Embedded Explicit Two-Step Runge-Kutta-Nyström for Solving Special Second-Order IVPs

Transcription

1 Applied Mathematical Sciences, Vol. 8, 14, no. 94, HIKARI Ltd, Embedded Explicit TwoStep RungeKuttaNyström for Solving Special SecondOrder IVPs L. M. Ariffin Department of Mathematics and Statistics, Faculty of Science, Technology and Human Development Universiti Tun Hussein Onn Malaysia 864 Batu Pahat, Johor, Malaysia N. Senu, M. Suleiman and Z.A. Majid Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia, 434 UPM Serdang Selangor, Malaysia. Copyright c 14 L. M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid. This is an open access article 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 A thirdorder threestage explicit Twostep RungeKuttaNyström TSRKN) method embedded into fourthorder threestage TSRKN method is developed to solve special secondorder initial value problems IVPs) directly. The stability of the method is investigated. Numerical results are obtained by solving a standard set of test problems, which then reduced to firstorder system when solved using RungeKutta RK) method and comparison are made with existing RK method with same order using variable stepsize. The results clearly showed the advantage and the efficiency of the new method. Mathematics Subject Classification: 65L5, 65L Keywords: Twostep RungeKuttaNyström methods, Second Order ODEs, Absolute Stability

2 4656 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid 1 Introduction Special secondorder ordinary differential equations ODE s) is given by with the initial conditions y x) =f x, y x)) 1) y x )=y,y x )=y where y x) f x, y) R n. Equation 1) can be solved when using standard RungeKutta RK) method provided that it has to be reduce to an equivalent firstorder system of twice the dimension. However, Sharp and Fine [1], Dormand et al. [] and ElMikkawy and ElDesouky [3] shows that they manage to solve equation 1) directly and efficiently by using RungeKuttaNyström RKN) method. In general, RK and RKN codes involving the embedded pairs of order q p) is efficient when the method of order q = p + 1 is used to obtain the numerical solutions of the problem where as the method of order p is used to obtain the local truncation error. Unlike twostep RK method used in Jackiewicz and Verner [4], RK method for the numerical solution of 1) requires many evaluations of the function fper step and hence is not as efficient as linear multistep methods, when the derivative evaluations are relatively expensive. According to Senu et al. [5], when solving 1) numerically, the algebraic order of the method is essential where it is the main criterion to achieve high accuracy and to have lower stage of RKN method with maximal order in order to reduce the computational cost. Thus, in this paper we derive an embedded pair which is explicit and twostep in nature. We consider the TSRKN method for the initial value problem 1) by Paternoster [6] which was derived as an indirect method from the twostep RK method presented by Jackiewicz et al. [7], as follows where Y j m y i+ = 1 θ) y i+1 + θy i + h v j y i + h m w j y i+1 + m h v j f x i + c j h, Y j ) i + wj f x i+1 + c j h, Y j ) i+1 m y i+ = 1 θ) y i+1 + θy i + h v j f x i + c j h, Y j ) i + w j f x i+1 + c j h, Y j ) i+1 i+1 = y i+1 + hc j y i+1 + h m s=1 a js f x i+1 + c s h, Y s i+1), j =1,...m ) 3)

