On Nonlinear Methods for Stiff and Singular First Order Initial Value Problems

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1 Nonlinear Analysis and Differential Equations, Vol. 6, 08, no., 5-64 HIKARI Ltd, On Nonlinear Methods for Stiff and Singular First Order Initial Value Problems A. O. Adesanya, T. P. Pantuvo and D. Umar Department of Mathematics Modibbo Adama University of Technology Yola, Nigeria Copyright c 08 A. O. Adesanya, T. P. Pantuvo and D. Umar. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper considers development of nonlinear method by interpolation and collocation of inverse polynomial approximate solution to give systems of equations, solving for the unknown parameters and substituting the results into the approximate solution gives a continuous method. We evaluated the continuous method at different grid points to give discrete methods which are implemented without developing separate predictors. The method is convergent and A-stable, numerical examples show that the method is good for stiff and singular problems. Mathematics Subject Classification: 65L05, 65L06 Keywords: continuous nonlinear method, inverse approximate solution, discrete method, stiff problems, singular problems Introduction Ordinary Differential Equations (ODEs frequently occur in mathematical models that arise in many branches of sciences, engineering and economics, unfortunately many of such models cannot be solved exactly, so it is common to seek approximate solutions. When the solutions to Initial Value Problems Corresponding author

2 54 A. O. Adesanya, T. P. Pantuvo and D. Umar (IVPs are known in advance to have some special properties, then approximate solution which make use of these properties can be used to develop method of solution. The special properties may be some analytical properties of the function, such as stiffness; it may be the solutions of the real life problems such as problems with oscillatory solutions or whose solutions possess singularity. For example, if an IVPs whose solutions are known to be periodic, or to oscillate with a known frequency, then numerical integration formulae based on trigonometric functions are appropriate. On the other hand, if the problems whose solutions posses singularity, then numerical integration formulae based on rational functions will be effective (Lambert [] This paper considers the solution to y = f(x, y, y (x n = η 0 x n x x N ( where f : R R m R m is a given real valued piecewise continuous function in the interval x [x n, x N ] and assumed to satisfies Lipschitz existence and uniqueness theorem using nonlinear method through inverse polynomial approximate solution. Among the authors that developed nonlinear methods for the solution of (. include; Otunta and Nwachukwu [] considered the approximate solution y n = a 0 + a x + a x + a x + b x + b x + b x + b x 4 ( to develop a one step L-stable method of order 7. The accuracy in the neighborhood of the singular point was impressive. Gadella and Lara [] considered three types of equations = n k=0 p jk (x x k j + n j=0 r jk (x x k j ( The first case was a rigid equation, the second case was one with a pole in the solution and the third equation was equation with a singular regular point. In all cases, it was shown that their method possesses better approximation, more efficient and needs shorter execution time than the Taylor method. Ying et al. [4] considered y n = a 0 + a x + a x (4 b 0 + b x to develop a rational method using three points x n, x, x. The method is of order, with the finite region of absolute stability not A-stable. Fatunla [5] considered A y (x = + k n= a (5 nx n

3 On nonlinear methods for stiff and singular IVP 55 within the interval (x n, x, x at the mesh points x j, j = 0(k. to developed a nonlinear method, the method was reported to be efficient on stiff and singular IVPs It most be noted that the methods reviewed above are developed through Taylor series expansion (discretely formulated. Continuous formulation of method has been well developed using linear methods but it application to nonlinear method has not been well established in literature, this paper therefore considers the continuous formulation of nonlinear method. kindly refer to Awoyemi et al. [6] for advantages of continuous formulation of methods Mathematical Background We consider an approximate solution in the form (5, a n R, are constants to be determined within the interval of integration π N < x < < x N+ < x N = b with a constant steplength given by h = x j+ x j, j = 0,,, N. Evaluating (5 at x j, j = 0,,..., r and it first derivative y = ( y (x k na n x n + y (x n= k a n x n n= (6 where at x j,, j = 0,,..., s gives a system of equation XB = U (7 B = [ ] T [ A a a a v, U = yn,,..., r, y n,,,... s (8 y n x n y k n n= a nx k n... r x r y k r n= X = a nx k r 0 ( ( x n y n + y nx k n y k n n= na nx n + y k n n= a nx k n... 0 ( x s s + sx k s (y k s n= na nx n s + s k n= a nx k s ] T v = r +s. Solving for the unknown parameters and substituting the results into the approximate solution, after factorization gives the continuous method.

