Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae

Transcription

1 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 doi: 648/ac74 Nuerical Approach or Solving Sti Dierential Equation Through the Extended Trapezoidal Rule Forulae Yohanna Sani Awari, *, Micah Geore Kuleng Departent o Matheatical Science, Taraba State Univerit, Jalingo, Nigeria Departent o Matheatic, Univerit o Jo, Jo, Nigeria Eail addre: awari4c@ahooco (Y S Awari), kuleng_g@ahooco (M G Kuleng) * Correponding author To cite thi article: Yohanna Sani Awari, Micah Geore Kuleng Nuerical Approach or Solving Sti Dierential Equation Through the Extended Trapezoidal Rule Forulae Aerican Journal o Matheatical and Coputer Modelling Vol, No 4, 7, pp -6 doi: 648/ac74 Received: Septeber 8, 7; Accepted: October 9, 7; Publihed: Noveber 8, 7 Abtract: The ot popular ethod or the olution o ti initial value proble or ordinar dierential equation are the backward dierentiation orulae (BDF) In thi paper, we ocu on the derivation o the ourth, ixth and eighth order extended trapezoidal rule o irt kind (ETR) orulae through Herite polnoial a bai unction which we naed FETR, SETR and EETR repectivel We then interpolate and collocate at oe point o interet to generate the deire ethod The tabilit anali on our ethod ugget that the are not onl convergent but poe region uitable or the olution o ti ordinar dierential equation (ODE) The ethod were ver eicient when ipleented in block or, the tend to peror better over exiting ethod Keword: Stine, Herite Polnoial, ETR, A-Stabilit, Ordinar Dierential Equation Introduction A ver iportant pecial cla o dierential equation taken up in the initial value proble tered a ti dierential equation reult ro the phenoenon with widel diering tie cale [, 7 There i no univerall acceptable deinition o tine Stine i a ubtle, diicult and iportant concept in the nuerical olution o ordinar dierential equation It depend on the dierential equation, the initial condition and the interval under conideration The initial value proble with ti ordinar dierential equation te occur in an ield o engineering cience, particularl in the tudie o electrical circuit, vibration, cheical reaction and o on Sti dierential equation are ubiquitou in atrocheical kinetic, an non-indutrial area like weather prediction and biolog A et o dierential equation i ti when an exceivel all tep i needed to obtain correct integration In other word we can a a et o dierential equation i ti when it contain at leat two tie contant (where tie i uppoed to be the oint independent variable) that dier b everal order o agnitude A ore rigorou deinition o tine wa alo given b Shapine and Gear: B a ti proble we ean one or which no olution coponent i untable (no Eigen-value o the Jacobian atrix ha a real part which i at all large and poitive) and at leat oe coponent i ver table (at leat one Eigen-value ha a real part which i large and negative) Further, we will not call a proble ti unle it olution i lowl varing with repect to ot negative part o the Eigen-value Conequentl a proble a be ti or oe interval and not or other We eek to propoe a new nuerical integrator or the olution o irt-order ordinar dierential equation o the or: = ( x, ), ( a) =, I = x, xn () where, i continuou within the interval o integration, we aue that atiie Lipchitz condition which guarantee the exitence and uniquene o olution o () A lot o work ha been conducted b an cholaron inding the olution o () The have adopted everal ethod uing dierent polnoial a bai unction, oe

2 4 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae o thee cholar include; [6, 7,, 4,,, 9, Derivation o the Method The approach in thi paper baicall entail ubtituting into () a trail olution o the or N R () : x, x where : x, xn R R an d I = a + h, are the interpolation and collocation point, =,, N, i the Herite polnoial generated b the orula: b a h = N For the ake o reporting, we preent oe ew ter o the Herite polnoial a k tep, Fro () = ( x n + ), ( xn, ( xn )) k k αn+ h β n+ = =, n Subtituting (4) into () we obtained Derivation o the Block ETR Method, n +, () e λt (4) λ (5) Fourth Order Block ETR (FETR) Interpolating () at Y, Y,, Y and collocating (5) at ( Y ), ( Y ),, ( Y ) give a te o equation which can be put in the or Y, Y,, Y r Solving (6) or the,,, Y Y Y ield r, Y Y, Y Y Y = Y (6) Y ( Y ) ( Y ) = ( Y ) ubtituting (7) into () or = and = give r r an equation which i evaluated at oe point o interet to generate the ollowing et o ain and additional equation Additional Equation,,, Y Y Y (7) Y = h A I Y + U I (8) ( ) Sixth Order Block ETR (SETR) Interpolating () at = h B I Y + V I and collocating (5) at we obtained ( ) [ n Y = ha Y + U (9) whoe olution give the coeicient o n = hb Y + V a [ A U M = B V k = α + α + h β + β α n+ q n+ q n+ q n+ q = = k β = (), β q( x ) ubtituting () into () or α q ( x ) () a q =, b r +, r =, and