3 Embedded explicit twostep RungeKuttaNyström 4657 Y j i = y i + hc j y i + h m s=1 a js f x i + c s h, Yi s ), j =1,...m 4) where θ, v j,w j, v j, w j,a js for j, s =1,...m are the coefficients of the methods with m is the number of stages for the method. Alternatively TSRKN )and 3) can be written as y i+ = m 1 θ) y i+1 + θy i + h v j y i + h m w j y i+1 + m h v j k j i + w j k j i+1 5) y i+ = m 1 θ) y i+1 + θy i + h v j k j i + w j k j i+1 6) where k j m ) i = f x i + c j h, y i + hc j y i + h a js ki s, j =1,...m s=1 k j m ) i+1 = f x i+1 + c j h, y i+1 + hc j y i+1 + h a js ki+1 s, j =1,...m. 7) Essentially, the methods derived should have the properties of zero stable and consistent. According to Paternoster [6], TSRKN is zero stable if 1 <θ 1 and it is consistent if m v j +w j )=1+θ. Fulfillment of these two properties implies that the method is convergent see Watt [8] and Jackiewicz et al. [7]). Thus, we apply both properties into the derivation of our new method with θ= that will guarantee zero stability of our method as proposed by Jackiewicz and Verner [4]. An embedded qp) pair of TSRKN methods is based on the method c, A, v, w, v, w) of order q and the other TSRKN method c, A, ˆv, ŵ, ˆ v, ˆ w) of order pp <q) which is similar with the embedded pair for onestep RKN derived by Senu et al. [5]. It can also be presented by the following Butcher array: where c =[c 1,c,...,c m ] T, A =[a ij ], θ =,v =[v 1,v,...,v m ] T, w = [w 1,w,...,w m ] T, v =[ v 1, v,..., v m ] T, w =[ w 1, w,..., w m ] T,ˆv =[ˆv 1, ˆv,...,ˆv m ] T, ŵ =[ŵ 1, ŵ,...,ŵ m ] T, ˆ v = [ˆ v ] T 1, ˆ v,...,ˆ v m and ˆ w = [ˆ w ] T 1, ˆ w,...,ˆ w m. The main motivation for the embedded pair of TSRKN is to obtain cheap local error estimation which is to be used in a variable stepsize algorithm. s=1

4 4658 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid Table 1: Embedded mstage step explicit RKN method c θ A v w v w = c a, c m a m,1 a m, θ v 1 v v m w 1 w w m v 1 v v m w 1 w w m ˆv ŵ ˆ v ˆ w ˆv 1 ˆv ˆv m ŵ 1 ŵ ŵ m ˆ v 1 ˆ v ˆ v m ˆ w 1 ˆ w ˆ w m Stability of the Method In this section the linear stability of the TSRKN method will be discussed. By substituting x i+1 = x i + h, y i+1 = y i + hy i + h y i,y i+1 = y i + hy i and apply the test equation y = f x, y) = λ y into TSRKN method ) and 3), we obtain y i+ = y i + hy i + h h m ) λ m m y i + h v j y i + h ) v j λ Y j ) i + wj λ Y j i+1 y i+ = y i + h λ y i ) + h m where v j λ Y j ) i + wj λ Y j ) i+1 w j y i + h λ y i )) + Y j i+1 = y i + hy i + h ) λ y i + hcj y i + h )) λ y i + Y j h m s=1 a js λ Y s i+1 i = y i + hy i c j + h m s=1 ) 8) 9) 1) ) a js λ Yi s. 11)

5 Embedded explicit twostep RungeKuttaNyström 4659 Multiply equation 9) by h gives hy i+ = hy i + h λ y i ) + h m v j λ Y j ) i + wj λ Yi+1) j. 1) The application of the test equation to equation 1) and 11) yields the recursion Y i and Y i+1 respectively. Y i = N 1 y i e + hy i c) and Y i+1 = N 1 y i e + e H + ch) + hy i e + c)). Elimination of the auxiliary vectors Y i and Y i+1 in equations 8) and 9) yields y i+ = y i 1+ H + Hwe + H vn 1 e + wn 1 e + H ) wn 1 e + H wn 1 c + hy i 1+ve + we + H vn 1 c + wn 1 e + wn 1 c ) 13) ) hy i+ = y i H + HvN 1 e + HwN 1 e + H wn 1 e + H wn 1 c + hy i 1+HvN 1 c + HwN 1 e + HwN 1 c ) 14) with v =v 1,v,...,v m ), w =w 1,w,...,w m ), v = v 1, v,..., v m ) and w = w 1, w,..., w m ). The resulting recursion is Z i+ = D H) Z i with Z i+ = y i+,hy i+) T and Zi = ) y i,hy T i ) AH) BH) D H) = A H) B H) with AH) = 1+ H + Hwe + H vn 1 e + wn 1 e + H wn 1 e + H wn 1 c BH) = 1+ve + we + H vn 1 c + wn 1 e + wn 1 c A H) = H + HvN 1 e + HwN 1 e + H wn 1 e + H wn 1 c B H) = 1+HvN 1 c + HwN 1 e + HwN 1 c where N = I HA, H = λh), c =c 1,..., c s ) T, e =1,..., 1) T with D H) is the stability matrix for the TSRKN methods above. The characteristic equation of D can be written as ξ traced)ξ + detd) = 15) which is the stability polynomial of the TSRKN method. Definition.1 Absolute stability interval) An interval H a, ) is called the interval of absolute stability of the method ) and 3) if for all H H a, ), ξ 1, < 1.

