ON A NUMERICAL SOLUTION OF THE LAPLACE EQUATION

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1 ON A NUMERICAL SOLUTION OF THE LAPLACE EQUATION Jasmina Veta Buralieva (joint work with E. Hadzieva and K. Hadzi-Velkova Saneva ) GFTA 2015 Ohrid, August 2015 (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

2 Outline - Classical Galerkin method for ODE - Wavelets and MRA - Wavelet-Galerkin method for ODE - Transformation of the Laplace DE - Application of W-G method on the Laplace DE (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

3 Classical Galerkin metod for ODE Classical Galerkin metod for ODE Sturm-Liouville equation with BC Lu(t) d dt ( g(t) du dt ) + h(t)u(t) = f (t), a t b (1) u(a) = c, u(b) = d. (2) 1) {v j } - complete orthonormal system for L 2 ([a, b]) 2) every v j C 2 ([a, b]) 3) v j (a) = c, v j (b) = d. Approximation u s of the exact solution u u s = k Λ x k v k. (3) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

4 Classical Galerkin metod for ODE Criterion for coefficients x k < Lu s, v j >=< f, v j >, j Λ. (4) If we substitute the equation (3) in (4) we obtain Lv k, v j x k = f, v j, j Λ. (5) k Λ A = [a j,k ] j,k Λ, a j,k = Lv k, v j ; X = (x k ) k Λ ; Y = (y k ) k Λ, y k = f, v k AX = Y. (6) Wavelet-Galerkin method: functions v j are wavelets (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

5 Wavelets Wavelets wavelet ψ: L 2 function which satisfy the admissibility condition C ψ = The condition (7) implies that ˆψ(0) = ˆψ(ω) 2 dω <. (7) ω ψ(t)dt = 0. wavelets ψ a,b (t) = 1 a ψ ( t b a ), a > 0, b R. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

6 Multiresolution analysis Multiresolution analysis (MRA) Multiresolution analysis of the space L 2 (R) consists of a sequence of closed subspace {V j } j= with the following properties: 1. V j V j+1 2. j Z V j = L 2 (R) 3. j Z V j = {0} 4. f (t) V j f (2t) V j+1 5. f (t) V j f (t k) V j, k Z 6. there exists a function φ (called scaling function or father wavelet) such that φ j,k (t) = 2 j/2 φ(2 j t k), k Z constitute orthonormal basis for corresponding subspace V j. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

7 Multiresolution analysis Let φ L 2 (R) be compactly supported scaling function of MRA. Then 1) φ(t)dt 0 2)φ(t) = k Z a kφ(2t k), where a k are real coefficients and a k 0 for only finitely many k Z (the number of nonzero coefficients a k is denoted by L). 3) φ j,k (t) = 2 j/2 φ(2 j t k), j, k Z are orthonormal in L 2 (R) i.e. φ(t n)φ(t k)dt = δ k,n, (8) where δ n,k = { 0, n k 1, n = k. (9) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

8 Multiresolution analysis One can construct wavelet ψ such that ψ j,k (t) = 2 j/2 φ(2 j t k), j, k Z constitute an orthonormal basis for L 2 (R). Daubechies scaling function Daubechies wavelet function ψ(t) = L 1 φ(t) = a k φ(2t k) (10) 1 k=2 L k=0 ( 1) k a 1 k φ(2t k) (11) where L is a positive even integer and denotes the genus of the Daubechies wavelet. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

9 Wavelet-Galerkin method for ordinary differential equations Wavelet-Galerkin method for ODE g(t)u (t) + g (t)u (t) + h(t)u(t) = 0, t [a, b], (12) with BC u(a) = c, u(b) = d. (13) Approximate solution u j (t) = 2 j k=1 L where φ is the scaling function of MRA. c k φ j,k (t), k Z, (14) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

10 Wavelet-Galerkin method for ordinary differential equations Remark There are no closed-form formulas for the Daubechies wavelets and scaling functions. W-G method with Daubechies scaling functions: homogeneous differential equations. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

11 Wavelet-Galerkin method for ordinary differential equations For j = 0 and L = 4 g(t) d 2 dt 2 u 0 (t) = 1 k= 3 1 k= 3 c k φ(t k), t [a, b]. (15) c k φ(t k) + g (t) d dt +h(t) 1 k= 3 1 k= 3 c k φ(t k)+ c k φ(t k) = 0. (16) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

12 Wavelet-Galerkin method for ordinary differential equations Taking inner product with φ(t n), n { 3, 2, 1, 0, 1}, we obtain c k Ω n k + c k a n,k + c k s n,k = 0, (17) k= 3 k= 3 k= 3 where Ω n k = a n,k = g(t)φ (t k)φ(t n)dt, (18) g (t)φ (t k)φ(t n)dt, (19) s n,k = 4 3 h(t)φ(t k)φ(t n)dt. (20) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