4 56 A. O. Adesanya, T. P. Pantuvo and D. Umar. Analysis of the method.. Order Definition We associate the operator l with the non-linear method defined by l [y (x : h] = t y (x t = 0 (9 where y (x is an arbitrary function continuously differentiable on [a, b]. Following Fatunla [5], we can write terms in (9 as a Taylor series expansion about the point x to obtain the expansion l [y (x : h] = c 0 y (x + c hy (x c p h p y p (x +... (0 where the constant coefficients c p = p = 0,,,... are given as [ r ] c p = j p r Φ j j p Ψ j p! (p! j= j= ( (9 has order p if l [y (x : h] = 0 ( h p+, c 0 = c =... = c p = 0, c p+ 0 ( therefore c p+ is the error constant and c p+ h p+ y p+ is the Local Truncation Error (LTE... Consistency Definition ( A methods is said to be consistent if (i it has order p (ii yj y lim n h 0 h = jy n... Zero stability Definition A methods is said to be zero stable if lim h 0 j = y n..4 Convergence Definition 4 A methods is said to be convergent if (i lim h 0 (j y n 0 (ii it is consistent and zero stable..5 A-stability Definition 5 A methods is said to be A-stable if lim z (R (z, this implies that the method is bounded. R (z is the stability function, z = λh, λ = f y

5 On nonlinear methods for stiff and singular IVP 57. Specification of the method Interpolating at x n and collocating at x j, j = 0, u, v,and ω, 0 < u < v < ω, (5 reduces to y (x = A + 4 n= a nx n ( solving for a ns using Crammer s rule and substituting into ( gives the continuous nonlinear which is evaluated at ω to give w = G H + H + H + H 4 + H 5 The parameters are given in the Appendix. In this paper, we considered four cases among others that belong to this family Case One step method ( u =, v =, ω =, the discrete method gives = y n 6 +6h f f 6y n +4h f +6h f n f 7hf n y + 7hf + h f f h f n f 4hy n f + h f f f + h y n f + 57h f f + h f + h f n f y f 57hy n f f (4 Case Two step hybrid method, the hybrid point is at x v ( u =, v =, ω =, the discrete method gives = y n 4 + 8hf 9h f +4h f f 7y n +4hf n + 8hf + h f f f + h f f + h f n f f 9h f n f + hy n f h f f 4h f n f 8h y n f f + 7hy n f (5

6 58 A. O. Adesanya, T. P. Pantuvo and D. Umar ( Case Two step hybrid method, the hybrid point is at x u u =, v =, ω = 44 y + 6hf y y + 47h f y +8h f y + 9h f f f +h f f = yn y + h f f y + h f f 44y n y 9h f n f f + h f n f y +h f n f 6hf n y 8hy n f + h y n f f 47hy n f (6 Case 4 Three step method (u =, v =, ω =, the discrete method gives 4 + 9hf + 8hf +9hf + 6h f f f = yn +h f f + 6h f f + h f f 4y n 6h f n f f + h f n f +6h f n f + h y n f f 9hf n 8hy n f 9hy n f (7 The results of analysis of the methods are shown in Table I. Table I: Results of Analysis of the Methods Method Order [ LTE ] R (z 0(y n 5 + y 9 0 h5 n 0y ny n 7 40(y n y ny n [ 0y ny n yn + 90y n(y n yn + 60(y n y n yn ] 4 0(y 4 n 5 + y 9 5 h5 n 0y ny n 7 40(y