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A New Multi-Step Approach Based on Top Order Methods (Toms) for the Numerical Integration of Stiff Ordinary Differential Equations

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A New Multi-Step Approach Based on Top Order Methods (Toms) for the Numerical Integration of Stiff Ordinary Differential Equations "Science Stays True Here" Journal of Mathematics and Statistical Science (ISSN 2411-2518), Vol.5, 1-14 | Science Signpost Publishing A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations Awari, Yohanna Sani Department of Mathematical Sciences, Taraba State University, Jalingo Nigeria. Kumleng, Geofrey Micah Department of Mathematics, University of Jos, Nigeria. Abstract This paper presents an entirely new approach to obtaining self-starting Top Order Methods (TOMs) which we shall called Extended Top Order Methods (ETOMs). ETOMs were obtained through hermite polynomial used as basis function. Stability analysis of the new approach shows a uniform order six method for k = 3, they also possess very good absolute stability regions which made them highly suitable for the numerical integration of stiff ordinary differential equations. Implementation of the method in block form eliminates the need for starters and hence, generating simultaneously approximate solutions yi, i = 1, 2, ..., 6on the go. To further observe the effect of the new approach, it was implemented on four numerical initial value problems of stiff ordinary differential equations occurring in real life and was shown to compete favorably with the work of existing scientists. Keywords: Trapezoidal Method, Hermite Polynomial, Top Order Methods, Stiff Equation, Block Method, Stiff ODEs. 1. Introduction Many field of applications, notably Science and Technology yields initial value problems of first order ordinary differential equations. Some of these equations may not be easily solved theoretically, therefore numerical methods are provided as a means to approximating their solutions. A potentially good numerical method for the solution of stiff system of ODEs of the form y′ = f (t, y), y′(t0 ) = y0 ,t ∈[t0 ,Tn ] (1) Corresponding author: Awari, Yohanna Sani, Department of Mathematical Sciences, Taraba State University, Jalingo Nigeria. E-mail: [email protected]. A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations must have good accuracy and some reasonably wide region of absolute stability [5, 12]. One of the first and most important stability requirements particularly for linear multistep method is A-stability which was proposed in [6, 12]. However, the requirement of A-stability put some limitations on the choice of suitable LMMs. Top Order Method (TOM) is group into a family of Boundary Value Methods (BVMs) and was introduced by [3]. According to these authors, the method belongs to the group of symmetric Schemes. Boundary Value Methods have also been extensively discussed by the following authors: [4, 7, 8]. A. Symmetric schemes Brugnano grouped as symmetric schemes BVMs having the following general properties: a. They have an odd number of steps, k = 2v −1, v ≥ 1, and must be used with (v,v −1) –boundary conditions (that is, they require v −1initial and v −1final additional methods). b. The corresponding polynomials ρ(z) have skew-symmetric coefficients. That is − z k ρ(z 1 ) = −ρ(z) ; c. The corresponding polynomialsσ (z) have symmetric coefficients. That is − z kσ (z 1 ) = σ (z) − d. Dv,v−1 ≡⊂ B. Lemma: The sixth order Top Order Method is symmetric. Proof: From the first discrete scheme, we get that 11 27 27 11 1 9 9 1 α = , α = , α = − , α = − and β = , β = , β = , β = 3 60 2 60 1 60 0 60 3 20 2 20 1 20 0 20 i. Consider the LHS of (b) in subsection A , we obtained − 11 − 27 − 27 − 11 ρ(z 1 ) = z 3 + z 2 − z 1 − 60 60 60 60 2 A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations Then, − 11 27 27 11 z 3 ρ(z 1 ) = + z1 − z 2 − z 3 60 60 60 60 Now, RHS of (b), yields 11 27 27 11 − ρ(z) = − z 3 − z 2 + z1 + 60 60 60 60 ii. From the LHS of (c), we have that 1 9 9 1 σ (z −1 ) = z −3 + z −2 + z −1 + 20 20 20 20 3 −1 3 1 −3 9 −2 9 −1 1 z σ (z ) = z z + z + z + 20 20 20 20 1 9 9 1 = + z1 + z 2 + z 3 20 20 20 20 From RHS of (c), yields 1 9 9 1 σ (z) = z 3 + z 2 + z1 + . In both cases, RHS=LHS. QED 20 20 20 20 So far, scholars are finding it difficult to derive schemes from the same continuous formulation in form of TOMs that can effectively be implemented in block form for the approximate solution of first