The Direct and Inverse Problems for the Hyperbolic Boundary Value Problem

Transcription

1 Al- Mustansiriyah J. Sci. Vol. 24, No 5, 203 The Direct and Inverse Probles for the Hyperbolic Boundary Value Proble Jail Air Ali Al-Hawasy and Halah Rahan Jaber 2,2 Deparent of atheatics, College of Science, University of Al-Mustansiriyah Received 6/3/203 Accepted 5/9/203 الخالصة خناول هذا انبحث حم ان سأنت ان باشزة ن عادنت حفاضه ت جزئ ت ين اننىع انزائذي يع شزوط ابخذائ ت و حذود ت باسخخذاو طز قت انعناصز ان حذدة و خناول ا ضا حم ان سأنت انعكس ت ن سأنت انق ى االبخذائ ت ان ذكىرة اعاله ال جاد انشزط االبخذائ ان صاحب ن سأنت انق ى االبخذائ ت ين اننىع انزائذي بخحى هها انى يسأنت ايثه ت غ ز خط ت وانخ خى حهها باسخخذاو طز قت هىك وج فزغ ز يشزوطت.اعط ج اننخائج عهى شكم جذاول او رسىياث ح ث اظهزث كفأه انطز قخ ن ف انحم. ABSTRACT This paper deals with solving the direct proble for partial differential equation of hyperbolic type with initial conditions and boundary conditions using finite eleent ethod. Also it deals with the direct ethod for solving the inverse proble to deterine the initial condition which associates the hyperbolic partial differential equation when the solution of the equation is given at finite nuber of points of the doain that the solution is defined. This proble is transfored to a nonlinear optiization proble which is solved by the unconstraint Hook and Jives ethod. The results are given by tables and/or figures and show the efficiency of these ethods. INTRODUCTION During the last three decades, inverse probles have been studied fro any researchers. Warin S. and Suabsagun Y. used the iterative ethod for Levenberg-Marquardt ethod to estiate the odel paraeters of conductivity variation of the ground []. Liao W. applied TAMC tool to solve the optiization proble which obtained fro the forulation of inverse proble to deterine the unknown acoustic coefficient (coefficient of 2D wave equation) [2]. Rashedi K. and Yousefi S.A., used a technique on the Ritz-Galrekin ethod to solve the inverse proble to deterine the coefficient of a parabolic equation [3]. Al- Hawasy Ali, J. A. used the direct ethod to solve the inverse proble to deterine the unknown region that the equation is defined [4]. In fact the differenence between the direct ethod which is used in [4] and which is used here are, first the varational ethod is used there to solve the direct proble while the finite eleents ethod is used here, second the inverse proble is used there to find the region that the equation is defined while here the inverse proble is used to find the initial condition. Since the inverse probles for hyperbolic partial differential equations arise naturally in geophysics, oil prospecting, in the design of optical devise, and in any other area. Hence our interest in this paper to study 39

2 The Direct and Inverse Probles for the Hyperbolic Boundary Value Proble Jail and Halah the inverse proble of hyperbolic differential equations to deterine an initial condition. The paper consists of two parts, in the first part, the direct proble is to solve the hyperbolic partial differential equation when the initial and the boundary conditions are given using the finite eleents ethod. While the second part deals with the direct ethod for solving the inverse proble to deterine the initial condition associated with hyperbolic differential equation when the solution of this equation are given at finite nubers of points of the doain that the equation is defined. This inverse proble is transfored to a nonlinear optiization proble which is solved by using the unconstraint Hook and Jives ethod. The results for both direct and inverse probles are given in tables and/or figures which show the efficiency of both ethods. STATEMENT OF THE DIRECT PROBLEM Let Ω R d be an open and bounded region with Lipischitz boundary Г= Ω and let I= (0, T) and Q=Ω I. The hyperbolic equation is given by: ( ) ( ) in Q, ( ) () with the boundary condition ( ) in, where (2) and the initial conditions ( ) ( ), in Ω (3) ( ) ( ), in Ω (4) where A(t) is the 2 nd order elliptic differential operator i.e.: ( ) d y aij ( x, t) i, jxi xi Now, we denote by (,.),and. the inner product and nor in Sobolev space V= ( ) by the duality bracket between V and its dual and by the nor in ( ). The weak for of the proble (-4) is given by: ( ) ( ( ) ), alost everywhere on I (5) ( ),in Ω (6) ( ),in Ω (7) with ( ) d y v aij( x, t) x (8) i, j j xi where the initial conditions ake sense if, ( ),and ( ) is the usual bilinear for associated with A(t),we suppose ( ) is syetric and for soe positive constants, satisfies ( ) and ( ). 40

