Reformulation of Block Implicit Linear Multistep Method into Runge Kutta Type Method for Initial Value Problem

Transcription

1 International Journal of Science and Technology Volume 4 No. 4, April, 05 Reformulation of Block Implicit Linear Multitep Method into Runge Kutta Type Method for Initial Value Problem Muhammad R., Y. A Yahaya, A.S Abdulkareem. & Department of Mathematic/Statitic, Federal Univerity of Technology, Minna. Department of Chemical Engineering, Federal Univerity of Technology, Minna. ABSTRACT In thi reearch work, we reformulated the block hybrid Backward Differentiation Formula (BDF) for k = 4 into Runge Kutta Type Method (RKTM) of the ame tep number for the olution of Initial value problem in Ordinary Differential Equation (ODE). The method can be ue to olve both firt and econd order (pecial or general form). It can alo be extended to olve higher order ODE. Thi method differ from conventional BDF a derivation i done only once. Keyword: Block, Implicit, Runge-Kutta Type, Reformulation.. INTRODUCTION In an earlier work (Muhammad and Yahaya (0)), the idea of multitep collocation wa adopted to obtain the continou form of ome derived hybrid Backward Differentiation Formulae (BDF) collated from literature. The continou form were evaluated at ome grid and off grid point which gave rie to block BDF (Hybrid and Non-hybrid) for k =,,.6. The Block Backward Differentiation Formulae y = f(x, y) y(x 0 ) = y () y = f(x, y) y(x 0 ) = y y (x 0 ) = β () y = f(x, y, y ) y(x 0 ) = y y (x 0 ) = β () We conider the numerical olution of the Initial Value Problem that ha benefit uch a elf tarting, high order, low error contant, atifactory tability property uch a A- tability and low implementation cot. We emphaize the combination of multitep tructure with the ue of off grid t m y(x) = j= α j (x) y n+j + h β j j= f(x j, y(x j)) (4) (Hybrid and Non-hybrid) were ued to obtain olution of ome tiff and non-tiff Initial Value Problem. In thi paper, we eek to reformulate the Block Backward Differentiation Formulae (Hybrid and Non-hybrid) for k = 4 into Runge Kutta Type Method for the olution of Initial Value Problem in Ordinary Differential Equation (ODE) of the form point and eek a method that i both multitage and multivalue. Thi will enable u to extend the general linear formulation to the high order Runge kutta cae (Butcher 00) by conidering a polynomial Where t denote the number of interpolation point x n+j, j = 0,. t and m denote the ditinct collocation point x jε[x n, x n+k ], j = 0,.. m choen from the given tep [x n, x n+k ]. Butcher defined an -tage Runge Kutta method for the firt order differential equation in the form y n+ = y n + h i,j= a ij k i (5) where for i =,. k i = f(x i + α j h, y n + h i,j= a ij k j ) (6) The real parameter α j, k i, a ij define the method. The method in Butcher array form can be written a α β b T Where a ij = β IJST 05 IJST Publication UK. All right reerved. 90

2 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, 05 The Runge Kutta Nytrom (RKN) method i an extenion of Runge Kutta method for econd order ODE of the form y = f(x, y, y ) y(x 0 ) = y 0 y (x 0 ) = y 0 (7) An S-tage implicit Runge Kutta Nytrom for direct integration of econd order initial value problem i defined in the form y n+ = y n + α i hy n + h i,j= a ij k j (a) = y n + h i,j= a ij k j (b) y n+ where for i =,. k i = f(x i + α j h, y n + α i hy n + h i,j= a ij k j, y n + h i,j= a ij k j ) (c) The real parameter α j, k j, a ij, a ij define the method, the method in butcher array form i expreed a α A A b T b A = a ij = β A = a ij = β β = βe. CONSTRUCTION OF THE BLOCK HYBRID BACKWARD DIFFERENTIATION FORMULA WHEN K = 4 (BHBDF4) Conider the approximate olution to () in the form of power erie y(x) = t+m j=0 α j x j (9) α R, j = 0()t + m, y C m (a, b) P(x) y (x) = t+m j=0 jα j x j (0) Where α j are the parameter to be determined, t and m are the point of interpolation and collocation repectively. When K = 4, we interpolate (t = 5) at j = 0,,,, and collocate (m = ) at j = 4. The equation can be expreed a y(x) = t+m j=0 α j x j = y n+j j = 0,,,, () y (x) = t+m j=0 jα j x j = f n+j j = 4 () The general form of the propoed method upon addition of one off grid point i expreed a; y (x) = α (x)y n + α (x)y n+ + α (x)y n+ + α 4 (x)y n+ + α 5 (x)y n+ + hβ 0 (x)f n+4 () The matrix D of dimenion (t + m) (t + m) of the propoed method i expreed a: D = x n x n x n + h (x n + h) [ 0 x n + h x n + h x n + h (x n + h) (x n + h) (x n + h) (x n + h) (x n + h) (x n + h) x n + h (x n + 4h) x n 4 x n 5 x n (x n + h) (x n + h) 4 (x n + h) 5 (x n + h) 4 (x n + h) 4 (x n + h)4 4(x n + 4h) (x n + h) 5 (x n + h) 5 (x n + h)5 5(x n + 4h) 4 ] IJST 05 IJST Publication UK. All right reerved. 9

