In this paper we discussed the numerical solution to the model of a freely suspended spring with a mass attached. Suitable nonstandard finite difference schemes were developed  for the solution of the resulting dynamical system which follows a free un-damped harmonic oscillator. The result of the numerical experiment  on the schemes are illustrated  with 3D graphs . Keywords: Numerical methods , Non-Local Approximations,  Non-standard finite difference schemes, Qualitative properties, Free un-damped Harmonic Oscillator,  Monotonicity of solution

Obayomi, Adesoji A.

Sunday, Adetolaju Olu

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  91 Construction of new Non-standard  Finite Difference Schemes for the Solution of a free un-damped Harmonic Oscillator Equation Adesoji A.Obayomi 1*  Adetolaju Olu Sunday1 Department of Mathematical Sciences,  Ekiti State University, Ado-Ekiti/ Nigeria. * Tel: +234-080-3719-7130,  xlconsummate@yahoo.com Abstract In this paper we discussed the numerical solution to the model of a freely suspended spring with a mass attached. Suitable nonstandard finite difference schemes were developed  for the solution of the resulting dynamical system which follows a free un-damped harmonic oscillator. The result of the numerical experiment  on the schemes are illustrated  with 3D graphs . Keywords:  Numerical methods , Non-Local Approximations,  Non-standard finite difference schemes, Qualitative properties, Free un-damped Harmonic Oscillator,  Monotonicity of solution.  1. Introduction The solutions of continuous dynamical systems given by systems of ordinary differential equations are typically computed by various numerical procedures defined on discretized time meshes. It is a well known fact that finite difference scheme is one  of the oldest and popular techniques for numerical solution of ordinary differential equations. In most of the equations in mathematical physics, engineering and in some physical sciences, finite difference schemes have been designed and investigated both from the theoretical point view, which is the convergence aspect, and the practical point of view which is the consistency and stability point of view (Anguelov and Lubuma, 2000). Non-standard finite difference schemes (NSFD) have emerged as an alternative method for solving a wide range of problems whose mathematical models involve algebraic, differential and biological models as well as chaotic systems (Mickens 2005). These techniques have many advantages over classical techniques and provide an efficient numerical solution. In fact, the non-standard finite difference method is an extension of the standard finite difference method. Non-standard schemes as introduced by Mickens (1989,1990,1994) are used to help resolve some of the issues related to numerical instabilities. Furthermore, Mickens (1999,2000,2005) introduced certain rules for obtaining the best difference equations, one of the most important is the  selection of a suitable denominator function ϕ(h) for the dicrete derivatives 1.1 Equation of motion of  a spring suspended freely with a mass attached The major reference for this model is Zill (2005). When a spring is suspended vertically from a rigid support, then we can assume that the spring do not exert any force. We can then consider a body of mass M attached to the free end  so that the amount of stretch depend on the value of M. If we displace the body by a distance  S then the spring exerts a restoring force F= -kS , where k>0  is the spring constant following Hooke’s Law. Let us neglect t the air Mathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  92 resistance and there are no retardation  force acting against the direction of motion(i.e free and un-damped motion). At  the equilibrium point the weight  M x g and the initial force exerted by the stiffness of the spring are  equal in magnitude i.e Mg = kS.  