13 Wavelet-Galerkin method for ordinary differential equations By using BC (13) we obtain and u 0 (a) = u 0 (b) = 1 k= 3 1 k= 3 c k φ(a k) = c (21) c k φ(b k) = d (22) We replace the first and the last equation of system (17) by (21) and (22), respectively and obtain the matrix equation TC = B (23) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

14 Wavelet-Galerkin method for ordinary differential equations T = φ(a + 3) φ(a + 2) Ω a 2, 3 + s 2, 3 Ω a 2, 2 + s 2, 2 Ω a 1, 3 + s 1, 3 Ω a 1, 2 + s 1, 2 Ω a 0, 3 + s 0, 3 Ω a 0, 2 + s 0, 2 φ(b + 3) φ(b + 2) φ(a + 1) φ(a) φ(a 1) Ω a 2, 1 + s 2, 1 Ω a 2,0 + s 2,0 Ω a 2,1 + s 2,1 Ω a 1, 1 + s 1, 1 Ω a 1,0 + s 1,0 Ω a 1,1 + s 1,1 Ω a 0, 1 + s 0, 1 Ω 0 + a 0,0 + s 0,0 Ω a 0,1 + a 0,1 φ(b + 1) φ(b) φ(b 1) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

15 Wavelet-Galerkin method for ordinary differential equations C = c 3 c 2 c 1 c 0 c 1, B = a b. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

16 Transformation of the Laplace DE Transformation of the Laplace DE The famous Laplace equation with substitutions u 2 u x u y u z 2 = 0 (24) has the form x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, u r ( r 2 u ) + 1 r sin θ ( sin θ u ) + 1 θ θ sin 2 θ 2 u = 0. (25) ϕ2 (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

17 Transformation of the Laplace DE Fourier method subsumes u = u(r, θ, ϕ) can be represented u(r, ϕ, θ) = R(r)Φ(ϕ)Θ(θ) ΦΘ d dr (r 2 R ) + RΦ 1 d sin θ dθ (sin θθ ) + 1 sin 2 RΘΦ = 0, θ 1 R d [ 1 dr (r 2 R ) = Θ 1 sin θ d dθ (sin θ Θ ) + 1 ] sin 2 θ Φ, (26) Φ 1 R d dr (r 2 R ) = λ, 1 Θ 1 sin θ d dθ (sin θ Θ ) + 1 sin 2 θ Φ Φ = λ. (27) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

18 Transformation of the Laplace DE System of three ODEs 1 Θ sin θ d dθ dθ (sin θ 1 R d dr (r 2 R ) = λ Φ Φ = µ dθ ) + λ sin2 θ = µ (28) The first ODE is Cauchy-Euler equation r 2 R + 2rR λr = 0. (29) Its exact solution is for R(r) = C 1 r n + C 2 1 r n+1 λ = n(n + 1). (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

19 Transformation of the Laplace DE The second ODE is Sturm-Liouville equation general solution sin θ d dθ Its exact solution is Φ + µφ = 0, (30) Φ(ϕ) = A cos mϕ + B sin mϕ, for µ = m 2, where m = 1, 2,.... The third ODE is Sturm-Liouville equation ( sin θ dθ dθ ) ( ) + Θ λ sin 2 θ µ = 0. (31) P n,m = (1 x 2 ) m d m P n(x) 2 dx m = (1 x 2 ) d n+m n!2 n dx n+m [(x 2 1) n ], (32) for λ = n(n + 1) and µ = m 2 where P n (x) = 1 n!2 n d n dx n [(x 2 1) n ], x = cos θ. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34 m 2

20 Application of the W-G method on the Laplace equation Application of the W-G method on the Laplace equation satisfies 1 2 t2, t [0, 1] t φ(t) = 2 3t 3 2, t [1, 2] 1 2 t2 3t + 9 2, t [2, 3] 0, t / [0, 3] (33) so L = 4. φ(t) = 1 4 φ(2t) φ(2t 1) φ(2t 2) + 1 φ(2t 3), 4 (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

21 Application of the W-G method on the Laplace equation 1. λ = 0 and µ = 4 The first ODE is with the BC: R(1) = 1, R(3) = 0. Approximate solution is r 2 R + 2rR = 0 (34) R 0 (r) = { c 1 φ(r + 1) + c 0 φ(r) + c 1 φ(r 1), r [1, 2] c 0 φ(r) + c 1 φ(r 1), r [2, 3] where c 1 = , c 0 = , c 1 = 0. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

22 Application of the W-G method on the Laplace equation Table: Comparison of results Case t numerical solution R 0 exact solution R absolute error (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

23 Application of the W-G method on the Laplace equation (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

24 Application of the W-G method on the Laplace equation The second ODE is with BC: Φ(0) = 1, Φ( π 4 ) = 1. Approximate solution is Φ + 4Φ = 0 (35) Φ 0 (ϕ) = c 2 φ(ϕ + 2) + c 1 φ(ϕ + 1) + c 0 φ(ϕ), ϕ [0, π 4 ] where c 2 = , c 1 = , c 0 = (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