3 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, 05 Uing the maple oftware package, we invert the matrix D, to obtain column which form the matrix C. The element of C are ued to generate the continou coefficient of the method a: α (x) = C + C x + C x + C 4 x + C 5 x 4 + C 6 x 5 α (x) = C + C x + C x + C 4 x + C 5 x 4 + C 6 x 5 α (x) = C + C x + C x + C 4 x + C 5 x 4 + C 6 x 5 (4) α 4 (x) = C 4 + C 4 x + C 4 x + C 44 x + C 54 x 4 + C 64 x 5 α 5 (x) = C 5 + C 5 x + C 5 x + C x + C 55 x 4 + C 65 x 5 β 0 (x) = C 6 + C 6 x + C 6 x + C 46 x + C 56 x 4 + C 66 x 5 The value of the continou coefficient (4) are ubtituted into () to give a the continou form of the four tep block hybrid BDF with one off tep interpolation point. h 5 (h5 + 96h 4 + 9x n h x n h + 95x 4 n h + 7x 5 n ) y(x)=( 499x n h + 906x n h + 4x 4 n )x h+90x n h 5 x h 5 x5 ) y n + ( 9 9h +99x n h +7x n h+70x n h 5 x 94 (x h n(56h x 5 n h + 75x n h + 9x n h + 4x 4 n ))x + 75h +56x n h+40x n x + 9h+570x n x 4 57 h 5 9 h 5 99h 5 x5 ) y n+ + ( + ( h 5 (444h4 + 9x n h + h +906x n h+90x n x + h 5 h 5 (x n(54h x n h + 75x n h + 9x n h + 4x n 4 )) h +55x n h +64x n h+40x n h 5 (x n(9x n h + 797x n h + 7x n h + 4x 4 n + 6h 4 )) + 94 h (9x 5 nh + 967x n h x n h + 75x 4 n + 06h 4 )x 9h + 94x n h + 70x n h + 40x n h 5 x h + 570x n h 5 x h 5 x5 ) y n+ h 5 (x n(46x n h + 9x n h + 0x 4 n + 40x n h + 0h 4 )) 5970 h (9x 5 n h + 7x n h + 50x 4 n + 75x 4 n + 076x n h + 0h 4 )x + 9x n h + 5x n h + 00x n + 40h 5970 h 5 x h + 50x n h 5 x h 5 x5 ) y n+ h 5 x 99h x n h + 570x n h 5 x 46h + 7x n h + 00x n h 5 x ( (x 95 h n(5x 4 5 n + 6x n h + 947x n h + x n h + 67h 4 )) + (5x 95 h 5 n x n h + 4x n h + 764x n h + 67h 4 )x (50x 95 h 5 n + 57x n h + 4x n h + h )x + (50x 95 h 5 n + 04x n h + 947h )x (5x 95 h 5 n + 6h)x h 5 x5 )y n+/ + ( x h 4 9 n(x 4 n + x n h + x n h + x n h + 6h 4 ) + ( (5x 99 n 4 + 6x n h + 4x n h + x n h + h 4 )) x ( (0x 9 n + 7x n h + 4x n h + h )) x + ( (5x 99 n + x n h + 7h )x ( (0x 9 n + h)x 4 ) + 99 x5 ) f n+4 ( 5) IJST 05 IJST Publication UK. All right reerved. 9

