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May  2019, 18(3): 1509-1521. doi: 10.3934/cpaa.2019072

Applications of generalized trigonometric functions with two parameters

Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma, Saitama 337-8570, Japan

* Corresponding author

Dedicated to Professor Yoshio Yamada on the occasion of his retirement

Received  November 2017 Revised  May 2018 Published  November 2018

Fund Project: The work of S. Takeuchi was supported by JSPS KAKENHI Grant Number 17K05336

Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the $p$-Laplacian. However, few applications to differential equations unrelated to the $p$-Laplacian are known. We will apply GTFs with two parameters to nonlinear nonlocal boundary value problems without $p$-Laplacian. Moreover, we will give integral formulas for the functions, e.g. Wallis-type formulas, and apply the formulas to the lemniscate function and the lemniscate constant.

Citation: Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964. Google Scholar

[2]

G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. Google Scholar

[3]

P. BindingL. BoultonJ. ČepičkaP. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3487-3494. doi: 10.1090/S0002-9939-06-08001-4. Google Scholar

[4]

F. D. Burgoyne, Generalized trigonometric functions, Math. Comp., 18 (1964), 314-316. doi: 10.2307/2003310. Google Scholar

[5]

P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions, Rocky Mountain J. Math., 42 (2012), 25-57. doi: 10.1216/RMJ-2012-42-1-25. Google Scholar

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L. Boulton and G. Lord, Approximation properties of the q-sine bases, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2690-2711. doi: 10.1098/rspa.2010.0486. Google Scholar

[7]

B. A. Bhayo and L. Yin, On generalized $(p, q)$-elliptic integrals, preprint, arXiv: 1507.00031.Google Scholar

[8]

C. CaoS. IbrahimK. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. Google Scholar

[9]

M. del PinoM. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(\vert u'\vert ^{p-2}u')'+f(t,u) = 0,\;u(0) = u(T) = 0,\;p>1$, J. Differential Equations, 80 (1989), 1-13. doi: 10.1016/0022-0396(89)90093-4. Google Scholar

[10]

O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005. Google Scholar

[11]

P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations, 12 (1999), 773-788. Google Scholar

[12]

D. E. EdmundsP. Gurka and J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012), 47-56. doi: 10.1016/j.jat.2011.09.004. Google Scholar

[13]

D. E. EdmundsP. Gurka and J. Lang, Basis properties of generalized trigonometric functions, J. Math. Anal. Appl., 420 (2014), 1680-1692. doi: 10.1016/j.jmaa.2014.06.015. Google Scholar

[14]

A. Elbert, A half-linear second order differential equation, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), pp. 153-180, Colloq. Math. Soc. Janos Bolyai, 30, NorthHolland, Amsterdam-New York, 1981. Google Scholar

[15]

T. Hyde, A Wallis product on clovers, Amer. Math. Monthly, 121 (2014), 237-243. doi: 10.4169/amer.math.monthly.121.03.237. Google Scholar

[16]

T. Kamiya and S. Takeuchi, Complete (p, q)-elliptic integrals with application to a family of means, J. Class. Anal., 10 (2017), 15-25. doi: 10.7153/jca-10-02. Google Scholar

[17]

J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, 2016. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18429-1. Google Scholar

[18]

P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290. Google Scholar

[19]

P. Lindqvist and J. Peetre, p-arclength of the q-circle, The Mathematics Student, 72 (2003), 139-145. Google Scholar

[20]

P. Lindqvist and J. Peetre, Comments on Erik Lundberg's 1879 thesis. Especially on the work of Göran Dillner and his influence on Lundberg, Memorie dell'Instituto Lombardo (Classe di Scienze Matem. Nat.) 31, 2004.Google Scholar

[21]

Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822. Google Scholar

[22]

E. Neuman, Some properties of the generalized Jacobian elliptic functions Ⅲ, Integral Transforms Spec. Funct., 27 (2016), 824-834. doi: 10.1080/10652469.2016.1210144. Google Scholar

[23]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. Google Scholar

[24]

D. Shelupsky, A generalization of the trigonometric functions, Amer. Math. Monthly, 66 (1959), 879-884. doi: 10.2307/2309789. Google Scholar

[25]

S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35. doi: 10.1016/j.jmaa.2011.06.063. Google Scholar

[26]

S. Takeuchi, The basis property of generalized Jacobian elliptic functions, Commun. Pure Appl. Anal., 13 (2014), 2675-2692. doi: 10.3934/cpaa.2014.13.2675. Google Scholar

[27]

S. Takeuchi, A new form of the generalized complete elliptic integrals, Kodai Math. J., 39 (2016), 202-226. Google Scholar

[28]

S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, J. Math. Anal. Appl., 444 (2016), 1000-1014. doi: 10.1016/j.jmaa.2016.06.074. Google Scholar

[29]

S. Takeuchi, Legendre-type relations for generalized complete elliptic integrals, J. Class. Anal., 9 (2016), 35-42. doi: 10.7153/jca-09-04. Google Scholar

[30]

S. Takeuchi, Complete $p$-elliptic integrals and a computation formula of $\pi_p$ for $p=4$, The Ramanujan Journal, 46 (2018), 309-321. doi: 10.1007/s11139-018-9993-y. Google Scholar