25 Application of the W-G method on the Laplace equation Table: Comparison of results Case t numerical solution Φ 0 exact solution Φ absolute error π/ (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

26 Application of the W-G method on the Laplace equation (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

27 Application of the W-G method on the Laplace equation The third ODE is with BC: Θ(1) = 1, Θ(2) = 2. sin 2 (θ)θ + cos(θ) sin(θ)θ 4Θ = 0 (36) (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

28 Application of the W-G method on the Laplace equation 2. For λ = 0 and µ = 2 The second ODE is with BC: Φ(0) = 1, Φ( π 4 ) = 1. Approximate solution is Φ + 2Φ = 0 (37) where Φ 0 (ϕ) = c 2 φ(ϕ + 2) + c 1 φ(ϕ + 1) + c 0 φ(ϕ), ϕ [0, π 4 ] c 2 = , c 1 = , c 0 = (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

29 Application of the W-G method on the Laplace equation The third ODE is with BC: Θ(1) = 1, Θ(2) = 2. Approximate solution is sin 2 (θ)θ + cos(θ) sin(θ)θ 2Θ = 0 (38) Θ 0 (θ) = c 1 φ(θ + 1) + c 0 φ(θ) + c 1 φ(θ 1), θ [1, 2] where c 1 = 2.427, c 0 = , c 1 = (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

30 Application of the W-G method on the Laplace equation Table: Numerical results Case t numerical solution Φ 0 Case t numerical solution Θ π/ // // // // 2 2 (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

31 Application of the W-G method on the Laplace equation 3. λ = 1 and µ = 4 The first ODE is with BC: R(1) = 1, R(3) = 0. Approximate solution is r 2 R + 2rR R = 0 (39) where R 0 (r) = { c 1 φ(r + 1) + c 0 φ(r) + c 1 φ(r 1), r [1, 2] c 0 φ(r) + c 1 φ(r 1), r [2, 3] c 1 = , c 0 = , c 1 = 0. (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

32 Application of the W-G method on the Laplace equation The third ODE is sin 2 (θ)θ + cos(θ) sin(θ)θ + Θ with BC: Θ(1) = 1, Θ(2) = 2. Approximate solution is ( ) sin 2 (θ) 4 = 0 (40) Θ 0 (θ) = c 1 φ(θ + 1) + c 0 φ(θ) + c 1 φ(θ 1), θ [1, 2] where c 1 = , c 0 = , c 1 = (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

33 Application of the W-G method on the Laplace equation Table: Numerical results Case t numerical solution Φ 0 Case t numerical solution Θ (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

34 References A. H. Siddiqi, Applied Functional Analysis: Numerical Methods, Wavelet Methods and Image Processing, Markel Dekker, New York, Anandita D., A wavelet-galerkin method for the solution of partial differential equation, master thesis, 2011 C. Qian, J. Weiss, Wavelets and the numerical solution of partial differential equations, Journal of Computational Physics, Vol. 106, Issue 1, 1993, pp D. S. Mitrinovič, J. D. Kečki: Jednačine matematičke fizike, Beograd, I. Daubeshies, Ten lectures on Wavelets, Philadelphia: SIAM, I. Daubechies, Orthonormal bases of compactly suppoted wavelets, Commun. Pure Appl. Math., 41, 1988, pp G. G. Walter, X. Shen, Wavelets and Other Orthogonal Systems With Application, CRS Press, Secon Editiotn,2000 M. W. Frazier, An Introduction to Wavelets Through Linear Algebra, Springer-Verlag, New York, S. Kostadinova, J. Veta Buralieva, K. Hadzi-Velkova Saneva (2013), Wavelet-Galerkin solution of some ordinary differential equation,proceedings of XI Internation Conference ETAI 2013, September 2013, Ohrid, Macedonia. S. Mallat, Multiresolution approximation and wavelets, Trans. Amer. Math. Soc., 315, 1989, pp T. Lofti, K. Mahdiani, Numerical solution of boundary value problem by using wavelet-galerkin methods, Mathematical Sciences, Vol. 1, No. 3, 2007, pp Vladimirov V. S., Uravneninija matematicheskori fiziki, Nauka, Moskva,1967. V. Mishra, Sabina, Wavelet Galerkin solutions of ordinary differential equations, Int. Joirnal of Math.Analysis, Vol. 5(9), 2001, (Faculty of Informatics,UGD-Stip) On a numerical solution of the Laplace equation GFTA, / 34

Ninth International Water Technology Conference, IWTC9 2005, Sharm El-Sheikh, Egypt 673

Ninth International Water Technology Conference, IWTC9 2005, Sharm El-Sheikh, Egypt 673 A NEW NUMERICAL APPROACH FOR THE SOLUTION OF CONTAMINANT TRANSPORT EQUATION Mohamed El-Gamel Department of Mathematical