3 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 5 evaluating at point o interet to generate: Additional Equation ξ 85 ξ 49 ξ 87 ξ 9 ξ 9 ξ φ = h h 66 h h h h ξ 768 ξ 876 ξ 9 ξ 54 ξ 46 ξ φ = h h h h h h ξ 6687 ξ 46 ξ 475 ξ 465 ξ 5 ξ φ = h 44 h h h h 44 h ξ 998 ξ 87 ξ 75 ξ 874 ξ 7996 ξ φ = + + () h h h h h h Eighth Order Block ETR (EETR) Following the ae procedure a in ourth and ixth order ETR, we obtained Additional Equation ξ 695 ξ 774 ξ ξ ξ ψ = + ξ h h h h h ξ 867 ξ 5584 ξ 55 4 ξ 66 ξ ψ = + + ξ h h h h h ψ = ξ ξ + ξ + ξ ξ ξ h 5 4 h h h h

4 6 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae I = A = Main Equation or FETR, SETR and EETR Repectivel B = B = () = ( x, ), () The cla o ethod we have developed hall each be ipleented in block or to generate approxiate olution ( a) = iultaneoul without the need or tarter Anali o Baic Propertie o the Newl Derived Block Method Order o the Block ETR Method Conider Equation (8) and It Main Equation in () Appling (5), we obtained Expanding (4) in Talor Serie give I = x, xn (4) N R (5) : x, x Following (5), we obtained the order and error contant o (8), including it ain equation in () a : x, xn R R and I = a + h, repectivel Conider Equation (6) Appling (5) on (), including it ain equation in (), ield =,, N, (6) Expanding (6) in Talor Serie give b a h = N k tep = k k αn+ h β n+ = = n + n + ( x n + ) (7) Following (7), we obtained a unior order ( xn+, ( xn+ )) or () and it ain equation in (), whoe error contant were calculated a e λt Conider Equation (7) Application o (5) on (), including it ain equation in () ield λ,,, Y Y Y (8) Expanding (8) in Talor Serie give [ [ ( Y ), Y,, Y [ ( ) n n n Y, Y,, Y r,,, Y Y Y, Y Y, r

5 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 7 Y Y Y = Y Y ( Y ) ( Y ) = ( Y ) (9) Fro (9), we get the order o equation () and ain equation () a = and error contant a r = r Zero Stabilit o the Main Method() Zero Stabilit o Main Method or SETR Conider the characteritic equation aociated with the ain dicrete chee o () given b,,, Y Y Y Y = h ( A I ) Y + U I () ( ) where = h B I Y + V I i the eigenvalue() o the Jacobian o () and n Y = ha Y + U are the irt and econd characteritic [ polnoial o the chee in () repectivel and which i given b [ n = hb Y + V () A U M = B V Solving (), we obtained k = α + α + h β + β n+ q n+ q n+ q n+ q = = k () () Zero-Stabilit propert require that the root o () ut ati α and ever root with β ut have x ultiplicit Zero Stabilit o Main Method or EETR Siilarl, the irt and econd characteritic polnoial o the ain chee in () i given b x = (), (4) β q( x ) (5) Fro (4), it can eail be hown that theain chee o the eighth order ETR i zero table Zero Stabilit o the Block Method() Deinition : The block ethod SETR i aid to be zerotable provided the root α q ( x ) o the irt characteritic polnoial atiie q a b = peciied b r +, r =, (6) ξ 85 ξ 49 ξ 87 ξ 9 ξ 9 ξ φ = h h 66 h h h h and or thoe root with ξ 768 ξ 876 ξ 9 ξ 54 ξ 46 ξ φ = h h h h h h the ultiplicit doe not exceed two Conitenc and Convergence: Since each o the block ethod (8), (), () and their individual ain equation in () ha order ξ 6687 ξ 46 ξ 475 ξ 465 ξ 5 ξ φ = h 44 h h h h 44 h the are all conitent and b Henrici (96) Convergence= Conitenc and zero tabilit

6 8 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae 5 4 Fourth order ETR Sixth order ETR Eighth order ETR I(z) Re(z) Figure Stabilit Region or the Fourth, Sixth and Eighth Order Block ETR Method 4 Nuerical Ipleentation o FETR, SETR Proble : conider a highl ti ODE o the or ξ 998 ξ 87 ξ 75 ξ 874 ξ 7996 ξ φ = + +, h h h h h h ξ 695 ξ 774 ξ ξ ξ ψ = + ξ h h h h h ξ 867 ξ 5584 ξ 55 4 ξ 66 ξ ψ = + + ξ +, 5 4 h h h h h ψ = ξ ξ + ξ + ξ ξ ξ, h 5 4 h h h h