6 466 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid 3 Construction of the Method The ETSRKN43) parameters must satisfy the following algebraic conditions as given in Ariffin et al. [9]. These order conditions was obtained in a more elementary way using the Taylor series expansion as proposed by Williamson [1]. Different approach was done by Jackiewicz et al. [7] where they used the theory of Hairer and Wanner [11] in order to derive the TSRK order conditions up to fourthorder. Here are the order conditions for TSRKN methods: Order conditions for y: order 1 : vi + w i = 1 16) order : order 3 : order 4 : wi + v i + w i = 3 wi + c i v i = 4 3, wi + c i v i + w i )= ) 18) 1 wi + c i v i = 3, 1 wi +a ij v i = 3, 19) 1 wi + c i v i + w i )+c i w i = 3, wi + c i v i + c i w i = 4 3. ) Order conditions for y order 1 : vi + w i = 1 1) order : wi + c i v i + w i )=, wi + c i v i = ) order 3 : 1 wi + c i v i = 4 3, 1 wi +a ij v i = 4 3, 3) 1 wi + c i v i + w i )+c i w i = 4 3, wi + c i v i + c i w i = 8 4) 3 order 4 : 1 wi +a ij c i v i =, 1 aij c i w i =, 6 1 wi + c 3 i 6 v i = 3, 5) wi + c 3 i v i + w i )+3c i w i +3c i w i = 3, 6)

7 Embedded explicit twostep RungeKuttaNyström wi + c i w i + c i w i + c 3 i v i =, 1 wi + c i w i +a ij c i v i =, 7) 1 wi + c i w i + c 3 i v i =, aij c j v i = 3 Paternoster [1] gives the following definition: 8) Definition 3.1 An mstage TSRKN method is said to satisfy the following simplifying conditions if its parameters satisfy m a ij c k j = c k i, i =1,...,m, k =1,...,q. 9) k k 1) Equation 9) allow the reduction of order conditions in the theory of TSRKN methods. For the fourthorder method, solving simultaneously the equations 16) 8) together with simplifying assumption 9). It involved equations with 17 parameters and by setting c,c 3 and v 3 as free parameters, the following solution of threeparameter family is obtained: a 1 = 1 c,a 31 = c c 3 4c c 3 +c c 3 +c 3 )c 3, c c ) a 3 = c 3 c c 3 + c + c 3 c 3 ),v 1 = c 3 +3c c 3 c +4), c c ) 3c c 3 c 3 ) v = 3 1+c )c c 3 c ),v c ) 3 = 3c 3 c c 3 + c + c 3 c 3 ), w 1 = 4c 3 +3c c c ),w =,w 3 =, 3 1+c ) 1+c 3 ) v 1 = [6 v 3 c 3 3c 6 v 3 c v 3 c 3 6 v 3 c 3c 6 v 3 c 3c 6 v 3 c 3 c + 1 v 3 c 3 c +11c 3c 15c 3c 6 v 3 c 3 +3c +6 v 3 c 6 v 3 c 15c ]/ [6c 1+c ) 1+c 3 )], v = v 3c 3 1+c 3 ) c 1+c ), w 1 = 4+4c 3 v 3 c 3 c +3 v 3 c 3, w =, w 3 =. 3 1+c ) According to Dormand [], the strategy to choose the free parameter is by minimizing the truncation error coefficients which are defined by τ p+1) np+1 = ) p+1) τ j and τ p+1) np+1 = ) τ p+1) j for yn and y n respectively. Thus, we use this approach to obtain the value of all free parameters obtained from the above solution.

