On the generalized solutions of a certain fourth order Euler equations

Liangprom, Amphon¹ ; Nonlaopon, Kamsing¹

¹ Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 8, p. 4077-4084.

Résumé

In this paper, using Laplace transform technique, we propose the generalized solutions of the fourth order Euler differential equations

$t^4y^{(4)}(t)+t^3y'''(t)+t^2y''(t)+ty'(t)+my(t)=0,$

where $m$ is an integer and $t\in\mathbb{R}$. We find types of solutions depend on the values of $m$. Precisely, we have a distributional solution for $m=-k^4-5k^3-9k^2-4k$ and a weak solution for $m=-k^4+5k^3-9k^2+4k,$ where $k\in\mathbb{N}.$

Détail

DOI : 10.22436/jnsa.010.08.04

Classification : 34A37, 44A10, 46F10, 46F12
Keywords: Generalized solution, distributional solution, Euler equation, Dirac delta function.

Bibliographie

[1] Akanbi, M. A. Third order Euler method for numerical solution of ordinary differential equations, ARPN J. Eng. Appl. Sci., Volume 5 (2010), pp. 42-49

[2] Boyce, W. E.; DiPrima, R. C. Elementary differential equations and boundary value problems, Seventh edition, John Wiley & Sons, Inc., New York-London-Sydney, 2001

[3] Bupasiri, S.; Nonlaopon, K. On the weak solutions of compound equations related to the ultra-hyperbolic operators, Far East J. Appl. Math., Volume 35 (2009), pp. 129-139 | Zbl

[4] Coddington, E. A. An introduction to ordinary differential equations, Prentice-Hall Mathematics Series Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961

[5] Coddington, E. A.; N. Levinson Theory of ordinary differential equations , McGraw-Hill Book Company, Inc., New York-Toronto-London, 1995

[6] Cooke, K. L.; Wiener, J. Distributional and analytic solutions of functional-differential equations, J. Math. Anal. Appl., Volume 98 (1984), pp. 111-129 | DOI

[7] Hernández-Ureña, L. G.; Estrada, R. Solution of ordinary differential equations by series of delta functions, J. Math. Anal. Appl., Volume 191 (1995), pp. 40-55 | DOI

[8] Kananthai, A. Distribution solutions of the third order Euler equation, Southeast Asian Bull. Math., Volume 23 (1999), pp. 627-631

[9] Kananthai, A. The distribution solutions of ordinary differential equation with polynomial coefficients, Southeast Asian Bull. Math., Volume 25 (2001), pp. 129-134 | DOI

[10] Kananthai, A.; Nonlaopon, K. On the weak solution of the compound ultra-hyperbolic equation, CMU. J., Volume 1 (2002), pp. 209-214

[11] Kananthai, A.; Suantai, S.; Longani, V. On the weak solutions of the equation related to the diamond operator, Vychisl. Tekhnol., Volume 5 (2000), pp. 61-67 | Zbl

[12] R. P. Kanwal Generalized functions, Theory and applications, Third edition, Birkhuser Boston, Inc., Boston, MA, 2004

[13] Krall, A. M.; Kanwal, R. P.; L. L. Littlejohn Distributional solutions of ordinary differential equations , Oscillations, bifurcation and chaos, Toronto, Ont., (1986), CMS Conf. Proc., Amer. Math. Soc., Providence, RI, Volume 8 (1987), pp. 227-246

[14] Kumar, D.; Singh, J.; Baleanu, D. A hybrid computational approach for Klein-Gordon equations on Cantor sets , Nonlinear Dynam., Volume 87 (2017), pp. 511-517 | Zbl | DOI

[15] Kumar, D.; Singh, J.; Baleanu, D. A new analysis for fractional model of regularized longwave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., Volume 2017 (2017), pp. 1-12 | DOI | Zbl

[16] Littlejohn, L. L.; Kanwal, R. P. Distributional solutions of the hypergeometric differential equation, J. Math. Anal. Appl., Volume 122 (1987), pp. 325-345 | DOI

[17] Nonlaopon, K.; Orankitjaroen, S.; Kananthai, A. The generalized solutions of a certain n order differential equations with polynomial coefficients, Integral Transforms Spec. Funct., Volume 26 (2015), pp. 1015-1024 | DOI | Zbl

[18] Sabuwala, A. H.; Leon, D. De; Particular solution to the Euler-Cauchy equation with polynomial non-homogeneities, , Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 8th AIMS Conference. Suppl. Vol. II (2011), pp. 1271-1278 | Zbl

[19] Sarikaya, M. Z.; Yildirim, H. On the weak solutions of the compound Bessel ultra-hyperbolic equation, Appl. Math. Comput., Volume 189 (2007), pp. 910-917 | DOI | Zbl

[20] L. Schwartz Théorie des distributions á valeurs vectorielles, I, (French) Ann. Inst. Fourier, Grenoble, Volume 7 (1957), pp. 1-141 | DOI

[21] Singh, J.; Kumar, D.; Qurashi, M. A.; Baleanu, D. Analysis of a new fractional model for damped Bergers equation, Open Phys., Volume 15 (2017), pp. 35-41

[22] Singh, J.; Kumar, D.; Swroop, R.; S. Kumar An efficient computational approach for time-fractional Rosenau–Hyman equation , Neural Comput. Appl., Volume 2017 (2017), pp. 1-8 | DOI

[23] Srisombat, P.; Nonlaopon, K. On the weak solutions of the compound ultra-hyperbolic Bessel equation, Selçuk J. Appl. Math., Volume 11 (2010), pp. 127-136 | Zbl

[24] Srivastava, H. M.; Kumar, D.; Singh, J. An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., Volume 45 (2017), pp. 192-204 | DOI

[25] J. Wiener Generalized-function solutions of differential and functional-differential equations, J. Math. Anal. Appl., Volume 88 (1982), pp. 170-182 | DOI

[26] Wiener, J. Generalized solutions of functional-differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1993

[27] Wiener, J.; Cooke, K. L. Coexistence of analytic and distributional solutions for linear differential equations, I, J. Math. Anal. Appl., Volume 148 (1990), pp. 390-421 | DOI | Zbl

[28] Wiener, J.; Cooke, K. L.; S. M. Shah Coexistence of analytic and distributional solutions for linear differential equations, II, J. Math. Anal. Appl., Volume 159 (1991), pp. 271-289 | Zbl | DOI

[29] Zemanian, A. H. Distribution theory and transform analysis, An introduction to generalized functions, with applications, McGraw-Hill Book Co., New York-Toronto-London-Sydney., 1965