If the body is displaced by a distance  x from the equilibrium position then the restoring force is k(X+S). Since there are no external forces acting on the body against the direction of motion , then we can use the Newton‘s second Law to represent the free motion of the displacement from the equilibrium point by the expression M𝑑2𝑥𝑑𝑡2   =  - k(X+S) + Mg              (1) If we substitute the fact that Mg = kS then the equation may be written as 𝑑2𝑥𝑑𝑡2   + k 𝑥 = 0               (2) Which can also be written as  𝑑2𝑥𝑑𝑡2   +𝑤2𝑥 = 0                  (3) Where  𝑤2 = 𝑘/𝑀 Any dynamic system whose function obey the equation above is said to perform a simple free undamped harmonic motion  2. Derivation of the Nonstandard Finite Difference Schemes There are quite a large number of numerical methods that may be used for obtaining approximate  solution of a second order ordinary differential equation among which Finite difference schemes  have been in the fore front.  While the aforementioned techniques are rather successful at dealing with generic differential equations, they can often come unstuck for some particular problems of interest that arise in applications, particularly when such problems exhibit special properties such as symmetries or conservation laws, or when there are solutions with some special structure. Non-standard schemes as introduced by Mickens (1989,1990,1994) are used to help resolve some of the issues related to numerical instabilities. Furthermore, Mickens (1999,2000,2005) introduced certain rules for obtaining the best difference equations. Many authors including [1], [2], [5], have been contributing to this new technique. Such techniques produce numerically stable schemes which provide reliable solution to the differential equation and carried along the dissipative & qualitative properties of the original  equation. In this paper we will employ Mickens five point rule to obtain a new numerical scheme  for the solution of the model of a suspended spring with mass attached  on a free motion.  2.1 Construction Consider the equation in  (3) The analytic solution is of the form  c1coswt + c2sinwt The component  of equation (3) may be approximated in the following manner: 𝑑2𝑥𝑑𝑡2   ≡ 𝑋𝑘+1−2𝑋𝑘+ 𝑋𝑘−1 h                  (4) 𝑤2𝑥 ≡ 𝑤2(𝑋𝑘)                  (5) Mathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  93 Using  Micken’s rules 2 and 3,and the subsequent non-local approximation technique introduced by Angueluv and Lubume (2003), the denominator function h must be replaced by  a much more complex function of the step size h. Experience has shown that such functions perform better when it is related to a particular solution of the ordinary differential equation. This is referred to as the renormalization of the denominator function.  Mickens (1994) in his paper has shown that such function must obey the following conditions to avoid numerical instability: ϕ(h) = h + O(ℎ2)  for first derivative ϕ(h) = ℎ2 + O(ℎ4) for second derivative For the first derivative the following are suitable ϕ(h) functions: Sin(h) , 1-e−h  etc. For the second  derivative the following are suitable  ϕ(h) functions: sin2(h) , h(e−λh  −1λ )  etc. Hence we may rewrite equation  (4) in the form 𝑑2𝑥𝑑𝑡2   ≡ 𝑋𝑘+1−2𝑋𝑘+ 𝑋𝑘−1 h   =𝑋𝑘+1−2𝑋𝑘+ 𝑋𝑘−1 ϕ(h)          (6) ϕ(h) = 4sin2(h2)               (7) The rule 3 also required that the subsequent terms be approximated as a linear combination of several point on the grid . Hence we may approximate the other term as follows: 𝑤2𝑥 ≡ 𝑤2(𝑎𝑋𝑘 + 𝑏𝑋𝑘+1),   a+b=1           (8) 𝑤2𝑥 ≡ 𝑤2(𝑎𝑋𝑘+1 + b𝑋𝑘 +  c𝑋𝑘−1 ),  a+b+c=1         (9) 𝑤2𝑥 ≡ 𝑤2 (2𝑋𝑘+𝑋𝑘+13)              (10) 3. Construction of  new schemes We will now  apply the above technique to construction of  new scheme s. Consider a slipping chain under free un-damped motion which obeys a second order ordinary differential equation given by:   𝑑2𝑥𝑑𝑡2  + 64x = 0 , y(0) = 2/3, y’(0) = -4/3               (11) It can be verified that the analytic solution is   X(t) = 23 cos8t- 16 sin8t                (12) 3.1 Construction of  Scheme 1 Applying  equations (6,7,8 and 11) we can write 𝑋𝑘+1−2𝑋𝑘+ 𝑋𝑘−1 ϕ(h)+  64 (𝑎𝑋𝑘+1 + 𝑏𝑋𝑘) = 0,   a+b=1         (13) ϕ(h) = 4sin2(h2) 𝑋𝑘+1 − 2𝑋𝑘 + 𝑋𝑘−1 +  64 ϕ(𝑎𝑋𝑘+1 + 𝑏𝑋𝑘) = 0         (14) 𝑋𝑘+1 (1+ 64aϕ) = (2- 64bϕ) 𝑋𝑘 − 𝑋𝑘−1             (15) (i). 