3 Al- Mustansiriyah J. Sci. Vol. 24, No 5, 203 We can rewrite equation (5) by ( ) ( ( ) ), alost everywhere on (5a) (5b) DESCRTIZATION OF THE CONTINUOUS EQUATION In this section we discrtize the weak for (5-7) by using the finite eleents ethod. We suppose for siplicity the operator ( ) is independent of, the doain Ω is polyhedron. For every integer n, let ( ) * + be an adissible regular triangulation of into closed disiplices [5],{ } be subdivision of the interval into N (n) intervals, where [ ] of equal lengths equal Set, Let ( ) be the space of continuous pricewise affine in Ω. Hence, the discrete state equations, for each v V n is written in the for: ( ) ( ) ( ( ) ), (9), (0) ( ) ( ) () ( ) ( ) (2) Where ( ) are given, and ( ), ( ), for Now, suppose the function f is defined on,( ) continuous w.r.t.. for here and up and for brevity we will drop soe ties the agreeent of dependent variable, and any others ters which contain this independent variable. SOLUTION OF THE WAVE EQUATION BY FINITE ELEMENT METHOD To find the solution ( ) for fixed any j ( ), the procedure utilized here, can be described by using the following steps: Step : for fixed any j,( ), let * ( )+ be a finite basis of (where ( ),for are continuous pricewise affine in Ω with ( ) are zero on the boundary Г), then equations(9-2) for any and,,,, can be written in the for: ( ) ( ) ( ( ) ), (3), (4) ( ) ( ), (5) ( ) ( ) (6) 4

4 The Direct and Inverse Probles for the Hyperbolic Boundary Value Proble Jail and Halah Step2: Rewriting (4) in the for Substituting (7) in (3), we have: ( ) ( ) ( ) ( ) ( ( ) ) (8) Step3: Fro the basis of, using Galerkin ethod we write k 0 c k v k j c k vk k j d k vk. k, j c k vk k, k d k v k 0, k d k v k, and (7) Where, ( ) and, ( ), are unknown constants, for each. Step4: Substituting,,,, and in equations (7,8,5,&6) we get the following linear syste of ordinary differential equations: (A+( ) B) =A + A +( ) ( ),.. (9) ( ), (20) AC 0 =e 0 (2) AD 0 =e (22) Where A=( ), ( ), B=( ), ( ) ( ), ( ), ( ), ( ( ) ) ( ), ( ), ( ), ( )and ( ), for each The above linear syste has a unique solution [6]. To solve the linear systes (9) and (20), first we solve the linear systes (2) and (22) to get the unknowns and, then we set in (9) and (20) to get and, then we repeat this procedure to solve (9) and (20), for to get the unknowns and (solution of the direct proble). THE INVERSE PROBLEM OF THE WAVE EQUATION DESCRIPTION OF THE INVERSE PROBLEM The previous section is devoted to the solution of the direct proble for the wave equation in, in which the solution of the proble 42

5 Al- Mustansiriyah J. Sci. Vol. 24, No 5, 203 (discrete wave for) is found over the region Q, when the initial condition y( )is given, while the inverse proble is to deterine the initial condition ( )when the solution of the wave equation is given on (where contains a finite nuber of the points of Q). Where *( ) + In general the unknown initial condition can be expressed by the polynoial ( ) a x i p i0 i (23) where ( ) are unknown constants) MATHEMATICAL STATEMENT OF THE INVERSE PROBLEM FOR THE WAVE EQUATION Before solving the inverse proble to find the unknown initial condition, the region of space variable Ω is assued be a square, hence the unknown initial condition (23) ust be in the for: ( ) ( )( )( )( ) (24) Therefore, our proble becoes to find the unknowns constants (a and b). Now, to solve this proble by using the direct ethod to find these unknowns the proble is transfored to the following discrete leastsquare approxiation M Min ( ) u ( x, x, t) u ( x, x, t) 2 i 0 i 2i ap i 2i (25) where ( ) are the given values of solution of the discrete wave equation at the point ( ) with when all the initial and boundary conditions are known (solution of the direct proble), uap ( x i, x2i, t) are the values of the approxiate solution of the sae proble but when ( ) has the for (24), i.e., the proble becoes to find the unknowns a and b which are in (24), the unconstrained Hook and Jives ethod [7], used to deterine these values. NUMERICAL EXAMPLES EXAMPLE () Consider the following wave equation : ( ), where ( ) associated with the initial and boundary conditions ( ) on where 43