7 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 9 Exact SolutionI = Table Maxiu Error or Exaple Skwae et al, 4 th Order Method A = B 49 I = a + h, =,, N, = ( x, ( x )) n+ n+ e λt = h B I Y + V I = (), q β x ξ 998 ξ 87 ξ 75 ξ 874 ξ 7996 ξ 487 φ = h h h h h h ψ = ξ ξ + ξ ξ + ξ ξ 5 4 h h h h h B = Y = ha 5 ( Y ) + U :[, I = a + h, x xn R R 976 ( x n + ) ( xn+, ( xn+ )) 5, Y Y, 44 [ 76 α β n Y = h A I ( Y ) + ( U I ) = h B I Y + V I ξ 768 ξ 876 ξ 9 ξ 54 ξ 46 ξ 4 6 φ = h h h h h h

8 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae 5 Skwae et al, 4 th Order Method ξ 6687 ξ 46 ξ 475 ξ 465 ξ 5 ξ φ = h 44 h h h h 44 h 9A = 6 5 :[, x xn R R B = I = a + h, ( x n + ) ( x, ( x )) 8 8, Y Y, n+ n+ 9 9 [ n Y = h A I ( Y ) + ( U I ) = + V I h B I Y k 95 = α + α + h β + β = = k n+ q n+ q n+ q n+ q α x Skwae et al, 6th Order Method B = b a 67h = k tep N 5 λ Y, Y,, Y 4 Y Y Y = Y Y [ 5 5 n Y = ha Y + U 6 5 α q( x ) a q = b ( ) ( ) Y Y = ( Y ) = ( x, ), [ n = hb Y + V

9 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 Skwae et al, 6th Order Method ξ ξ ξ ξ ξ ψ = + + ξ + ψ 5 4 = ξ ξ + ξ + ξ ξ ξ 5 4 h h h h h h h h h h [ 8 9 n Y = ha Y + U M ( z) = V + zb( I za) U b a 9 45 =,, N, h = N 5 t e λ 45 Y λ Y Y = Y [ 7 n Y = ha Y + U 5 = (), β q ξ 998 ξ 87 ξ 75 ξ 874 ξ 7996 ξ φ = ψ = ξ ξ + ξ ξ + ξ ξ 5 4 h h h h h h h h h h h B = 5 Φ ( w, z) = det( wi M ( z)) b a 6 8 =,, N, h = N 7 45 t e λ 8 5 λ Y Y Y = Y [ 9 45 n Y = ha Y + U 4 β = (), FETR 5576a I = x x 7 k = [, N k α = h β n + n+ n+ = =

10 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae FETR [ [ ( ) [ ( ) Y n, Y n,, Y n Y, Y,, Y r = r = r k A U 5 455M = B V = α + α + h β + β = = k n+ q n+ q n+ q n+ q r + 5 ξ 85 ξ 49 ξ 87 ξ 9 ξ 9 ξ, r =, φ = h h h h h h I = = ( x, ), ( a) = k k k tep αn+ = h β n+ = = 84Y, Y,, Y ( Y ), ( Y ),, Y Y ( ) ( ) Y Y = ( Y ) = r

11 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 FETR [ [ [ n n n 5467 = hb Y + V α q ( x ) a q = b A U M = B V ψ = + + ξ + ψ 5 4 = ξ ξ + ξ + ξ ξ ξ 5 4 h h h h h h h h h h = ( x, ), ( a) = 6 k k k tep αn+ = h β n+ = = 7 444Y, Y,, Y ( Y ), ( Y ),, Y ( Y ) Y Y = ( Y ) [ 9 69 n = hb Y + V β q( x ) α q x = r A U M = B V SETR 8986 :[ x, x R :[, 58 n N + ( x + ) n,,, 79Y Y Y r 4 Y Y Y 5 96 α β x xn R R Y, Y,,,, Y h( A I ) Y = + U I ξ 768 ξ 876 ξ 9 ξ 54 ξ 46 ξ 47 ξ 6687 ξ 46 ξ 475 ξ 465 ξ 5 ξ φ = φ = h h h h h h h 44 h h h h 44 h 7 A = I = [ x x :[ x, x, N B = N R n + n + 8Y, Y,, Y r Y, Y,, Y r