[31]

L. Yin and L.-G. Huang, Inequalities for the generalized trigonometric and hyperbolic functions with two parameters, J. Nonlinear Sci. Appl., 8 (2015), 315-323. doi: 10.22436/jnsa.008.04.04. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964. Google Scholar

[2]

G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. Google Scholar

[3]

P. BindingL. BoultonJ. ČepičkaP. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3487-3494. doi: 10.1090/S0002-9939-06-08001-4. Google Scholar

[4]

F. D. Burgoyne, Generalized trigonometric functions, Math. Comp., 18 (1964), 314-316. doi: 10.2307/2003310. Google Scholar

[5]

P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions, Rocky Mountain J. Math., 42 (2012), 25-57. doi: 10.1216/RMJ-2012-42-1-25. Google Scholar

[6]

L. Boulton and G. Lord, Approximation properties of the q-sine bases, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2690-2711. doi: 10.1098/rspa.2010.0486. Google Scholar

[7]

B. A. Bhayo and L. Yin, On generalized $(p, q)$-elliptic integrals, preprint, arXiv: 1507.00031.Google Scholar

[8]

C. CaoS. IbrahimK. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. Google Scholar

[9]

M. del PinoM. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(\vert u'\vert ^{p-2}u')'+f(t,u) = 0,\;u(0) = u(T) = 0,\;p>1$, J. Differential Equations, 80 (1989), 1-13. doi: 10.1016/0022-0396(89)90093-4. Google Scholar

[10]

O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005. Google Scholar

[11]

P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations, 12 (1999), 773-788. Google Scholar

[12]

D. E. EdmundsP. Gurka and J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012), 47-56. doi: 10.1016/j.jat.2011.09.004. Google Scholar

[13]

D. E. EdmundsP. Gurka and J. Lang, Basis properties of generalized trigonometric functions, J. Math. Anal. Appl., 420 (2014), 1680-1692. doi: 10.1016/j.jmaa.2014.06.015. Google Scholar

[14]

A. Elbert, A half-linear second order differential equation, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), pp. 153-180, Colloq. Math. Soc. Janos Bolyai, 30, NorthHolland, Amsterdam-New York, 1981. Google Scholar

[15]

T. Hyde, A Wallis product on clovers, Amer. Math. Monthly, 121 (2014), 237-243. doi: 10.4169/amer.math.monthly.121.03.237. Google Scholar

[16]

T. Kamiya and S. Takeuchi, Complete (p, q)-elliptic integrals with application to a family of means, J. Class. Anal., 10 (2017), 15-25. doi: 10.7153/jca-10-02. Google Scholar

[17]

J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, 2016. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18429-1. Google Scholar

[18]

P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290. Google Scholar

[19]

P. Lindqvist and J. Peetre, p-arclength of the q-circle, The Mathematics Student, 72 (2003), 139-145. Google Scholar

[20]

P. Lindqvist and J. Peetre, Comments on Erik Lundberg's 1879 thesis. Especially on the work of Göran Dillner and his influence on Lundberg, Memorie dell'Instituto Lombardo (Classe di Scienze Matem. Nat.) 31, 2004.Google Scholar

[21]

Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822. Google Scholar

[22]

E. Neuman, Some properties of the generalized Jacobian elliptic functions Ⅲ, Integral Transforms Spec. Funct., 27 (2016), 824-834. doi: 10.1080/10652469.2016.1210144. Google Scholar

[23]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. Google Scholar

[24]

D. Shelupsky, A generalization of the trigonometric functions, Amer. Math. Monthly, 66 (1959), 879-884. doi: 10.2307/2309789. Google Scholar

[25]

S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35. doi: 10.1016/j.jmaa.2011.06.063. Google Scholar

[26]

S. Takeuchi, The basis property of generalized Jacobian elliptic functions, Commun. Pure Appl. Anal., 13 (2014), 2675-2692. doi: 10.3934/cpaa.2014.13.2675. Google Scholar

[27]

S. Takeuchi, A new form of the generalized complete elliptic integrals, Kodai Math. J., 39 (2016), 202-226. Google Scholar

[28]

S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, J. Math. Anal. Appl., 444 (2016), 1000-1014. doi: 10.1016/j.jmaa.2016.06.074. Google Scholar

[29]

S. Takeuchi, Legendre-type relations for generalized complete elliptic integrals, J. Class. Anal., 9 (2016), 35-42. doi: 10.7153/jca-09-04. Google Scholar

[30]

S. Takeuchi, Complete $p$-elliptic integrals and a computation formula of $\pi_p$ for $p=4$, The Ramanujan Journal, 46 (2018), 309-321. doi: 10.1007/s11139-018-9993-y. Google Scholar

[31]

L. Yin and L.-G. Huang, Inequalities for the generalized trigonometric and hyperbolic functions with two parameters, J. Nonlinear Sci. Appl., 8 (2015), 315-323. doi: 10.22436/jnsa.008.04.04. Google Scholar

Figure 1.  Graphs of solutions of (4) with $H = 1$ for $m = 0.5, \ 1.0$ and $10.0$.
Figure 2.  Graphs of solutions of (13) for $p = 1.1, \ 2.0$ and $5.0$.
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