4 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, 05 Evaluating (5) at point x = x n+4 and it derivative at x = x n+, x = x n+,, x = x n+, x = x n+/ yield the following five dicrete hybrid cheme which are ued a a block integrator; y n y n y n+ + y n y n+/ = y n hf n+4 y n y n y n y n+/ = y n hf n hf n y 05 n+ y n+ 94 y 05 n+ 96 y 05 n+/ = 5 y 05 n 95 hf 05 n+ hf 05 n y n y n+ y n y n+/ = 9 y n hf n hf n+4 (6) y n y n+ 5 5 y n+ + y n+ = 505 y 5 n 9550 hf 5 n+ 0 5 hf n+4 It i of order [5,5,5,5,5] T with error contant [ , , 4 0, 955, ]T Re-arranging the block implicit hybrid cheme imultaneouly we obtained the following block cheme y n+ = y n + h 00 {0976f n+ 405f n f n+ 4f n+ + 5f n+4 } y n+ = y n + h 600 {6064f n+ 665f n f n+ 659f n+ + 95f n+4 } y n+ = y n + h 5 {56f n+ + 40f n+ + 45f n+ 6f n+ + 55f n+4 } y n+ = y n + h 400 {7f n+ 05f n+ + 05f n+ + 57f n+ + 5f n+4 } y n+4 = y n + h 5 {56f n+ + 40f n+ + 60f n+ + 64f n+ + 0f n+4 }. REFORMULATION OF THE BHBDF4 INTO RUNGE KUTTA TYPE METHOD (RKTM) Reformulating the block hybrid method with the coefficient a characterized by the butcher array (7) α β b T Where a ij = β Give IJST 05 IJST Publication UK. All right reerved. 9

5 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, Uing Equation (5) we obtained an implicit 6-tage block Runge kutta type method of uniform order 4. y n+ =y n + h(0k k 407 k + 97 k k k 6) y n+ = y n + h (0k k k k k k 6) y n+ = y n + h (0k k + 4 k + 4 k 4 6 k k 6) y n+ = y n + h (0k + 6 k 40 k k k k 6) y n+4 = y n + h (0k k + k + 5 k 4 + k k 6) Where k = f(x n, y n ) k = f(x n + h, y n + h (0k k 407 k + 97 k k k 6)) k = f(x n + h, y n + h (0k k k k k k 6)) k 4 = f(x n + h, y n + h (0k k + 4 k + 4 k 4 6 k k 6)) k 5 = f(x n + h, y n + h (0k + 6 k 40 k k k k 6)) k 6 = f(x n + 4h, y n + h (0k k + k + 5 k 4 + k k 6)) Extending the method () with the coefficient a hown in the butcher array () α A A b T b A = a ij = β A = a ij = β β = βe IJST 05 IJST Publication UK. All right reerved. 94

6 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, NOTE: The butcher table i being rearranged with the off grid point appearing firt, followed by the c i in decending order. Thi i done in other to atify the conitency condition. Uing Equation (), we obtained an implicit 6 tage block Runge kutta Type method of uniform order 5. y n+ = y n + hy n + h (0k + 5 k k k k k 0 6), y n+ = y n + h (0k + 4 k k + 97 k 4 47 k k 6) y n+ = y n + hy n + h (0k k 5 k k k k 6), y n+ = y n + h (0k + 00 k 5 79 k k k k 50 6) y n+ = y n + hy n + h (0k + 95 k 0 5 k + 76 k 4 4 k k 6), y n+ = y n + h (0k + 56 k k + 4 k 4 6 k 5 + k 5 6) y n+ = y n + hy n + h (0k + 9 k 7 00 k k k k 6), y n+ = y n + h (0k + 6 k 40 k k k k 6) () IJST 05 IJST Publication UK. All right reerved. 95

