Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr METİN DEMİRALP İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) demiralp@be.itu.edu.tr Abstract: A numerical method based on Fluctuationlessness Approximation, which was developed recently, is constructed or solving Boundary Value Problems o Ordinary Dierential Equations on appropriately deined Hilbert Spaces. The numerical solution is written in the orm o a Maclaurin series. The unknown coeicients o this series are determined by constructing an n 2) unknown containing linear system o equations. The eigenvalues o the independent variable s matrix representation are used in the construction o the matrices and the vectors o the linear system. The numerical solution obtained by Fluctuationlessness Approximation is then compared with the Maclaurin coeicients o the analytical solution to observe the quality o the convergence. Some illustrative examples are presented in order to give an idea about the eiciency o the method explained here. Key Words: Boundary Value Problems, Eigenvalues, Fluctuationlessness Approximation, Hilbert Spaces. 1 Introduction Ordinary Dierential Equations ODEs) are among the important subjects o the Applied Mathematics. They are mathematical models o the physical and engineering problems. Amongst these, Boundary Value Problems BVP) occupy a large portion. They may be the model o the problem itsel, or they may arise ater using the method o separation o variables or Partial Dierential Equations PDEs). The Initial Value Problems IVPs) have unique solutions at least or the case o linearity and or the many cases o nonlinear structures showing no biurcative nature. BVPs may have a unique solution, no solution, or in certain cases, ininitely many solutions even in the case o nonlinearity, depending on the structures o the equations and the accompanying conditions. Despite some BVPs may have easily obtainable analytical solutions the others may involve equations that can not be solved analytically, or even they are solved, the solutions may be quite inconvenient or practical utilization. In the cases where a practically employable analytical solution can not be constructed the numerical solutions help us to ind the very close values to the solution at certain ixed points in the domain o the independent variable. Certain numerical schemes like Shooting Method and Finite Dierence Method have been developed in the past. However, as many numerical solutions have, these solutions may have some drawbacks such as needing a large number o nodes or producing unsatisactory erroneous values. Fluctuationlessness Theorem, which was proven recently, has an impressive eature that, even using ew number o nodes, the numerical solution approximates the analytical solution satisactorily. This theorem simply states that, in the absence o luctuations, the matrix representation o a unction operator, whose action is simply to multiply its operand by the unction value, in any n dimensional subspace o the Hilbert Space under consideration, is equal to the image o the independent variable s matrix representation in the same subspace under that unction. There are three types o BVPs. I only a value to the normal derivative o the unknown unction is given in the boundary conditions then it is a Neumann type boundary condition. I only a value to the unknown unction is given in the boundary conditions then it is a Dirichlet type boundary condition. I the boundary has the orm o a curve or surace that gives a value to the normal derivative o and itsel o the unction then it is a Cauchy type boundary condition or sometimes it is called Mixed condition. The method we develop in this paper does not have any dierence in the type o the boundary conditions o the problem. It is constructed to solve BVPs generally. BVPs can also be deined or nonlinear ODEs, but we will restrict ourselves to a consideration o linear ISSN: 1790-5117 87 ISBN: 978-960-474-083-3