n y ny n [ 0y ny n yn + 90y n(y n yn + 60(y n y n yn ] 4 0(y 4 n 5 + y 9 5 h5 n 0y ny n 7 40(y n y ny n [ 0y ny n yn + 90y n(y n yn + 60(y n y n yn ] 9 0(y 4 n 5 + y 9 0 h5 n 0y ny n 7 40(y n y ny n 0y ny n yn + 90y n(y n yn + 60(y n y n yn 54z+z +z z z +z 08 7z+z +z +4 z 7z +z 4 48z+9z +z z 9z +z 48 8z+z +z + 8z z +z Numerical Examples The following notations are used in the tables ST IM, ST IM, ST IM, and ST IM are Cases,,, and 4 respectively, tt ( ττ = tt 0 ττ

7 On nonlinear methods for stiff and singular IVP 59 Example linear stiff system within the interval [0; 0] ( ( ( y (x 00y (x y y = (x y (0, (x 0.y (x y (x y (0 = ( the exact solution is given as ( ( y (x 0.0 exp ( 0.99x + exp ( x = y (x 0 exp ( 0.99x We solved this problem at h = 0.0, 0.00, 0.000, the comparison of our results with the Block Extended Backward Differentiation Formula (BEBDF of Musa et al. [7] as shown in Table II confirmed the accuray of our methods Table II: Results of Example h STIM STIM STIM STIM BEBDF ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 004 Example Nonlinear stiff system, x [0, 0], N = 500 ( ( ( y (x 00y (x + 000y y = (x y (0, (x y (x y (x ( + y (x y (0 The exact solution ( y (x y (x = ( exp ( x exp ( x = ( We compared our results with the Second Derivative Multistep Method of order 0 (SDMM 0 by Yakubu and Marcus [8]. Our methods gave better approximation despite the low order as shown in Table III Table III: Results of Example N y i STIM STIM STIM STIM SDMM 0 50 y ( ( ( ( ( 04 y ( ( ( ( ( 0 50 y.675 ( ( ( ( ( 05 y.8589 ( ( ( ( ( y.7077 ( (.7077 ( (.608 ( 09 y.5760 ( (.5760 ( (.608 ( 09 Example A linear stiff problem ( ( y (x y y = (x (x 00y (x 0y (x ( y (0, y (0 = (

8 60 A. O. Adesanya, T. P. Pantuvo and D. Umar with the exact solution ( y (x y (x ( 0.0e = 00x + e x e 00x e x The comparison of our results with the one step nonlinear method of Ying and Yacoob (OSNLM [9] in Table IV confirmed that our methods are better. Table IV: Results of Example N STIM STIM STIM STIM OSNLM ( (.8779 ( ( ( ( ( ( ( ( ( ( ( ( ( 0 Example 4 A singular problem in the interval 0 x,with the singular point at x = with the exact solution y (x = y (x, y (0 = y (x = x The comparison of our results with the 4 Step Rational Method (4SRM of Adeboye and Umar [0], Table V confirmed that our methods are better. Table 5: Results of Example 4 x STIM STIM STIM STIM 4SRM ( ( ( ( 5.0 ( ( ( ( ( 5.0 ( ( ( ( ( 4.0 ( ( ( ( ( 4 6. ( ( ( ( (.5 ( 05 4 Discussion of Results We have discussed the development of continuous formulation of nonlinear methods for the solution of singular and stiff IVPs. The method adopted in this paper enables more methods to be developed without cost. All the methods are convergent with good stability properties. Results of numerical examples justified the supremacy of the methods over the existing methods.

