uteshova, R.E.: Weighted limit solution of a nonlinear ordinary differential equation at a singular point and its property[J]. Ukr. Math. J. 69(12), 1997–2004 (2018). https://doi.org/10.1007/s11253-018-1483-2

    Article  MATH  Google Scholar 

  4. Guo, Y., Liu, T.: Bubble solutions for a supercritical polyharmonic Hénon-type equation[J]. Manuscrip. Math. (2020). https://doi.org/10.1007/s00229-020-01266-3

    Article  Google Scholar 

  5. Yuan, Y.: Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass[J]. Sci. China Math. (2021). https://doi.org/10.1007/s11425-021-1859-8

    Article  Google Scholar 

  6. Wei, H., Ji, S.G.: Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions[J]. Sci. China Math. (2022). https://doi.org/10.1007/s11425-020-1900-5

    Article  Google Scholar 

  7. Naito, M., Usami, H.: On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations[J]. J. Differ. Equ. 318, 359–383 (2022). https://doi.org/10.1016/j.jde.2022.02.025

    Article  MATH  Google Scholar 

  8. Marshall, A.: Principles of economics: an introductory volume[J]. Macmillan, 102-116 (1920)

  9. Wiratnim, W., Kono, H.O.: Determination of intrinsic minimum bubbling velocity in fine powder aerations based on experimentally measured elastic deformation coefficients[J]. Canad. J. Chem. Eng. 83(3), 418–424 (2005). https://doi.org/10.1002/cjce.5450830304

    Article  Google Scholar 

  10. Tugade, M.M., Fredrickson, B.L., et al.: Psychological resilience and positive emotional granularity: examining the benefits of positive emotions on coping and health[J]. J. Pers. 72(6), 1161–1190 (2004). https://doi.org/10.1111/j.1467-6494.2004.00294.x

    Article  Google Scholar 

  11. Huang, K., Xu, Q., et al.: Study on the relationship between mental flexibility and anxiety and depression in patients undergoing chemotherapy after breast cancer surgery[J]. J. Nurs. Sci. 28(2), 89–91 (2013)

    Google Scholar 

  12. Zhang, S., Wang, H.F., et al.: Diagnostic value of ultrasound Elastography and contrast-enhanced ultrasound in benign and malignant thyroid nodules[J]. Hainan Medicine 27(015), 2476–2479 (2016)

    Google Scholar 

  13. Dong, P.P.: The mathematical definition and practical significance of margin and elasticity in economics[J]. Enterpr. Technol. Develop. Mid-term J. 33(11), 121–123 (2014). https://doi.org/10.3969/j.issn.1006-8937.2014.07.069

    Article  Google Scholar 

  14. Li, S.C., Zhao, C.C., et al.: Analysis of oil and gas flow characteristics in the reservoir with the elastic outer boundary[J]. J. Petrol. Ence Eng. 175, 280–285 (2018). https://doi.org/10.1016/j.petrol.2018.12.042

    Article  Google Scholar 

  15. Li, S.C., Guo, H., et al.: The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir[J]. Comput. Appl. Math. 39(2), 1–16 (2020). https://doi.org/10.1007/s40314-020-1104-1

    Article  Google Scholar 

  16. Li, S.C., Zhou, M., et al.: Analysis of seepage pressure in dual-porosity reservoir under elastic boundary[J]. J. Math. Comput. Sci. 10(2), 316–338 (2020)

    Google Scholar 

  17. Liu, S.D., Liu, S.S.: Special functions (second edition) [M]. China Meteorological Press, Beijing (2002)

    Google Scholar 

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Author notes

  1. Jie Tang, Lihui Leng and Shunchu Li have contributed equally to this work.

Authors and Affiliations

  1. Institute of Applied Mathematics, Xihua University, Chengdu, 610039, China

    Pengshe Zheng, Jie Tang, Lihui Leng & Shunchu Li

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  1. Pengshe Zheng
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  2. Jie Tang
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  3. Lihui Leng
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Contributions

All authors contributed to the study conception and design. Material preparation. The first draft of the manuscript was written by Jie Tang. The rest of the authors have made some corrections to the article manuscript. All authors read and approved the final manuscript. All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Jie Tang.

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Zheng, P., Tang, J., Leng, L. et al. Solving nonlinear ordinary differential equations with variable coefficients by elastic transformation method. J. Appl. Math. Comput. 69, 1297–1320 (2023). https://doi.org/10.1007/s12190-022-01791-2

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