12 4 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae SETR = r Y, Y,, Y 69 M z = V + zb I za U k x = α ( x ) + α ( x ) + h β + β n+ q n+ q n+ q n+ q = = k r + 5 ξ 85 ξ 49 ξ 87 ξ 9 ξ 9 ξ, r =, φ = h h 66 h h h h I = = :[ x, x 5 I [ x x n , N n + N R Y, Y,, Y r Y, Y,, Y = Y, Y,, Y r 9 7 = + ( ) Φ ( w, z) = det( wi M ( z)) M z V zb I za U a r q =, r =, b r

13 Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 5 Proble ξ 85 ξ 49 ξ 87 ξ 9 ξ 9 ξ φ = h h 66 h h h h, ξ 768 ξ 876 ξ 9 ξ 54 ξ 46 ξ φ = h h h h h h, ξ 6687 ξ 46 ξ 475 ξ 465 ξ 5 ξ φ = h 44 h h h h 44 h, ξ 998 ξ 87 ξ 75 ξ 874 ξ 7996 ξ φ = + + h h h h h h ξ 695 ξ 774 ξ ξ ξ Exact Solutionψ = + ξ h h h h h Table Maxiu Error or Exaple Exact Solution Khadiah 5 4 th Order Method FETR Abolute Error ψ = ξ ξ + ξ + ξ ξ + ξ 5 4 h h h h h ψ = ξ ξ + ξ + ξ ξ ξ 5 4 h h h h h I =

14 6 YohannaSaniAwari and Micah Geore Kuleng: Nuerical Approach or Solving Sti Dierential Equation through the Extended Trapezoidal Rule Forulae Exact Solution Khadiah 5 4 th Order Method FETR Abolute Error A = B = Φ ( w, z) = det( wi M ( z)) 7 B = Φ ( w, z) = det( wi M ( z)) Φ ( w, z) = det( wi M ( z)) 5 Concluion We derived block Extended Trapezoidal Rule o Firt (ETR) that are A-table o up to order 8 The ethod were hown tocopete avorablthan other exiting ethodin ter o accurac (ee table and ) Our newl derived ethod in block or are hown to have extenive region o tabilit and in particular are A-table up to order 8 and o ver uitable or ti te o ordinar dierential equation The continuou orulation or each tep nuber k i evaluated at the end point o the interval to recover the individual ain ethod or FETR, SETR and EETR in () and their counterpart chee in (8), () and () repectivel, to thi end, the idea o additional condition i dicarded Furtherore, we do not need an pair in our ipleentation Our block ethod preerve the A-tabilit propert o the trapezoidal rule (reer to igure ), the are alo le expenive in ter o the nuber o unction evaluation per tep Reerence [ Awari, Y S (7) A-Stable Block ETR/ETR Method or Sti ODE, MAYFEB Journal o Matheatic Vol 4: Page -5 [ Awari, Y S, Garba, E J D, Kuleng, M G and Taku, N (6) Eicient ipleentation o Soe Block Extended Trapezoidal rule o irt kind (ETR) cla o ethod Applied to Firt Order Initial Value Proble o Dierential Equation, Global Journal o Matheatic, Vol 7 () [ Biala, T A, Jator, S N, Adenii, R B and Nduku, P L (5) Block Hbrid Sipon Method with Two O-grid Point or Sti Ste International Journal o Nonlinear Science Vol (), pp - [4 Happ K, Shell, Arora and Nagaich, R K () Solution o Non Linear Singular Perturbation Equation Uing Herite Collocation Method, Applied Matheatical Science, Vol 7,, no 9, [5 Khadiah, M Abualnaa, (5) A Block Procedure with Linear Multitep Method Uing Legendre Polnoial or Solving ODE, Applied Matheatic 6:77-7 [6 Miletic, E and Moln arka, G (9) Talor Serie Method with Nuerical Derivative or Nuerical Solution o ODE Initial Value Proble, HU ISSN 48-78: HEJ Manucript no: ANM--B [7 Okuonghae, R I andikhile, M N O () A(α)-Stable Linear Multitep Method or Sti IVP in ODE Acta Univ Palacki Olouc, Fac rer nat, Matheatica 5 (), 7 9 [8 Skwae, Y, Sunda, J, Ibiola, E A () L-Stable Block Hbrid Sipon Method or Nuerical Solution o Initial Value Proble in Sti Ordinar Dierential Equation, International Journal o Pure and Applied Science and Technolog ():45-54 [9 Tahabi, A (8) Nuerical Solution or Sti Ordinar Dierential Equation Ste, International Matheatical Foru,, 8, no 5, 7 7 [ Yao, N M, Akinenwa, O A, Jator, S N () A linear ultitep hbrid Method with continuou coeicient or olving ti ordinar dierential equation, Int J Cop Math 5(): 47-5