9 On nonlinear methods for stiff and singular IVP 6 References [] J. D. Lambert, Computational Methods For Ordinary Differential Equations, John Wiley, New York, 97. [] F. O. Otunta and G, C. Nwachukwu, Rational one-step Numerical Integrator for Initial Value Problems in Ordinary Differential Equations, Journal of the Nigerian Association of Mathematical Physics, 9 (005, [] M. Gadella and L. P. Lara, A Numerical Method for Solving ODE by Rational Approximation, Applied Mathematical Sciences, 7 (0, no., [4] T. Y. Ying, O. Zurmi and K. H. Mansor, Rational Block Method for the Numerical Solution of First Order Initial Value Problem : concept and Ideas, Global Journal of Pure and Applied Mathematics, 4 (006, no., [5] S. O. Fatunla, Nonlinear Multistep Methods for Initial Value Problems, Comput. Math. Appl., 8 (98, [6] D. O. Awoyemi, E. A. Adebile, A. O. Adesanya and T. A. Anake, Modified block method for the direct solution of second order ordinary differential equations, Intern. J. Appl. Math. Comp., (0, no., [7] H. Musa, M. B. Suleiman and N. Senu, Fully Implicit -Point Block Extended Backward Differentiation Formula for Stiff Initial Value Problems, Applied Mathematical Sciences, 6 (0, no. 85, [8] D. G. Yakubu and S. Marcus, The efficiency of second derivative multistep methods for the numerical integration of stiff systems, J. Nig. Math. Soc., 5 (06, [9] T. Y. Ying and N. Yaacob, A new class of rational multistep methods for solving initial value problem, Malaysian Journal of Mathematical Sciences, 7 (0, no., -57. [0] K. R. Adeboye and A. E. Umar, Generalized Rational Approximation Method via Pade Approximants for the Solutions of IVPs With Singular Solution and Stiff Differential Equations, Journal of Mathematical Sciences, (0, no., Received: April 5, 08; Published: July 8 08

10 6 A. O. Adesanya, T. P. Pantuvo and D. Umar G = H = H = Appendix 4uv yuyvyω + 4u vyuyvyω + 4uω yuyvyω 4u ωyuyvyω 4vω yuyvyω +4v ωyuyvyω h u v fufvyω + h u v fufvyω 6huv fvyuyω + 6hu vfuyvyω +h u ω fuyvfω h u ω fuyvfω h v ω fvyufω + h v ω fvyufω + 6huω yuyvfω 6hu ωfuyvyω 6hvω yuyvfω + 6hv ωfvyuyω 8hu v fuyvyω + 8hu v fvyuyω +8hu ω fuyvyω 8hu ω yuyvfω 8hv