equations only in this work. These problems may also involve an unknown parameter which deines an eigenvalue problem o a linear ordinary dierential operator i appropriate boundary conditions are imposed. Even i this is done the solution may not exist or all values o the unknown parameter. Instead, certain speciic values or the parameter and the structures or the corresponding unction are obtained as solution. What we get as solution is a set o speciic parameter values which are called eigenvalues and corresponding unctions which are called eigenunctions. The set o eigenunctions generally span a Hilbert space which is the domain o the linear ordinary dierential operator mentioned above. We do not consider these types BVPs here in the perspective o our approach. They are let or some uture works. The rest o the paper is organized as ollows. In the second section the Fluctuationlessness Theorem is explained briely. The third section is devoted to the application o this theorem on the numerical solution o BVPs without unknown parameters. The ourth section involves numerical examples with illustrations. The ith section involves the concluding remarks. 2 Fluctuationlessness Theorem We take an n dimensional subspace H n o the Hilbert Space H spanned by the unctions u 1 x), u 2 x),..., u n x) which are analytic and thereore square integrable on the given inite interval a, b. These unctions are irst n elements o the basis unction set or H. We deine the inner product o any two unctions,, g, rom H as ollows:, g) b a dxwx)x)gx), 1) where wx) stands or a weight unction. For an arbitrary unction gx) in H n we can write the ollowing identity g x) P g x). 2) Here P is an operator which projects rom H to the subspace H n. P becomes the unit operator o H n i it is considered rom H n to H n. Now we can consider a new operator, x which multiplies its operand by x and its domain is H. The action o this operator on a unction gx) rom H n can be expressed as ollows: xg x) = g j xu j x) = xp gx). 3) j=1 We can see rom this equation that the unction xg x) may not remain in the subspace H n although P gx) is in this subspace. In such cases x operator causes a space extension. In order to avoid rom this situation, it is better to use P x instead o x. So we can write P xgx) = P xp gx). 4) The operator x res which is the restriction o x rom H n to H n can be explicitly given as ollows: x res P xp. 5) Now we can write the ollowing equation or the operator x. x ) P + x ) P + = P xp + xp +P x + x 6) I we deine x luc as the last three additive terms o the right hand side in the above equality, xp + P x x luc + x, 7) then we can express the operator x as the sum o x res and x luc. Here the operator approaches to 0 operator as n goes to ininity. The approximations by ignoring the terms which contain this operator is called as Fluctuationlessness Approximation. Thereore the x operator can now be expressed as x x res P xp 8) in the luctuationlessness limit. The matrix representation o this operator is X, which has the general term X ij deined by the ollowing expression X ij = u i, P xp ) u j, 1 i, j n. 9) Now we can deine a unction operator which multiplies its operand by a unction x) analytical on the interval a, b as ollows: x) 10) Hence is an algebraic operator in terms o x and its luctuationlessness approximation is given below. x res ) P xp ) 11) ISSN: 1790-5117 88 ISBN: 978-960-474-083-3
The matrix representation o the operator s restriction rom H n to H n is written as ollows X ), 12) M where M stands or the matrix representation o the operator s restricted orm mapping rom H n to H n. 1 5 3 Numerical Solution o Boundary Value Problems in the Fluctuationlessness Theorem Perspective We consider the ollowing BVP composed o a second order linear and inhomogeneous ODE and two accompanying linear and inhomogeneous boundary conditions each o which is given at a dierent endpoint o the interval 0, 1 y + p x) y + q x) y = r x), 13) a 1 y 0) + a 2 y 0) = c, 14) b 1 y 1) + b 2 y 1) = d, 15) where 0 < x < 1. The ODE given by 13) has two linearly independent homogeneous solutions we denote by φ 1 x) and φ 2 x). These are assumed not to have any arbitrary constants. In other words, they are constructed by imposing speciic values to the arbitrary constants o the general solution to the homogeneous orm o 13). The inhomogeneous ODE in 13) has an additional particular solution φ p x) which identically vanishes when the right hand side becomes zero. Thereore the general solution o 13) can be written 6 as yx) = C 1 φ 1 x) + C 2 φ 2 x) + φ p x) 16) The arbitrary constants C 1 and C 2 should take speciic values to satisy the boundary conditions given by 14) and 15) when they are imposed. I this happens then the ollowing equations are obtained where A 11 C 1 + A 12 C 2 = B 1, A 21 C 1 + A 22 C 2 = B 2, 17) A 11 a 1 φ 1 0) + a 2 φ 10) A 12 a 1 φ 2 0) + a 2 φ 20) A 21 a 1 φ 1 1) + a 2 φ 11) A 22 a 1 φ 2 1) + a 2 φ 21) 18) B 1 c a 1 φ p 0) a 2 φ p0) B 2 d a 1 φ p 1) a 2 φ p1) 19) These mean that the existence