8 466 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid Letting c = 5 gives τ 5) 6 in term of c 3. By using the minimize command in MAPLE, τ 5) has a minimum value zero at c 3 = 395. Since TSRKN 473 is a twostep method, we may consider for the case c 3 [, ]. Obtaining the value of c 3 gives τ 5) in terms of v 3. Again, by using the minimize command τ 5) has a minimum value zero at v 3 = The above obtained value of the free parameters gives τ 5) = and τ 5) = The stability interval of the fourthorder formula is approximately.185,). For the thirdorder method, using the same values for a ij and c i obtained in the fourthorder method, solving the first four equations for y n and the first seven equations for y n simultaneously, the following solution of fourparameter is obtained: v 1 = v 3, v = v 3, ˆ v = 35 6ˆ v ˆ v ˆv 3, ŵ 1 = ˆv 3, ŵ =, ŵ 3 =, ˆ w = ˆ w ˆ v ˆ v ˆv 3, ˆ w 3 = ˆ w ˆ v 1 +5ˆ v ˆv 3. Using the minimize error approach; we obtain the value for all parameters. By minimizing τ 4), we obtain the minimum value zero at ˆv 3 = However, we choose ˆv 3 = 13 and it gives τ 4) 1 = Letting ˆ v 1 = 3 and ˆ w 1 = 38113, we obtain 5 the value of ˆ v 3 with τ 4) = The coefficients in Table are generated using MAPLE where the significant digits is set to by the command Digits. 4 Problems Tested Below are some of the problems tested: Problem 1 Homogeneous) d yx) dx = 64yx),y) = 1,y ) =. Exact solution: yx) = 1 sin8x) + cos8x). 4 Source: van der Houwen and Sommeijer [13]

9 Embedded explicit twostep RungeKuttaNyström 4663 Table : Coefficients for ETSRKN43) method represented by Butcher array

10 4664 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid The first order system: The new variable are y 1 = y and y = y. y 1 = y, y = 64y 1, y 1 ) = 1, y ) =, x 5. Exact solutions are y 1 x) = 1/4 sin8x) + cos8x),y x) = cos8x) 8 sin8x). Problem Inhomogeneous) d yx) = v yx)+v 1) sinx), dx y) = 1, y ) = v +1, x, where v 1, x 5. Exact solution is yx) = cosvx) + sinvx) + sinx). Numerical result is for the case v =1. Source: van der Houwen and Sommeijer [13] The first order system: y 1 = y, y = 1y sinx), y 1 ) = 1, y ) = 11, x 5. Exact solutions are y 1 x) = cos1x)+sin1x)+sinx),y x) = 1 sin1x)+ 1 cos1x) + cosx). Problem 3 d y 1 x) + y dx 1 x) =.1 cosx), y 1 ) = 1, y 1 ) = d y x) + y dx x) =.1 sinx), y ) =, y ) =.9995, x 1 Exact solutions are y 1 x) = cosx)+.5x sinx),y x) = sinx).5x cosx) Source: Stiefel and Bettis [14]. The first order system: y 1 = y,y = y cosx),y 1 ) = 1,y ) =, y 3 = y 4,y 4 = y sinx),y 3 ) =,y 4 ) =.9995, x 1. Exact solutions are y 1 x) = cosx)+.5x sinx),y x) = sinx)+.5x cosx)+.5 sinx), y 3 x) = sinx).5x cosx),y 4 x) = cosx)+.5x sinx).5 cosx). Problem 4

11 Embedded explicit twostep RungeKuttaNyström 4665 y 1 = y y 1, y 1 ) = 1, y 1) y =, 1 + y) 3 = y, y ) =, y y 1 + y) 3 ) = 1, x 15π. Exact solutions are y 1 x) = cosx),y x) = sinx). Source: Dormand et al. []. The first order system: y 1 = y,y = y 1 ) 3,y 1 ) = 1,y ) =, y1 + y3 y 3 = y 4,y 4 = y 3 ) 3,y 1 ) =,y ) = 1, x 15π. y1 + y3 Exact solutions are y 1 x) = cosx),y x) = sinx),y 3 x) = sinx),y 4 x) = cosx). Problem 5 d yx) = yx)+x, y) = 1,y ) =, x 5. dx Exact solution yx) = sinx) + cosx) +x. Source: Allen and Wing [15]. The first order system: y 1 = y,y = y 1 + x, y 1 ) = 1,y ) =, x 5. Exact solutions are y 1 x) = sinx) + cosx)+x, y x) = cosx) sinx)+1. Problem 6 Inhomogeneous System) d y 1 x) dx = v y 1 x)+v fx)+f x),y 1 ) = a + f),y 1 ) = f ), d y x) dx = v y x)+v fx)+f x),y ) = f),y ) = va + f ), x 1. Exact solutions are y 1 x) =a cosvx)+fx),y x) =a sinvx)+fx),fx) is chosen to be e 1x and parameters v and a are 4 and.1 respectively.