𝑋𝑘+1  =  (2−  4 ϕ) 𝑋𝑘− 𝑋𝑘−11+  4 ϕ               (16) Mathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  94 3.2 Construction of  Scheme 2 Since cos(h2) is very close to unit “1” for small h we may : (ii). Substitute   a=1/4  and  b = 3ψ/4,  ψ= cos(h )   , ϕ =11 sin2(4h) in (16) to get a new scheme 𝑋𝑘+1  =  (2− 4  ) 𝑋𝑘− 𝑋𝑘−11+  4 ϕ            (17)  3.3 Construction of  Scheme 3 Another scheme can be obtained by  applying equations  6,7 and 9 in equation (11) thus 𝑋𝑘+1−2𝑋𝑘+ 𝑋𝑘−1 ϕ(h)+  64 (𝑎𝑋𝑘+1 + b𝑋𝑘 +  c𝑋𝑘−1 ) = 0,   a=b=c=1/3        (18) So that (iii). 𝑋𝑘+1 (1+ 6 3ϕ) = (2- 6 3ϕ) 𝑋𝑘 − ( +6 3ϕ)𝑋𝑘−1         (19) 𝑋𝑘+1  =  (2− 6 3 ϕ) 𝑋𝑘− (1+6 3 ϕ)𝑋𝑘−1  (1+ 6 3 ϕ)         (20)     ϕ =14sin2(2h)  or  h(e−λh  −1λ ) 3.4 Construction of  Scheme 4 We can also use a direct substitution of equation 6 in equation (11)  to obtain: 𝑋𝑘+1−2𝑋𝑘+ 𝑋𝑘−1 ϕ(h)+  64 (𝑋𝑘) = 0,           (21) So that  (iv).          𝑋𝑘+1  =  (2 −  64bϕ) 𝑋𝑘 − 𝑋𝑘−1         (22)   ϕ =14sin2(2h)  or  h(  e−λh  −1λ ) The  schemes will now be tested for consistency with the analytic solution in a numerical experiment    Mathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  95  4. Performance of the schemes  Figure1:  Graph showing scheme 1 and the analytic solution   Figure 2 Graph showing scheme 2 and the analytic solution   Figure 3:  Graph showing scheme 3 and the analytic solution -1-0.500.511 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49Y1ANALYTIC-1-0.500.511 2 3 4 5 6 7 8 9 10Y2ANALYTIC-1-0.500.511 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Y3ANALYTICMathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  96  Figure 4:  Graph showing scheme 4 and the analytic solution              Figure 5:  Graph showing all the schemes  and the analytic solution  5. Summary and Conclusion It can be observed from that all the schemes produced solution curves whose behavior is consistent with the analytic solution for the first 20 iterations using a step length  of  h =0.1. The curves of  schemes  1 and 4 are more consistent for a larger number of iterations (see figure 1,4,5) . For all oscillations, the amplitude remains constant with respect to time.  We can conclude that the schemes carried along the qualitative properties of the free un-damped harmonic oscillator . With carefully chosen parameters a,b, and  step function ϕ we can get as close as possible to the original curve of the particular harmonic oscillator . We can also conclude that,  for  a linear second order equation like the free  un-damped harmonic  motion it is enough to select parameters a and b such that a + b =1 for non-local approximation. It can be observed that  the choice of  h is restricted to a very small  h for scheme ‘2’  because of  the term cosine(h).   -1-0.500.511 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46Y4ANALYTIC-202114274053667992105118131144157170183196Y1Y2Y3Y4ANALYTICMathematical Theory and Modeling                                                     www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.2, 2013  97 References  [1]. Anguelov R , Lubuma J.M.S. (2000). Nonstandard finite difference method, by Non-local Approximation Keynote address  at the South Africa Mathematical Society, Pretoria, South Africa 16-18 October, Notices of S. A. Math. Soc. 31: 143-543.  [2]. Anguelov R , Lubuma J.M.S. (2003). Nonstandard finite difference method  by nonlocal approximation, Mathematics and Computers in simulation 6: 465-475.  [3]. Ibijola E.A and Obayomi A.A. (2012). A New Family Of Numerical Schemes  for Solving the  Combustion Equation. Journal of Emerging Trends in Engineering and Applied Sciences  (JETEAS) 3 (3): 387-393  [4]. Ibijola E.A and Obayomi A.A.  (2012).A New Family Of Numerical Schemes  for Solving Equation of the Type    =    ( −  ). Innova Sciencia vol 4 (6)  pages 34-46 [5]. Mickens R.E. (1994).  Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore. Pp 1-115, 144-162  [6]. Potts R.B. (1982) American Mathematically monthly 89, 402-407. Differential and Difference equations. [7]. Zill D.G and Cullen R.M.  (2005) Differential Equations with boundary value problems   (sixth Edition) Brooks      

Construction of new Non-standard  Finite Difference Schemes for the Solution of a free un-damped Harmonic Oscillator Equation

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Construction of new Non-standard Finite Difference Schemes for the Solution of a free un-damped Harmonic Oscillator Equation

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