and uniqueness related properties in the C 1 and C 2 constants are determined completely by the character o the equation set in 17). The equation set may or may not have solutions depending on the coeicient matrix composed o the elements A 11, A 12, A 21, and A 11, and the right hand side vector whose elements are B 1 and B 2. The solution can exist only when the right hand side vector o the equation set is orthogonal to the let nullspace o the coeicient matrix. Otherwise there is an incompatibility in the set o the equations. This means that the given boundary conditions are incompatible with the ordinary dierential equation they accompany. In the case o compatibility there may be two possibilities: 1) The let and right nullspaces o the coeicient matrix are empty. This makes the solutions or C 1 and C 2 and thereore or whole BVP unique; 2) The rank o the coeicient matrix is just 1. This means that the let and right nullspaces are one dimensional and a one parameter arbitrariness appear in the solutions or C 1 and C 2 and thereore or the solution o the BVP. The case where the rank o the coeicient matrix corresponds to the situation where no boundary conditions are imposed. We will ocus on the BVPs where the boundary conditions are compatible to the ODE and produces unique solution. To ind the solution numerically, we propose the ollowing structure: yx) = x) = k x k 20) k=0 I we substitute this solution in the equations 13), 14) and 15), we obtain the ollowing equations: r x) = 0 q x) + 1 q x) x + p x) + 2 q x) x 2 + 2p x) x + 2 + 3 q x) x 3 + 3p x) x 2 + 6x +... + n q x) x n + np x) x n 1 +n n 1) x n 2 21) c = a 1 0 + a 2 1 22) d = b 1 0 + b 1 + b 2 ) 1 + b 1 + kb 2 ) k 23) k=2 For the numerical solution we will apply Fluctuationlessness Theorem in the interval 0, 1. We write the ollowing equation or 13) in terms o operators as ISSN: 1790-5117 89 ISBN: 978-960-474-083-3
ollows: M r e 1 = M + M p M +M q M e 1, 24) where M stands or the matrix representation o the operator in the Hilbert Space H n. By Fluctuationlessness Theorem we know that M X ). 25) The matrix X is symmetric and its spectral representation can be written as ollows: X = ξ i x i x T i 26) i=1 Here ξ i is the i th eigenvalue and x i is its eigenvector with unit norm. Substituting 25) in 24) we obtain the ollowing result. i=1 ξ i ) + p ξ i ) ξ i ) +q ξ i ) ξ i ) r ξ i ) x T i e ) 1 x i = 0 27) Since the eigenvectors are linearly independent, this can be satisied only when the coeicients o x i s at the let hand side are set equal to zero. So we can write the ollowing equations: ξ i )+p ξ i ) ξ i )+q ξ i ) ξ i ) r ξ i ) = 0 28) To ind the unknown constants, i s, 2 i n) in 21) we construct a set o vectors and matrices as ollows q ξi+1 ) ξ K 1 ij i+1, i = j p ξi+1 ), i = j K 2 ij i + 1, i = j K 3 ij i i + 1), i = j K 4 ij V 1 V 2 ξ 2 ξ2 2... ξ2 n 1 ξ 3 ξ3 2... ξ n 1 3......, ξ n ξn 2... ξn n 1 1 ξ 2 ξ2 2... ξ2 n 2 1 ξ 3 ξ3 2... ξ n 2......... 1 ξ n ξn 2... ξn n 2 3, a i r T T 0 q ξ i+1 ) +, 1 p ξ i+1 ) + q ξ i+1 ) ξ i+1 ) r ξ 2 )... r ξ n ), 2... n, 29) where the matrices K 1, K 2, K 3, K 4, and the vector a are given through elementwise deinitions. We can write the equation 21) in terms o these matrices and vectors as ollows: r = K 1 V 1 + K 2 V 1 K 3 + V 2 K 4 ) + a 30) Since the eigenvalues are distinct, the matrices V 1 and V 2 are invertible. Thereore we can ind by using the ollowing ormula: = K 1 V 1 + K 2 V 1 K 3 + V 2 K 4 ) 1 r a) 31) When we substitute the values o in the equation 16) we obtain the ollowing equation. x) = 0 + 1 x + k x k 32) k=2 Here all the values except 0 and 1 are known. To ind 0 and 1, the equations 22), 23) and 32) can be solved together. Hence we obtain the numerical solution. By comparing the coeicients o this solution with the Maclaurin coeicients o the analytical solution, we can see the quality o the approximation. 4 Numerical Implementation In this section we are going to give certain numerical examples. For the implementations some linear second order Boundary Value Problems are chosen, their exact and numerical solutions are presented in igures comparatively. The mesh points we used or calculations are chosen as the eigenvalues o the independent variable s matrix representation X. As can be shown by using Rayleigh ratios and integrals in the deinitions o the elements o this matrix, these eigenvalues lie inside the interval 0, 1. We used Mathematica 5.2 or calculations. We obtain the results on 10 grid points or the solutions o the dierential equations. Our irst problem is deined as ollows: y x) = 6x, y 0) = 0, y 1) = 1. 33) The analytical solution or this equation is easily ound to be y x) = x 3. The coeicients o the numerical solution are given as x) = 0.0x 2.07 10 16 x 2 + 1.0x 3 +1.0 10 16 x 4 + 8.5 10 17 x 5. 34) ISSN: 1790-5117 90 ISBN: 978-960-474-083-3