7 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, 05 y n+4 = y n + 4hy n + h (0k k 4 5 k k k k 6), y n+4 = y n + h (0k + 56 k 5 + k + 5 k 4 + k k 6) where k = f(x n, y n, y n ) k = f(x n + h, y n + hy n + h (0k k k k k k 6), y n+ = y n + h (0k + 4 k k + 97 k 4 47 k k 6)) k = f(x n + h, y n + hy n + h (0k k 5 k k k k 6), y n+ = y n + h (0k + 00 k 5 79 k k k k 50 6)) k 4 = f(x n + h, y n + hy n + h (0k + 95 k 0 5 k + 76 k 4 4 k k 6), y n + h (0k k + 4 k + 4 k 4 6 k k 6)) k 5 = f(x n + h, y n + hy n + h (0k + 9 k 7 00 k k k k 6), y n + h (0k + 6 k 40 k k k k 6)) k 6 = f(x n + 4h, y n + 4hy n + h (0k k 4 5 k k k k 6), y n + h (0k k + k + 5 k 4 + k k 6)) 4. NUMERICAL EXAMPLES To tudy the efficiency of the method we preent ome numerical example widely ued by reknown author uch a Yahaya and Adegboye (0), Sunday etal (0). Example : (SIR Model) The SIR model i an epidemiological model that compute the theoretical number of people infected with a contagiou illne in a cloed population over time. The name of thi cla of model derive from the fact that they involve coupled equation relating the number of uceptible people S(t), the number of infected I(t), and the number of people who have recovered R(t). Thi i a good and imple model for many infectiou dieae including meale, mump and rubella. It i given by the following three coupled equation ds dt = μ( S) βis di = μi γi + βis dt IJST 05 IJST Publication UK. All right reerved. 96

8 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, 05 dr = μr + γi dt Where μ, γ and β are poitive parameter. Define y to be y = S + I + R And adding equation (), () and () we obtain the following evolution equation for y y = μ( y) Taking μ = 0.5 and attaching an initial condition y(0) = 0.5 (for a particular cloed population), we obtain y = 0.5( y), y(0) = 0.5 Exact Solution y(t) = 0.5e 0.5t Applying the Runge Kutta Type Method (RKTM) to thi problem yield the following reult a indicated in table. Problem y xy + 4y = 0, y(0) =, y (0) = 0, h = < x < 0.4 Exact Solution y(x) = x 4 6x + Applying the Runge Kutta Type Method (RKTM) to thi problem yield the following reult a indicated in table TABLE : PERFORMANCE OF RKTM ON PROBLEM t Exact Solution Computed Solution Error E E E E E E E E E E 0 IJST 05 IJST Publication UK. All right reerved. 97

9 International Journal of Science and Technology (IJST) Volume 4 No. 4, April, 05 Table : Performance of RKTM on problem : x Exact Solution Computed Solution Error E E E E CONCLUSIONS Our computed reult in both table are very cloe to the exact olution. Thi indicate the efficiency of the method. The reult are obtained at once which ave computer time and human effort. Alo in thi propoed method, the derivation i done only once and we can extend to higher order unlike in conventional Backward Differentiation Formula. Acknowledgement We wih to expre our profound gratitude to the Almighty God Who make all thing poible. Our appreciation goe to all our colleague in Mathematic/Statitic department and Chemical Engineering Department for peaceful coexitence and providing an enabling environment uitable for undertaking a reearch work. To all whoe work we have found indipenable in the coure of thi reearch. All uch have been duly acknowledged. REFERENCES [] Butcher, J.C (00). Numerical method for ordinary differential equation. John Wiley & Son. [] Butcher, J.C & Hojjati, G. (005). Second derivative method with runge-kutta tability. Numerical Algorithm, 40, [] Kulikov, G. Yu. (00). Symmetric Runge Kutta Method and their tability. Ru J. Numeric Analyze and Math Modelling. (): -4 [4] Muhammad, R., & Yahaya, Y.A. (0). A Sixth Order Implicit Hybrid Backward Differentiation Formula (HBDF) for Block Solution of Ordinary Differential Equation. Scientific and Academic Publihing, American Journal of Mathematic and Statitic (4), [5] Sunday J.,Odekunle M.R & Adeanya A.O. (0). Order ix block integrator for the olution of firt order ordinary differential equation. International Journal of Mathematic and Soft Computing (), 7-96 [6] Yahaya, Y.A. & Adegboye, Z.A. (0). Reformulation of quade type four-tep block hybrid multtep method into runge-kutta method for olution of firt and econd order ordinary differential equation. Abacu, (), 4-4. [7] Yahaya Y.A. and Adegboye Z.A. (0). Derivation of an implicit ix tage block runge kutta type method for direct integration of boundary value problem in econd order ode uing the quade type multitep method. Abacu, 40(), -. IJST 05 IJST Publication UK. All right reerved. 9