order ordinary differential equations of the form (1). The approach by [3], yields self starting block TOMs however, with some boundary conditions provided, called initial additional equation(s) and final additional equations. The equations were obtained from three or more different formulae. The author [1], constructed block methods by pairing k -step Top Order Methods (TOMs) with 2k -step Backward Differentiation Formulas (BDF) and then shifting the equations 2k − 2 times. The approach which has been adopted in this paper allows for flexibility of obtaining all the discrete schemes from the same continuous formulation of the main method and then implementing in block form, thereby 3 A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations eliminating the need for starters and pairing. The boundary condition imposed by some referred scholars (that is, the v −1initial and v −1final additional equations stated above) is also eliminated. 2. Derivation of the Method The approach adopted in this paper entails substituting into (1) a trail solution of the form s+r−1 j y(x) = ∑a j H (x) (2) j=0 where s and r are the interpolation and collocation points, H j (x) is the hermite polynomial generated by the formula: x2 x2 n − 2 d 2 H = (−1)n e e (3) n dx n For sake of reporting, we present some few terms of the hermite polynomial as 2 3 4 2 H 0 (x) =1 , H1 (x) = x , H 2 (x) = x −1 , H 3 (x) = x − 3x , H 4 (x) = x − 6x + 3 , 5 3 6 4 2 H 5 (x) = x −10x +15x , H 6 (x) = x −15x + 45x −15 From (2) s+r−1 ′ j−1 y (x) = ∑ ja j H (x) (4) j=0 putting (4) into (1) we obtained s+r−1 j−1 f (x, y) = ∑ ja j H (x) (5) j=0 Interpolating (2) at xi ,i =1 and collocating (5) at xi ,i = 0,1,...,3 gives a system of equation which can be put in the form M 1z1 = u1 (6) where, 4 A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations 1 0 −1 0 3 0 −15 1 h (h2 −1) (h3 − 3h) (h4 − 6h2 + 3) (h5 −10h3 +15h) (h6 −15h4 + 45h2 −15) 2 3 4 2 5 3 6 4 2 1 2h (12h − 3) (8h − 6h) (16h − 24h + 3) (32h − 80h + 30h) (64h − 240h +180h −15) M = 0 1 0 − 3 0 15 0 1 2 − 3 − 4 − 2 + 5 − 3 + 0 1 2h (3h 3) (4h 12h) (5h 30h 15) (6h 60h 90h) 2 3 4 2 5 3 0 1 4h (12h − 3) (32h − 24h) (80h −120h +15) (192h − 480h +180h) 2 3 4 2 5 3 0 1 6h (27h − 3) (108h − 36h) (405h − 270h +15) (1458h −1620h + 270h) T z1 = [a0 a1 a2 a3 a4 a5 a6 ] and T u1 = [yn yn+1 yn+2 hf n hf n+1 hf n+2 hf n+3 ] Solving (6) for the a , j = 0,1,...,6 yields j 27 ψ 2 45ψ 3 15ψ 4 9 ψ 5 1 ψ 6 a = − + − +1+ − 0 4 h 2 4 h3 2 h 4 4 h5 4 h6 36 ψ 2 12 ψ 5 15ψ 4 20 ψ 3 2 ψ 6 = + − − − a1 2 5 4 3 6 11 h 11 h 11 h 11 h 11 h 147 ψ 5 415ψ 3 153ψ 2 19 ψ 6 195ψ 4 = − − + + + a2 5 3 2 6 4 44 h 44 h 44 h 44 h 22 h 82 ψ 4 183ψ 3 5 ψ 6 109 ψ 2 31ψ 5 = − + − − + a3 3 2 5 4 33 h 44 h 66 h 33 h 44 h 467 ψ 4 145ψ 3 19 ψ 6 63ψ 2 79 ψ 5 = − + − − + a4 3 2 5 4 44 h 11 h 44 h 11 h 22 h 37 ψ 4 151ψ 3 2 ψ 6 27 ψ 2 59 ψ 5 a = − + − − + 5 11 h3 44 h 2 11 h5 22 h 44 h 4 13 ψ 4 1 ψ 3 1 ψ 6 1 ψ 2 1 ψ 5 a = − + + − (7) 6 132 h3 11 h 2 132 h5 33 h 22 h 4 5 A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations substituting (7) into (2) for s = 3 and r = 4 gives (8) Definition 1: A numerical method is said to be of order p if c0 = c1 = c2 = ... = c p and c p+1 ≠ 0, c p+1 is called the error constant. Evaluated (8) at some points of interest yields the desired sixth order Extended Top Order Methods 1 h (11y + + 27y + − 27y + −11y ) = ( f + 9 f + + 9 f + + f + ) 60 n 3 n 2 n 1 n 20 n n 1 n 2 n 3 1 order p = 6, c = − p 2800 1 h (281y + + 27y + − 297y + −11y ) = (43 f + +123 f + + 35 f + − f + ) 600 n 3 n 2 n 1 n 200 n 1 n 2 n 3 n 4 47 order p = 6, c = − p 1540 6 A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations 1 h (23436y + − 4725y + −18700y + −11y ) = (105 f + + 450 f + +153 f + − 5 f + ) 42180 n 5 n 4 n 3 n 703 n 3 n 4 n 5 n 6 345 order p = 6, c = − p 77 1 h (y + − y + ) = (− f + 34 f + +114 f + + 34 f + − f + ) 2 n 3 n 1 180 n n 1 n 2 n 3 n 4 5 order p = 6, c = p 42 1 h (93y + −16y + − 77y + ) = (−11f + 2360 f + + 9800 f + + 3255 f + −104 f + ) 170 n 4 n 3 n 2 15300 n n 2 n 3 n 4 n 5 328 order p = 6, c = p 231 1 h (131y + − 32y + − 99y + ) = (−11f + 8770 f + + 39825 f + +13986 f + − 470 f + ) order 230 n 5 n 4 n 3 62100 n n 3 n 4 n 5 n 6 1191 p = 6, c = p 154 The first equation is shifted n = n + 3 times and in combination with other discrete schemes; they are perfectly netted together in block form to generate simultaneously the values of yi ,i = 1,2,...,6 for the numerical solution of (1).
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