Approximate solution of fourth order differential equation in Neumann problem

Subject Areas : Approximations and expansions

J. Rashidinia ¹ (School of Mathematics, Iran University of Science and Technology, Tehran, Iran)
D. Kalvand ² (Faculty of Mathematics, Yerevan state University , Yerevan, Armenia)
L. Tepoyan ³ (Faculty of Mathematics, Yerevan state University , Yerevan, Armenia)

Received: 2013-08-19 Accepted : 2013-12-04 Published : 2013-12-13

Keywords: Neumann problem, quintic spline function, Error analysis, Fourth order ordinary differential equation, generalized solution,

Abstract :

Generalized solution on Neumann problem of the fourth order ordinary differentialequation in space $W^2_\alpha(0,b)$ has been discussed, we obtain the condition on B.V.P when thesolution is in classical form. Formulation of Quintic Spline Function has been derived and theconsistency relations are given.Numerical method,based on Quintic spline approximation hasbeen developed. Spline solution of the given problem has been considered for a certain valueof $\alpha$. Error analysis of the spline method is given and it has been tested by an example.

References:

[1] Berezanski.J.M,Expansion in Eigenfunctions of Selfadjoint Operators.,Transl.Math. Monographs 17, American Mathematical Soc, Providence,1968.

[2] Bicadze.A.V, Equations of mixed type.,M. Izd. AN SSSR,1959 (Russian).

[3] Burenko. V.V, Sobolev Spaces on Domains., Teubner, 1999.

[4] Dezin. A.A,Partial Differential Equations.(An Introduction to a General Theory of Linear Boundary Value Problems),Springer,1987.

[5] Fichera. G, On a unied theory of boundary value problems for elliptic-parabolic equations of second order., Boundary Problems of Differential Equations, The Univ. of Wisconsin Press,pp. 97-120 , 1960.

[6] Kalvand. Daryoush, Neumann problem for the degenerate differential-operator equations of the fourth order., Vestnik RAU, Physical-Mathematical and Natural Sciences, No. 2,pp. 34-41, 2010 (Russian).

[7] Kalvand. Daryoush, Tepoyan. L, Neumann problem for the fourth order degenerate ordinary differential equation., Proceedings of the Yerevan State University, Physical and Mathematical Sciences, No. 1,pp. 22-26, 2010.

[8] Kalvand. Daryoush, Tepoyan. L, Rashidinia. J, Existence and uniqueness of the fourth order boundary value problem and quintic Spline solution., Proceeding of 9th Seminar on Differential Equations and Dynamical Systems, 11-13 July, Iran,pp. 133-136, 2012.

[9] Keldi. M. V, Fis, On certain cases of degeneration of equations of elliptic type on the boundary of a domain., Dokl. Akad. Nauk. SSSR, 77,pp. 181-183, 1951 (Russian).

[10] Rashidinia,J.Direct methods for solution of a linear fourth-order two-point boundary value problem.,J. Intern.Eng.Sci., Vol.13,pp.37-48(2002).

[11] Rashidinia, J.Jalilian,R. Non-polynomial spline for solution of boundary value problems in plate defection theory., J. Comput. Math., 84(10), pp.1483-1494.(2007)

[12] Rashidinia,J.Mahmoodi,R.Jalilian,R.Quintic spline solution of Boundary value problem in plate Defection.,

J. Comput. Sci.Eng.,Vol.16,No.1,pp.53-59(2009).

[13] Romanko. V.K, On the theory of the operators of the form, Differential Equations,Vol. 3,No. 11, pp. 1957-1970, 1967 (Russian).

[14] Showalter. R.E, Hilbert Space Methods for Partial Differential Equations., Electronic Journal of Differential Equations, Monograph 01, 1994.

[15] Tepoyan. L, Degenerate fourth-order dierential-operator equations.,Differ. Urav, Vol. 23(8), 1987, pp. 1366- 1376, (Russian); English Transl. in Amer. Math. Soc.,No. 8, 1988.

[16] Tepoyan. L, On a degenerate dierential-operator equation of higher order., Izvestiya Natsionalnoi AkademiiNauk Armenii. Matematika, Vol.34(5), pp. 48-56,1999.

[17] Tepoyan. L, On the spectrum of a degenerate operator., Izvestiya Natsionalnoi Akademii Nauk Armenii. Matematika,Vol. 38,No. 5,pp. 53-57, 2003.

[18] Tepoyan. L, The Neumann problem for a degenerate dierential-operator equation., Bulletin of TICMI (Tbil-isi International Centre of Mathematics and Informatics),Vol. 14, pp. 1-9, 2010.

[19] Tricomi, F, On linear partial dierential equations of second order of mixed type., M., Gostexizdat, 1947 (Russian).

254 J. Rashidinia et al. / J. Linear. Topological. Algebra. 02(04) (2013) 243-254.

[20] Usmani,R.A. Discrete methods for boundary value problems with applications in plate defection the-

ory.,J.Appl.Math.Phys., 30 ,pp.87-99(1979).

[21] Visik. M.I, Boundary-value problems for elliptic equations degenerate on the boundary of a region., Mat. Sb.,

35(77), pp. 513-568,1954 (Russian); Amer. Math. Soc,Vol. 35,No. 2,(English) 1964.

[22] Zahra ,W.K,Ashraf, M.El,Mhlawy. Numerical solution of two-parameter singularly perturbed boundary vproblems via exponential spline, Journal of King Saudi University Science January(2013).