12 4666 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid Source: Lambert and Watson [16]. The first order system: y 1 = y,y = 16y 1 +16e 1x + 1e 1x,y 1 ) = 1.1,y ) = 1, y 3 = y 4,y 4 = 16y 3 +16e 1x + 1e 1x,y 3 ) = 1,y ) = 9.6. Exact solutions are y 1 x) =.1 cos4x)+e 1x,y x) =.4 sin4x) 1e 1x,y 3 x) =.1 sin4x)+e 1x,y 4 x) =.4 cos4x) 1e 1x. Problem 7 Nonlinear System) d y 1 x) dx = 4x y 1 d y x) dx = 4x y + y y 1 + y y 1 y 1 + y,y 1 t )=,y 1 x )= π/ Exact solutions are y 1 x) = cosx ),y t) = sinx ). Source: Sharp et al. [17]. The first order system:,y x )=1,y x )=, π/ x 5 y 1 = y,y = y 3 4x y 1,y 1 ) =,y ) = π, y1 + y3 y 3 = y 4,y = 4x y 1 y 3 +,y 1 ) = 1,y ) =. y1 + y3 Exact solutions are y 1 x) = cosx ),y x) = x sinx ),y 3 x) = sinx ),y 4 x) = x cosx ). 5 Implementation and Numerical Results The set of test problems in section 4 is solved using the new method and the results are compared with the numerical results when the same set of test problems are reduced to first order system twice the dimension and solve using method by Fehlberg [18] and Butcher [19]. For the new method and the existing methods, the next step size is determined by ) TOL p+1 h new =.4 h old LT E where TOL is the chosen tolerance, h old is the current step size, p is the order of the method.

13 Embedded explicit twostep RungeKuttaNyström 4667 The numerical results are given in Figs. 17 and the notations used as follows: MAXE maximum global error max y n y x n ) ), that is the computed solution minus the true solution. ETSRKN 43) pair derived in this paper. ERK 43) pair by Butcher [19]. ERK 43) pair by Fehlberg [18]. TSRKN4 ) log1ma ) Timeseconds)... Figure 1: The efficiency curves of the ETSRKN43) method and its comparisons for Problem 1 with x end =5

14 4668 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid TSRKN4 ) log1ma ) Timeseconds) Figure : The efficiency curves of the ETSRKN43) method and its comparisons for Problem with x end =5 TSRKN4 ) log1ma ) Timeseconds) Figure 3: The efficiency curves of the ETSRKN43) method and its comparisons for Problem 3 with x end = 1

15 Embedded explicit twostep RungeKuttaNyström 4669 TSRKN4 ) log1ma ) Timeseconds) Figure 4: The efficiency curves of the ETSRKN43) method and its comparisons for Problem 4 with x end =15π TSRKN4 ) log1ma ) Timeseconds) Figure 5: The efficiency curves of the ETSRKN43) method and its comparisons for Problem 5 with x end = 5

16 467 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid TSRKN4 ) log1ma ) Timeseconds) Figure 6: The efficiency curves of the ETSRKN43) method and its comparisons for Problem 6 with x end = 1 TSRKN4 ) log1ma ) Timeseconds) Figure 7: The efficiency curves of the ETSRKN43) method and its comparisons for Problem 7 with x end =5

