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Solution of BVPs Using bvp4c and bvp5c of MATLAB

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Boundary Value Problems for Engineers

Abstract

MATLAB provides a platform to solve BVPs which consist of two residual control based, adaptive mesh solvers named as bvp4c and bvp5c .

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Notes

  1. 1.

    MATLAB demo program twoode solves given BVP twice, using the guess \( y(x) = 1,y^{\prime } (x) = 0 \) and then \( y(x) = - 1,y^{\prime } (x) = 0 \).

  2. 2.

    The numbers (even their ratios) are likely to change with MATLAB version/operating system/processor model . The numbers were obtained using MATLAB version 2016a, 64 bit, run on a computer with 2.5 GHz Intel CoreTM i5-2430 M processor.

  3. 3.

    Private communication with Dr. Jacek Kierzenka, Mathworks Inc.

  4. 4.

    (Elapsed time) The code was run on a laptop with 2.5 GHz Intel Core i5TM 2430 M single core computer.

  5. 5.

    For more detailed analysis and derivations of these equations, readers may refer to the book by Longuski et al. [24].

  6. 6.

    In 1744, Euler wrote his treatise on variational techniques in which he devoted an entire chapter, “De Curvis Elasticis”, working out a complete characterization of those curves which are solutions to the elastica problem, and these are nowadays known as “elasticae”. Remarkably, the elastica appears as yet another shape of the solution of a fundamental physics problem, the capillary. Laplace investigated the equation for the shape of the capillary in 1807. Since then, the subject has attracted many researchers and it is still an active field of investigation.

    In a broad sense elasticae are curves which are stationary points of the elastic energy functional. The elastic energy of a smooth curve is the integral of its squared curvature.

    The closed-form solutions of the elastica rely heavily on Jacobi elliptic functions [26,27,28.

  7. 7.

    These boundary conditions and used coefficient values differ from those reported in [39].

  8. 8.

    See, [47] for the application of this code using “function” functions, which is somehow seems an accustomed way of writing BVP codes using bvp4c (Incidently, the reader may notice that their equations contain minor mistakes—subscript of y5 is written twice as y3 in the equations -, but it is in correct form within the code).

  9. 9.

    This problem is derived from [48], referred to there as Swirling Flow III (SWF-III).

  10. 10.

    Llewellyn Hilleth Thomas (1903–1992) was a British physicist and applied mathematician.

  11. 11.

    Enrico Fermi (1901–1954) was an American physicist of Italian origin and the creator of the world’s first nuclear reactor .

  12. 12.

    This is an alternative solution of the equation which has been solved earlier in this chapter.

  13. 13.

    Israel Moiseevich Gelfand (1913–2009) was a prominent Russian mathematician.

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    Ali Ümit Keskin

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Keskin, A.Ü. (2019). Solution of BVPs Using bvp4c and bvp5c of MATLAB. In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_10

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