ω fvyuyω + 8hv ω yuyvfω h uv ω fufvfω +h uv ω fufvfω + h u vω fufvfω h u v ωfufvfω h u vω fufvfω + h u v ωfufvfω 4h uv ω fufvyω + 4h uv ω fuyvfω + 4h u vω fufvyω 4h u vω fvyufω 4h u v ωfuyvfω +4h u v ωfvyufω huvω fuyvyω + huvω fvyuyω + huv ωfuyvyω huv ωyuyvfω hu vωfvyuyω + hu vωyuyvfω h uvω fuyvfω + h uvω fvyufω + h uv ωfufvyω h uv ωfvyufω h u vωfufvyω + h u vωfuyvfω h u w ω 4 fnyuyvfω h u w ω 4 ynfvyufω 4h u w ω fnfuyvyω + 4h u w ω ynfufvyω +h u w 4 ω fnfuyvyω h u w 4 ω fnyuyvfω h u w 4 ω ynfufvyω + h u w 4 ω ynfvyufω +4h u w ω fnfuyvyω 4h u w ω fnyuyvfω 4h u w ω ynfufvyω + 4h u w ω ynfvyufω +4h u w ω fnyuyvfω 4h u w ω ynfvyufω h u 4 w ω fnfuyvyω + h u 4 w ω ynfufvyω h v w ω 4 fnyuyvfω + h v w ω 4 ynfuyvfω + 4h v w ω fnfvyuyω 4h v w ω ynfufvyω h v w 4 ω fnfvyuyω + h v w 4 ω fnyuyvfω + h v w 4 ω ynfufvyω h v w 4 ω ynfuyvfω 4h v w ω fnfvyuyω + 4h v w ω fnyuyvfω + 4h v w ω ynfufvyω 4h v w ω ynfuyvfω 4h v w ω fnyuyvfω + 4h v w ω ynfuyvfω + h v 4 w ω fnfvyuyω h v 4 w ω ynfufvyω 6huv ω 4 ynyuyvfω 8huv ω ynfvyuyω + 8huv ω ynyuyvfω + 6huv 4 ω ynfvyuyω 4uv ω ynyuyvyω + 4uv ω ynyuyvyω + 4u vω ynyuyvyω 4u v ωynyuyvyω 4u vω ynyuyvyω + 4u v ωynyuyvyω + h u v ω 4 ynfufvfω h u v 4 ω ynfufvfω h u 4 v ω ynfufvfω + h u 4 v ω ynfufvfω h u w ω 4 fnfuyvfω + h u w ω 4 ynfufvfω +h u w 4 ω fnfuyvfω h u w 4 ω ynfufvfω + h u w ω 4 fnfuyvfω h u w ω 4 ynfufvfω h u w 4 ω fnfuyvfω + h u w 4 ω ynfufvfω h u 4 w ω fnfuyvfω + h u 4 w ω ynfufvfω +h u 4 w ω fnfuyvfω h u 4 w ω ynfufvfω + h v w ω 4 fnfvyufω h v w ω 4 ynfufvfω h v w 4 ω fnfvyufω + h v w 4 ω ynfufvfω h v w ω 4 fnfvyufω + h v w ω 4 ynfufvfω +h v w 4 ω fnfvyufω h v w 4 ω ynfufvfω + h v 4 w ω fnfvyufω h v 4 w ω ynfufvfω h v 4 w ω fnfvyufω + h v 4 w ω ynfufvfω h u v ω 4 ynfuyvfω + h u v ω 4 ynfvyufω 4h u v ω ynfufvyω + 4h u v ω ynfuyvfω + h u v 4 ω ynfufvyω h u v 4 ω ynfvyufω +4h u v ω ynfufvyω 4h u v ω ynfvyufω 4h u v ω ynfuyvfω + 4h u v ω ynfvyufω h u 4 v ω ynfufvyω + h u 4 v ω ynfuyvfω + 4h u v w ynfuyvfω

