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• W2256172976 abstract "The hyperbolic equation (1D problem) supplemented by adequate boundary and initial conditions is considered. This equation is solved using the combined variant of the boundary element method. The problem is also solved in analytical way. The comparison of the results obtained by means of these two methods confirms the effectiveness and accuracy of the BEM. 1. Formulation of the problem The following equation is considered 2 2 2 2 ) , ( ) , ( ) , ( x t x U t t x U t t x U ∂ ∂ = ∂ ∂ + ∂ ∂ (1) where U (x, t) is an unknown function, x is the spatial co-ordinate and t is the time. The equation (1) is supplemented by the boundary conditions 0 ) , 1 ( : 1 , 0 0 ) , 0 ( : 0 , 0 = = > = = > t U x t t U x t (2) and the initial ones 0 ) , ( : 0 , 1 0 0 ) 0 , ( : 0 , 1 0 0 0 = ∂ ∂ = = = < < = t t t x U t x U x U t x (3) This type of boundary and initial conditions allows to solve the problem analytically and in this way the results obtained by means of the boundary element method using discretization in time can be compared with the analytical solution. Please cite this article as: Maria Lupa, Ewa Ładyga, Application of the Boundary Element Method using discretization in time for numerical solution of hyperbolic equation, Scientific Research of the Institute of Mathematics and Computer Science, 2008, Volume 7, Issue 1, pages 83-92. The website: http://www.amcm.pcz.pl/ M. Lupa, E. Ładyga 84 2. Boundary element method To solve the equation (1), the BEM using discretization in time is applied [1, 2]. So, the time grid ∞ < < < < < < < = − − F f f f t t t t t t 1 2 1 0 0 (4) with constant step 1 − − = f f t t t ∆ is introduced. For the time interval [ ] f f t t , 2 − the following approximations of time derivative can be taken into account t t x U t x U t t x U f f t t f ∆ ) , ( ) , ( ) , ( 1 − = − = ∂ ∂ (5) and t t x U t x U t t x U f f t t f ∆ 2 ) , ( ) , ( ) , ( 2 − = − = ∂ ∂ (6) or t t x U t x U t x U t t x U f f f t t f ∆ 2 ) , ( ) , ( 4 ) , ( 3 ) , ( 2 1 − − = + − = ∂ ∂ (7) The second time derivative is approximated in following way 2 2 1 2 2 ) ( ) , ( ) , ( 2 ) , ( ) , ( t t x U t x U t x U t t x U t f f ∆ − − + − = ∂ ∂ (8) Let t ∆ β 1 = and ). , ( t f x U U f ∆ = At the f-th time step ) 2 ( ≥ = f t f t ∆ the equation (1) can be approximately rewritten as 0 2 1 2 2 = − + − ∂ ∂ − − f f f f U C U B U A x U (9) where for the first variant (equation (5)) 2 2 2 , 2 , β β β β β = + = + = C B A (10) for the second variant (equation (6)) β β β β β 2 1 , 2 , 2 1 2 2 2 − = = + = C B A (11) Application of the boundary element method using discretization in time for numerical solution ... 85 and for the third variant (equation (7)) β β β β β β 2 1 ), ( 2 , 2 3 2 2 2 + = + = + = C B A (12) For equation (9) the weighted residual criterion is applied [1] ∫ =       − + − ∂ ∂ ∗ − − 1 0 2 1 2 2 0 ) , ( dx x U U C U B U A x U f f f f ξ (13) where ξ∈(0, 1) is the observation point, U * (ξ, x) is the fundamental solution and this function should fulfil the equation ) , ( ) , ( ) , ( * 2 2 x x U A x x U ξ δ ξ ξ − = − ∂ ∂ ∗ (14) where δ (ξ, x) is the Dirac function. It can be check that the following function ( ) ) ( exp 2 1 , * A x A x U ξ ξ − − = (15) fulfills the equation (14). Additionally, the function q (ξ, x) resulting from fundamental solution is defined x x U x q ∂ ∂ − = ) , ( ) , ( * * ξ ξ (16) and it can be calculated analytically ) ( exp 2 ) ( sgn ) , ( * A x x x q ξ ξ ξ − − − = (17) where sgn(⋅) is the sign function. Integrating twice by parts the first component of equation (13) and taking into account the property (14) of fundamental solution one obtains ∫ = − + + −       ∂ ∂ −       ∂ ∂" @default.
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• W2256172976 title "Application of the Boundary Element Method using discretization in time for numerical solution of hyperbolic equation" @default.
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