Since the analytical solution is a polynomial, its Maclaurin series equals to itsel. Hence we can observe that a high quality approximation, an almost exact match is achieved successully. A comparative igure o the solutions is displayed below: 0.6 0.5 0.4 0.3 y Analytical Numerical y 1 0.8 0.6 0.4 0.2 Analytical Numerical 0.2 0.4 0.6 0.8 1 x Figure 2: yx) = eπ2 x) e πx e 2π 1 0.2 0.1 0.2 0.4 0.6 0.8 1 x Figure 1: yx) = x 3 The second problem we consider here is given as y x) π 2 y x) = 0, y 0) = 1, y 1) = 0. 35) The analytical solution or this equation is y x) = eπ2 x) e πx e 2π. 36) 1 Numerical solution obtained by the Fluctuationlessness Theorem is ound to be as ollows: x) = 1 3.1533481x + 4.9348015x 2 5.18702x 3 + 4.0583x 4 2.557x 5 + 1.32x 6 0.58x 7 +0.20x 8 0.05x 9 + 0.006x 10 37) Maclaurin coeicients o the analytical solution are given through the ollowing equality: y x) = 1 3.15334809x + 4.934802x 2 5.18704x 3 + 4.0587x 4 2.559x 5 +1.33x 6 0.60x 7 + 0.23x 8 0.08x 9 + 0.02x 10 + O x 11) 38) The analytic and approximate solutions are depicted together in Figure 2. The last example we study in this section is given via the ollowing equation y x) + 2y x) = 0, y 0) = 1, y 1) + y 1) = 0. 39) The analytical solution o this problem is ) sin 2x y x) = ) sin 2 + ) x 2 cos 2 2 40) The numerical solution is obtained as x) = 0.670412404x 2.3 10 10 x 2 0.39013745x 3 1.12 10 7 x 4 +0.039014x 5 2.8 10 6 x 6 0.001851x 7 9.6 10 6 x 8 +6 10 6 x 9 4.2 10 6 x 10. 41) Maclaurin coeicients o the analytical solution can be taken rom the coeicients o the ollowing equality y x) = 0.670412405x 0.390137468x 3 +0.039013x 5 0.001857x 7 +5.1 10 6 x 9 + O x 11). 42) The comparison o the exact and approximate solutions can be seen in Figure 3. 0.35 0.3 0.25 0.2 0.15 0.1 0.05 y Figure 3: yx) = 5 Conclusion Analytical Numerical 0.2 0.4 0.6 0.8 1 x sin 2x) sin 2)+ 2 cos 2) x 2 In this work, we developed a new method based on the Fluctuationlessness Theorem or approximating the solution o ODE Boundary Value Problems over the interval 0, 1 numerically. The coeicients in the numerical solution are obtained very close to the ISSN: 1790-5117 91 ISBN: 978-960-474-083-3
Maclaurin coeicients o the exact solution. The results obtained at the grid points have the precision about 10 9. The method described here is a new study or the Boundary Value Problems o ordinary dierential equations. For the uture work our goal is to solve numerically Boundary Value Problems containing an unknown parameter and are in act eigenvalue problems. Reerences: 1 Altay, N., Demiralp, M., Numerical Solution o Ordinary Dierential Equations in Fluctuationlessness Theorem Perspective, Proceedings o the 1st WSEAS international Conerence on Multivariate Analysis and its Application in Science and Engineering MAASE 08), May 27 30, Istanbul, Turkey, 2008, pp.162-167. 2 Altay, N., Demiralp, M., Application O Fluctuationlessness Theorem On The Numerical Solution O Higher Order Linear Ordinary Dierential Equations, Proceedings o the 6th International Conerence o Numerical Analysis and Applied Mathematics ICNAAM 2008), September 16-20, Kos, Greece, 2008, pp. 52-55. 3 Demiralp, M., A Fluctuation Expansion Method or the Evaluation o a Function s Expectation Value, Int. Con. on Numer. Anal. and Appl. Math., Wiley, Rhodes, Greece, Sept. 16 20, 2005, pp. 711-714. 4 Demiralp, M., Determination o Quantum Expectation Values Via Fluctuation Expansion, Lecture Series on Computer and Computational Sciences, Selected Papers rom Int. Con. o Comput. Methods in Sci. and Eng. ICCMSE 2005), Loutraki, Greece, Sept. 16 20, Redderbrick BV, 4A, 2005, pp. 146-149. 5 Baykara, N. A., Demiralp, M., Fluctuation Expansion in the Quantum Optimal Control o One Dimensional Perturbed Harmonic Oscillator, Lecture Series on Computer and Computational Sciences, Selected Papers rom Int. Con. o Comput. Methods in Sci. and Eng. ICCMSE 2005), Loutraki, Greece, Sept. 16 20, Redderbrick BV, 4A, 2005, pp. 56-59. 6 Boyce, W. E., DiPrima, R. C., Elementary Dierential Equations And Boundary Value Problems, John Wiley and Sons Inc., 1997, pp. 625-628. ISSN: 1790-5117 92 ISBN: 978-960-474-083-3