11 On nonlinear methods for stiff and singular IVP 6 H = H4 = 4h u v w ynfvyufω + h u 4 v w fnfuyvyω h u 4 v w ynfuyvfω 6huv w 4 fnyuyvyω +6huv w 4 ynyuyvfω + 8huv w fnyuyvyω 8huv w ynyuyvfω + 6hu vw 4 fnyuyvyω 6hu vw 4 ynyuyvfω 8hu vw fnyuyvyω + 8hu vw ynyuyvfω + h 4 uv w ω 4 fnfufvfω h 4 uv w 4 ω fnfufvfω h 4 uv w ω 4 fnfufvfω + h 4 uv w 4 ω fnfufvfω + h 4 uv 4 w ω fnfufvfω h 4 uv 4 w ω fnfufvfω h 4 u vw ω 4 fnfufvfω + h 4 u vw 4 ω fnfufvfω + h 4 u v wω 4 fnfufvfω h 4 u v w 4 ωfnfufvfω h 4 u v 4 wω fnfufvfω + h 4 u v 4 w ωfnfufvfω + h 4 u vw ω 4 fnfufvfω h 4 u vw 4 ω fnfufvfω h 4 u v wω 4 fnfufvfω + h 4 u v w 4 ωfnfufvfω + h 4 u v 4 wω fnfufvfω h 4 u v 4 w ωfnfufvfω h 4 u 4 vw ω fnfufvfω + h 4 u 4 vw ω fnfufvfω + h 4 u 4 v wω fnfufvfω h 4 u 4 v w ωfnfufvfω h 4 u 4 v wω fnfufvfω + h 4 u 4 v w ωfnfufvfω h uv w ω 4 fnfuyvfω +4h uv w ω fnfufvyω h uv w 4 ω fnfufvyω + h uv w 4 ω fnfuyvfω 4h uv w ω fnfufvyω +4h uv w ω fnfuyvfω 4h uv w ω fnfuyvfω + h uv 4 w ω fnfufvyω + h u vw ω 4 fnfvyufω 4h u vw ω fnfufvyω + h u vw 4 ω fnfufvyω h u vw 4 ω fnfvyufω + h u v wω 4 fnfuyvfω h u v wω 4 fnfvyufω h u v w 4 ωfnfuyvfω + h u v w 4 ωfnfvyufω + 4h u v wω fnfufvyω 4h u v wω fnfuyvfω + 4h u v w ωfnfuyvfω 6h u 4 v wωfnfuyvyω 6hu vω 4 ynyuyvfω 6hu v 4 ωynfvyuyω + 8hu vω ynfuyvyω 8hu vω ynyuyvfω 8hu v ωynfuyvyω + 8hu v ωynfvyuyω 6hu 4 vω ynfuyvyω + 6hu 4 v ωynfuyvyω 8huw ω fnyuyvyω + 8huw ω ynfvyuyω + 6huw 4 ω fnyuyvyω 6huw 4 ω ynfvyuyω 6hu w 4 ωfnyuyvyω + 6hu w 4 ωynfvyuyω + 8hu w ωfnyuyvyω 8hu w ωynfvyuyω +8hvw ω fnyuyvyω 8hvw ω ynfuyvyω 6hvw 4 ω fnyuyvyω + 6hvw 4 ω ynfuyvyω +6hv w 4 ωfnyuyvyω 6hv w 4 ωynfuyvyω 8hv w ωfnyuyvyω + 8hv w ωynfuyvyω h uv w 4 fnfvyuyω + h uv w 4 ynfvyufω + h uv 4 w fnfvyuyω h uv 4 w ynfvyufω +h u vw 4 fnfuyvyω h u vw 4 ynfuyvfω h u 4 vw fnfuyvyω + h u 4 vw ynfuyvfω hu v w fnyuyvyω + hu v w ynyuyvfω + hu v w fnyuyvyω hu v w ynyuyvfω h uv ω 4 ynfvyufω + h uv 4 ω ynfvyufω + h u vω 4 ynfuyvfω h u v 4 ωynfufvyω h u 4 vω ynfuyvfω + h u 4 v ωynfufvyω h uw ω 4 fnyuyvfω + h uw ω 4 ynfvyufω +h uw 4 ω fnyuyvfω h uw 4 ω ynfvyufω h u w 4 ωfnfuyvyω + h u w 4 ωynfufvyω +h u 4 w ωfnfuyvyω h u 4 w ωynfufvyω + h vw ω 4 fnyuyvfω h vw ω 4 ynfuyvfω h vw 4 ω fnyuyvfω + h vw 4 ω ynfuyvfω + h v w 4 ωfnfvyuyω h v w 4 ωynfufvyω h v 4 w ωfnfvyuyω + h v 4 w ωynfufvyω hu v ω ynfuyvyω + hu v ω ynfvyuyω +hu v ω ynfuyvyω hu v ω ynyuyvfω hu v ω ynfvyuyω + hu v ω ynyuyvfω +hu w ω fnyuyvyω hu w ω ynfvyuyω hu w ω fnyuyvyω + hu w ω ynfvyuyω hv w ω fnyuyvyω + hv w ω ynfuyvyω + hv w ω fnyuyvyω hv w ω ynfuyvyω h u v w 4 fnfufvyω + h u v w 4 ynfufvfω + h u v 4 w fnfufvyω h u v 4 w ynfufvfω +h u v w 4 fnfufvyω h u v w 4 ynfufvfω h u v 4 w fnfufvyω + h u v 4 w ynfufvfω h u 4 v w fnfufvyω + h u 4 v w ynfufvfω + h u 4 v w fnfufvyω h u 4 v w ynfufvfω h u v w 4 fnfuyvyω + h u v w 4 fnfvyuyω + h u v w 4 ynfuyvfω h u v w 4 ynfvyufω +4h u v w fnfuyvyω 4h u v w ynfuyvfω h u v 4 w fnfvyuyω + h u v 4 w ynfvyufω 4h u v w fnfvyuyω + 4h u v w ynfvyufω 4h u v w fnfuyvyω + 4h u v w fnfvyuyω

12 64 A. O. Adesanya, T. P. Pantuvo and D. Umar H5 = h u v 4 wω fnfufvyω + h u v 4 wω fnfvyufω h u v 4 w ωfnfvyufω + 4h u vw ω fnfufvyω 4h u vw ω fnfvyufω + 4h u vw ω fnfvyufω 4h u v wω fnfufvyω + 4h u v wω fnfvyufω 4h u v w ωfnfvyufω + 4h u v wω fnfuyvfω 4h u v wω fnfvyufω 4h u v w ωfnfuyvfω +4h u v w ωfnfvyufω h u 4 vw ω fnfufvyω + h u 4 v wω fnfufvyω h u 4 v wω fnfuyvfω +h u 4 v w ωfnfuyvfω h uv w ω fnfuyvyω + h uv w ω fnfuyvyω + h u vw ω fnfvyuyω +h u v wω fnfuyvyω h u v wω fnfvyuyω h u v wω fnfuyvyω + h u v wω fnyuyvfω h u v w ωfnyuyvfω h u vw ω fnfvyuyω + h u v wω fnfvyuyω h u v wω fnyuyvfω +h u v w ωfnyuyvfω + 4huv wω fnyuyvyω 4huv wω fnyuyvyω 4hu vwω fnyuyvyω +4hu v wωfnyuyvyω + 4hu vwω fnyuyvyω 4hu v wωfnyuyvyω + h uvw ω 4 fnfuyvfω h uvw ω 4 fnfvyufω h uvw 4 ω fnfuyvfω + h uvw 4 ω fnfvyufω + h uv wω 4 fnfvyufω +h uv w 4 ωfnfufvyω h uv w 4 ωfnfvyufω h uv 4 wω fnfvyufω h uv 4 w ωfnfufvyω +h uv 4 w ωfnfvyufω h u vwω 4 fnfuyvfω h u vw 4 ωfnfufvyω + h u vw 4 ωfnfuyvfω +h u v 4 wωfnfufvyω + h u 4 vwω fnfuyvfω + h u 4 vw ωfnfufvyω h u 4 vw ωfnfuyvfω h u 4 v wωfnfufvyω + 8h uvw ω fnfuyvyω 8h uvw ω fnfvyuyω 6h uvw 4 ω fnfuyvyω +6h uvw 4 ω fnfvyuyω + 6h uv wω 4 fnyuyvfω + 6h uv w 4 ωfnfuyvyω 6h uv w 4 ωfnyuyvfω +8h uv wω fnfvyuyω 8h uv wω fnyuyvfω 8h uv w ωfnfuyvyω + 8h uv w ωfnyuyvfω 6h uv 4 wω fnfvyuyω 6h u vwω 4 fnyuyvfω 6h u vw 4 ωfnfvyuyω + 6h u vw 4 ωfnyuyvfω +6h u v 4 wωfnfvyuyω 8h u vwω fnfuyvyω + 8h u vwω fnyuyvfω + 8h u vw ωfnfvyuyω 8h u vw ωfnyuyvfω + 8h u v wωfnfuyvyω 8h u v wωfnfvyuyω + 6h u 4 vwω fnfuyvyω