17 Embedded explicit twostep RungeKuttaNyström Discussion and Conclusion From Figures 17, we observed that ETSRKN43) is more efficient when compared to ERK43)B and ERK43)F in terms of computational time. This is due to the fact that when secondorder problems is solve using method ERK43)B and ERK43)F, it has to be reduce to firstorder system that is twice its dimension. In terms of global error, ETSRKN43) produced smaller error compared to ERK43)B and ERK43)F for all problems except for problem 4 where ETSRKN43) global error is comparable with ERK43)B and ERK43)F. Thus we can conclude here that ETSRKN43) is more efficient than the existing technique where all the special secondorder problems are solved directly whereby for other techniques, the problems need to reduce to firstorder system of ODEs. Hence less time is needed to solve the same set of problems. References [1] P.W. Sharp and J.M. Fine, Some Nyström pairs for the general secondorder initialvalue problem, Journal of Computational and Applied Mathematics, 4 199), [] J.R. Dormand, M.E.A. ElMikkawy and P.J. Prince, Families of Runge KuttaNyström Formulae, IMA Journal of Numerical Analysis, ), 355. [3] M.E.A. ElMikkawy and R. ElDesouky, A new optimized nonfsal embedded RungeKuttaNyström algorithm of orders 6 and 4 in six stages, Applied Mathematics and Computation, 145 3), [4] Z. Jackiewicz and J.H. Verner, Derivation and Implementation of Two Step RungeKutta Pairs, Japan J. Indust. Appl. Math., 19 ), [5] N. Senu, M. Suleiman and F. Ismail, An embedded explicit RungeKutta Nyström method for solving oscillatory problems, Phys. Scr., 8 9), 155 7pp) [6] B. Paternoster, Two Step RungeKuttaNyström Methods for y = f x, y) and PStability, Computational Science ICCS, Lecture Notes in Computer Science 331, Part III, edited by P.M.A. Sloot et al. Eds.), Berlin Heidelberg: SpringerVerlag: ),

18 467 L.M. Ariffin, N. Senu, M. Suleiman and Z.A. Majid [7] Z. Jackiewicz,R. Renaut and A. Feldstein, TwoStep RungeKutta Methods, SIAM J. Numer. Anal, ), [8] J.M. Watt, The asymptotic discretization error of a class of methods for solving ordinary differential equations, Proc. Cambridge Philos. Soc., ), [9] L.M. Ariffin, N. Senu and M. Suleiman, A threestage explicit twostep RungeKuttaNyström method for solving secondorder ordinary differential equations, National Symposium on Mathematical Sciences, AIP Conf. Proc. 15, 13), [1] R.A. Williamson Numerical Solution of Hyperbolic Partial Differential Equations, Ph.D. Thesis, Churchill College, Cambridge, 1984). [11] E. Hairer and G. Wanner, MultistepMultistageMultiderivative Methods for Ordinary Differential Equations, Computing, ), [1] B. Paternoster, Two Step RungeKuttaNyström Methods for Oscillatory Problems Based on Mixed Polynomials, P.M.A. Sloot et. al. Eds): ICCS 3, LNCS 658 3), [13] P.J. van der Houwen and B.P. Sommeijer, Explicit RungeKutta Nystrom) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal ), [14] E. Stiefel and D.G. Bettis, Stabilization of Cowell s methods, Numerische Mathematik, ), [15] R.C. Allen Jr. and G.M. Wing, An invariant imbedding algorithm for the solution of inhomogeneous linear twopoint boundary value problems, Journal of Computational Physics, ), 458. [16] J.D. Lambert and I.A. Watson, Symmetric multistep methods for periodic initial value problems, IMA Journal of Applied Mathematics, ), 189. [17] P.W. Sharp, J.M. Fine and K. Burrage, Twostage and Threestage Diagonally Implicit RungeKuttaNyström methods of orders three and four, IMA Journal of Numerical Analysis, 1 199), [18] E. Fehlberg, Classical RungeKutta fourthorder and lower order formulas with step size control and their application to Waermeleitungs problem, Computing, 6 197), 6171.

19 Embedded explicit twostep RungeKuttaNyström 4673 [19] J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations: RungeKutta and General Linear Methods, John Wiley & Sons, Great Britain, 1987). Received: June 5, 14