Dual Integral Equations and Singular Integral. Equations for Helmholtz Equation
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1 Int.. Contemp. Math. Sciences, Vo. 4, 9, no. 34, Dua Integra Equations and Singuar Integra Equations for Hemhotz Equation Naser A. Hoshan Department of Mathematics TafiaTechnica University P.O. Box 179, Tafia, ordan Abstract Paper is devoted for Soving a Hemhotz equation in cyinica coordinates under mixed discontinuous boundary conditions of the third kind (Robin's boundary conditions for a semi-space and for infinite soid pate. A dua integra equations (DIE method were reduce the given mixed boundary vaue probem to a Fredhom singuar integra equation of the second kind. Keywords: Dua integra equations, mixed boundary conditions, Hemhotz equation 1. Introduction Hemhotz partia differentia equation arises in the study of mathematica physics equations with severa coordinate systems. In this paper we present a Hemhotz equation in axia symmetrica cyinica coordinates under mixed discontinuous boundary conditions of the third kind acted on the surface of semi-infinite space and on the surface of infinite pate, the soution of the of the Hemhotz mixed boundary vaue probem is obtained with the aid of the DIE method, the DIE are reduced to some type of Fredhom singuar integra equation of the second kind. Theory reated to mixed probem which discussed in this paper is due to Ufyand [8], the technique was used with detais to investigate the soution of some DIE for a Lapace's equation with some mixed boundary vaue probems.
2 1696 N. A. Hoshan. Formuation of the probem Suppose that for instance we wish to determine the function u u( r, z, satisfies Hemhotz partia differentia equation in a axia symmetrica semi-space of a soid cyinder subject to the mixed boundary conditions of the third kind acted on the eve surface z of the semi-space in cyinica coordinates a ( 1u z + b1u f1 r, r (, (.1 a ( u z + bu f r, r (, (. ai, bi, i 1, are constants u z u / z, f i ( r, i 1, known continuous functions accepted Hanke integra transform, is the ine of discontinuity (the radius of the disk. Separation variabes to the Hemhotz equation u + k u, k is a constant, we assume that u ( r, z is bounded at zero and infinity, a genera soution becomes u( r, z C( ( prexp( z p + k (.3 Use a mixed conditions (.1,(. for (.3, we obtain a pair of DIE for determination the unknown function C ( C ( 1 ( pr ( a1 p + k + b1 f ( r, r (, (.4 C ( pr ( a p + k + b f ( r, r (, (.5 ( The DIE (.4 and (.5 shoud be reduced to the standard form ( pr[1 ] F ( r, r (, (.6 A ( ( pr, r (, (1.7 Where F ( r ( a a1 p + k + ( b b1 a p + k + b f1 ( r p v( ( pr[1 ] A ( C( ( pr( a p + k + b v ( ( pr f( r Rewrite (.6 as and v( p r f ( r ( pr A ( ( pr F ( r + ( pr (.8
3 Dua integra equations 1697 Repacing a function A ( by another unknown continuous function φ (t with the hep of the reation [8] p φ ( t cos( pt dt (.9 For instance put g ( in (.8, then use the Hanke's inverse transform [9] p rf ( r ( pr (.1 Use the known reation [8,9] r cos( pt ( pr dt (.11 r t Put (.11 into (.1 then interchange the order of integration we have rf ( r cos( pt dt (.1 t r t Now compare (.1 and (.9, we discover that rf ( r φ ( t (.13 t r t Now when g ( in (.8, expressions (.13 and (.9 yied a Fredhom singuar integra equation of the second kind to determine the unknown function φ (t φ 1 ( x H ( x + K( x, t φ( t dt (.14 Where the free term rf( r H ( x and x r x r and the kerne K( x, t p ( prcos( pt. x r x The kerne and the free term shoud be satisfied the properties [4, 6] H ( x dx <, K ( x, t dxdt < (.15 In simiar manner, the DIE invoving mixed boundary conditions to the Hemhotz equation for unbounded pate high h when on the surface z a boundary conditions (.1, (. are given, whereas on another surface z h a homogeneous unmixed boundary condition is u ( r, h (.16 Separation variabes to the Hemhotz equation in cyinica coordinates under the assumption that u is bounded at zero and infinity, the genera soution is sh[( h z p + k ] u ( r, z C( ( pr (.17 ch[ h p + k ]
4 1698 N. A. Hoshan sh ( x, ch( x sine and cosine hyperboic functions, Appy the mixed (.1,(. to (.17, we get a DIE for determination A ( ( pr[1 ] F ( r, r (, (.18 A ( ( pr, r (, (.19 ( a a1 p + k + ( b b1 th( h p + k, a p + k + b th( h p + k A ( C( ( a p + k + bth( h p + k th(x is the tangent hyperboic function. The ast DIE (.18 and (.19 shoud be reduced to a Fredhom singuar integra equation of the form (.14 by appying the same technique mentioned above. Integra equation (.14 can be soved numericay with the use of some math software [3,7]. As h, the DIE (.18 and (.19 correspond the DIE invove a semi-space (.6 and (.7. Method of treating DIE iustrated in this paper can be used widey to sove severa types of mixed boundary vaue probems for heat and wave equations in different coordinates. In particuar for some of a constants ai, bi, i 1, is zero, a dua equations (.6 and (.7 were discussed with the use of some discontinuous integras,integra transform and operationa cacuus methods[1,,5] References [1] N. A Abderazaq. The Dua Integra Equations Method for Nonstationary Heat Conduction Equation. ourna Of Engineering Thermophysics,vo 17,No 1, 5 p 13-11, SB RAS. [] N. A. Abderazaq., The Soution Of Heat Equation With Mixed Boundary Conditions, ourna of Mathematics and Statistics, Science Pubications, New York USA.,6, (, , [3] Hacia L., Computationa resuts for integra equation in space- time Computationa methods in science and technoogy, 8((1,7-15,. [4] W. Hackbusch,,Integra Equations, Theory and Numerica Treatment, Birkhäuser Verag, Boston, [5] N. A. Hoshan, Exact Soution of Certain Dua Integra Equations Invoving Heat Equation, Far East ourna of Appied Math.,v 35,N1,Pupsh, p [6] K Kythe, P Puri, Computationa Methods for Linear Integra Equations Birkhhauer..
5 Dua integra equations 1699 [7] E.G. Ladopouos, Singuar Integra Equations, Linear and Non-Linear Theory and its Appications, Springer-Verag, Berin, New York [8] Ufyand. S. Dua Equations in Mathematica Physics Equations, Leningrad Nauka, 1977.(in Russian [9] N. A. Virchenco, N. A., Dua (Tripe Integra Equations. Kiev University, Kiev, (in Russian Received: May, 9
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