On the generalized solutions of a certain fourth order Euler equations
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Authors
Amphon Liangprom
- Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand.
Kamsing Nonlaopon
- Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand.
Abstract
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 \(k\in\mathbb{N}.\)
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ISRP Style
Amphon Liangprom, Kamsing Nonlaopon, On the generalized solutions of a certain fourth order Euler equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4077--4084
AMA Style
Liangprom Amphon, Nonlaopon Kamsing, On the generalized solutions of a certain fourth order Euler equations. J. Nonlinear Sci. Appl. (2017); 10(8):4077--4084
Chicago/Turabian Style
Liangprom, Amphon, Nonlaopon, Kamsing. "On the generalized solutions of a certain fourth order Euler equations." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4077--4084
Keywords
- Generalized solution
- distributional solution
- Euler equation
- Dirac delta function.
MSC
References
-
[1]
M. A. Akanbi, Third order Euler method for numerical solution of ordinary differential equations, ARPN J. Eng. Appl. Sci., 5 (2010), 42–49.
-
[2]
W. E. Boyce, R. C. DiPrima, Elementary differential equations and boundary value problems, Seventh edition, John Wiley & Sons, Inc., New York-London-Sydney (2001)
-
[3]
S. Bupasiri, K. Nonlaopon, On the weak solutions of compound equations related to the ultra-hyperbolic operators, Far East J. Appl. Math., 35 (2009), 129–139.
-
[4]
E. A. Coddington, An introduction to ordinary differential equations, Prentice-Hall Mathematics Series Prentice-Hall, Inc., Englewood Cliffs, N.J. (1961)
-
[5]
E. A. Coddington, N. Levinson , Theory of ordinary differential equations , McGraw-Hill Book Company, Inc., New York-Toronto-London (1995)
-
[6]
K. L. Cooke, J. Wiener, Distributional and analytic solutions of functional-differential equations, J. Math. Anal. Appl., 98 (1984), 111–129.
-
[7]
L. G. Hernández-Ureña, R. Estrada, Solution of ordinary differential equations by series of delta functions, J. Math. Anal. Appl., 191 (1995), 40–55.
-
[8]
A. Kananthai, Distribution solutions of the third order Euler equation, Southeast Asian Bull. Math., 23 (1999), 627–631.
-
[9]
A. Kananthai, The distribution solutions of ordinary differential equation with polynomial coefficients, Southeast Asian Bull. Math., 25 (2001), 129–134.
-
[10]
A. Kananthai, K. Nonlaopon, On the weak solution of the compound ultra-hyperbolic equation, CMU. J., 1 (2002), 209–214.
-
[11]
A. Kananthai, S. Suantai, V. Longani, On the weak solutions of the equation related to the diamond operator, Vychisl. Tekhnol., 5 (2000), 61–67.
-
[12]
R. P. Kanwal , Generalized functions, Theory and applications, Third edition, Birkhuser Boston, Inc., Boston, MA (2004)
-
[13]
A. M. Krall, R. P. Kanwal, L. L. Littlejohn , Distributional solutions of ordinary differential equations , Oscillations, bifurcation and chaos, Toronto, Ont., (1986), CMS Conf. Proc., Amer. Math. Soc., Providence, RI, 8 (1987), 227– 246.
-
[14]
D. Kumar, J. Singh, D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets , Nonlinear Dynam., 87 (2017), 511–517.
-
[15]
D. Kumar, J. Singh, D. Baleanu, A new analysis for fractional model of regularized longwave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 2017 (2017), 12 pages.
-
[16]
L. L. Littlejohn, R. P. Kanwal, Distributional solutions of the hypergeometric differential equation, J. Math. Anal. Appl., 122 (1987), 325–345.
-
[17]
K. Nonlaopon, S. Orankitjaroen, A. Kananthai, The generalized solutions of a certain n order differential equations with polynomial coefficients, Integral Transforms Spec. Funct., 26 (2015), 1015–1024.
-
[18]
A. H. Sabuwala, D. De Leon, , 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), 1271–1278.
-
[19]
M. Z. Sarikaya, H. Yildirim, On the weak solutions of the compound Bessel ultra-hyperbolic equation, Appl. Math. Comput., 189 (2007), 910–917.
-
[20]
L. Schwartz , Théorie des distributions á valeurs vectorielles, I, (French) Ann. Inst. Fourier, Grenoble, 7 (1957), 1–141.
-
[21]
J. Singh, D. Kumar, M. A. Qurashi, D. Baleanu, Analysis of a new fractional model for damped Bergers equation, Open Phys., 15 (2017), 35–41.
-
[22]
J. Singh, D. Kumar, R. Swroop, S. Kumar , An efficient computational approach for time-fractional Rosenau–Hyman equation , Neural Comput. Appl., 2017 (2017), 1–8.
-
[23]
P. Srisombat, K. Nonlaopon, On the weak solutions of the compound ultra-hyperbolic Bessel equation, Selçuk J. Appl. Math., 11 (2010), 127–136.
-
[24]
H. M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., 45 (2017), 192–204.
-
[25]
J. Wiener , Generalized-function solutions of differential and functional-differential equations, J. Math. Anal. Appl., 88 (1982), 170–182.
-
[26]
J. Wiener, Generalized solutions of functional-differential equations, World Scientific Publishing Co., Inc., River Edge, NJ (1993)
-
[27]
J. Wiener, K. L. Cooke, Coexistence of analytic and distributional solutions for linear differential equations, I, J. Math. Anal. Appl., 148 (1990), 390–421.
-
[28]
J. Wiener, K. L. Cooke, S. M. Shah , Coexistence of analytic and distributional solutions for linear differential equations, II, J. Math. Anal. Appl., 159 (1991), 271–289.
-
[29]
A. H. Zemanian, Distribution theory and transform analysis, An introduction to generalized functions, with applications, McGraw-Hill Book Co., New York-Toronto-London-Sydney. (1965)