6 Y Y The Direct and Inverse Probles for the Hyperbolic Boundary Value Proble Jail and Halah ( ), ( )( ) ( ) ( ) - The exact solution of this proble is: ( ) ( )( ) By using the finite eleent ethod for M=9, N=00, we get the results which are shown in Table () and Figure () at =0.3, the table shows the approxiate solution ( ) and the exact solution ( ) and the absolute error at x and x 2. Table-: Coparison between exact and approxiation solutions Exact solution Approxiate solution Absolute Error a X X b X X Figure-: (a) shows the exact solution (b) shows the approxiation solution EXAMPLE (2) Consider the following wave equation: ( ), where ( ) associated with the initial and boundary conditions 44

7 Al- Mustansiriyah J. Sci. Vol. 24, No 5, 203 ( ) on Where ( ),( )( ) ( ) ( )- The exact solution of this proble is: ( ) ( )( ) By using the finite eleents ethod for M=9, N=00, we get the results which are shown in Table (2) and Figure (2) at =0.3, which shows the approxiate results ( ) and the exact solution ( ) and the absolute error at the values of x and x 2 which are given in the table. Table-2: Coparison between exact and approxiation solution Exact solution Approxiate solution Absolute Error Exact solution Approxiate solution Absolute Error

8 Y Y The Direct and Inverse Probles for the Hyperbolic Boundary Value Proble Jail and Halah a X X b X X Figure-2: (a) shows the exact solution (b) shows the approxiation solution EXAMPLE (3) Consider the following hyperbolic equation: ( ), where ( ) With the boundary condition ( ) on Г=I əω And the initial conditions ( )( ) Where ( ), ( )( ) ( ) ( ) - In this exaple the inverse proble is to find the unknown (a and b) using the ethod of Hook and Jives when the solution of the direct ethod is given at the points of the set. By using the unconstraint Hook and Jives ethod with step length k=0.5, and different initial values of ( a and b). the results are shown in table (3). Table-3: Different initial values of unknowns ( a and b) and their final values K=0.5, Initial Final Z= ( Z= ( Values Values ) ) a b a b

9 Al- Mustansiriyah J. Sci. Vol. 24, No 5, 203 EXAMPLE (4) Consider the following hyperbolic equation: ( ), where ( ) With the boundary condition ( ) on Г=I əω And initial conditions ( )( ) Where ( ),( )( ) ( ) ( )- By using the ethod of Hook and Jives with step length k= and different initial values of (a and b). The results are shown in table (4). Table-4: Different initial value of unknowns ( a and b) and their final values Initial Values a b Z= ( ) K= Final Values a b Z= ( CONCLUSION In this paper we conclude that the finite eleents ethod is suitable and efficient to solve the direct proble and in the other hand the finite eleents ethod associated with unconstraint Hook and Jives ethod for solving a nonlinear optiization proble at certain tie with different values of space variable is efficient to deterine initial condition associated with the given partial differential equation. It is iportant to ention here that the value of is chose arbitral in the interval I, one can take any other value of provided this value belong to I. REFERENCES. Warin S., and Subsagun Y., Matheatical Inverse Proble of Magnetic Field fro Exponentially Varying Conductive Ground, Applied Matheatic Science, Vol. 6, No. 3, Lio W., An Accurate and Efficient Algorith for Paraeter Estiation of 2D Acoustic Wave Equation, International Journal of Applied Physics, Vol. No. 2, 20. )

10 The Direct and Inverse Probles for the Hyperbolic Boundary Value Proble Jail and Halah 3. Rashedi k., and Yousefi S. A., Ritz Galerkin Method for Solving a Class of Inverse Proble in the Parabolic Equation, International Journal of Nonlinear Science, Vol. 2 No. 4, Al-Hawasy J. A., On the Matheatical Inverse Probles with Applications to the Acoustic Wave Scattering, College of Science, Al-Nahrain University, M. Sc. Thesis, Thoee V., Galerkin Finite Eleent ethods for Parabolic Proble, 997 Springer Verlag Berlin Heidelberg, New York. 6. Al-Hawasy J. A., The Discrete Classical Optial Control Proble of Nonlinear Hyperbolic Partial Differential Equation (DCOCP) Journal Al-Nahrain, Vol.3, No Rao S. S., Optiization: Theory and Application, 2 nd, 984, Wiely, New York. 48